Reciprocal space 266 INDEX

The reciprocal space indexing can be implemented in several different ways. Two of them are trial-and-error and zone search methods. The first one is more efficient in high symmetry crystal systems (from cubic to orthorhombic) but becomes slow for low symmetry crystal systems (especially triclinic), while the second method works quite effectively and is fast with low symmetries (from triclinic to orthorhombic). 

Reciprocal spacings (1 jdm — qm = (2 sin 6)1 X) were calculated from the positions of the powder lines, and the lines were indexed on the basis of a face-centered cubic lattice. Analysis of a few lines in the back-reflection region gave a preliminary value of 0-081409 A-1 for 1 ja0. A refinement of l/a0 was carried out by... 

The indexing proeedure of any diffraction pattern requires the knowledge of positions of some reflections on the pattern. The relations between the position of each peak on a two-dimensional diffraction pattern and the corresponding point in reciprocal space can be established only after a successful indexing. The combined use of the reciprocal coordinates and the determined peak-shape are essential for extracting integrated intensities from diffraction data. 

An essential part of the data reduction is the indexing. In this procedure all positions on the recorded image where intensity from reflected beams can be expected have to be is determined irrespectively whether the reflection is strong or weak. In the two-dimensional reciprocal space every reflection can be indicated by a vector H which has two integer elements, h and k, the indices. All position in reciprocal space can be described as ... 

Figure 7.8 A scattering map in reciprocal space. Equal intensity contours are shown schematically, and the Ewald sphere is represented as a plane near reciprocal lattice points 0 and h. The dynamical diffraction from the specimen is displaced slightly from the relp and from the centre of the diffuse scatter by the refractive index effect...

One of the drawbacks of ellipsometry is that the raw data cannot be directly converted from the reciprocal space into the direct space. Rather, in order to obtain an accurate ellipsometric thickness measurement, one needs to guess a reasonable dielectric constant profile inside the sample, calculate A and and compare them to the experimentally measured A and values (note that the dielectric profile is related to the index of refraction profile, which in turn bears information about the concentration of the present species). This procedure is repeated until satisfactory agreement between the modeled and the experimental values is found. However, this trial-and-error process is complicated by an ambiguity in determining the true dielectric constant profiles that mimic the experimentally measured values. In what follows we will analyze the data qualitatively and point out trends that can be observed from the experimental measurements. We will demonstrate that this... 

Fig. 13a and b. Intensity contour maps around the 5.9-nm and 5.1-nm actin layer lines (indicated by arrows) a resting state b contracting state. Z is the reciprocal-space axial coordinate from the equator. M5 to M9 are myosin meridional reflections indexed to the fifth to ninth orders of a 42.9-nm repeat, (c) intensity profiles (in arbitrary units) of the 5.9- and 5.1-nm actin reflections. Dashed curves, resting state solid curves, contracting state. Intensity distributions were measured by scanning the intensity data perpendicular to the layer lines at intervals of 0.4 mm. The area of the peak above the background was adopted as an integrated intensity and plotted as a function of the reciprocal coordinate (R) from the meridian... 

FIG. 9.15 Path of a diffracted electron beam in low-energy electron diffraction (LEED), and indexing of points in reciprocal space, (a) interaction of a diffracted beam with a photographic plate for small angles of incidence and (b) illustration of the indexing of points in reciprocal space relative to the primary beam, labeled 00. 

The progress of iterative real- and reciprocal-space refinement is monitored by comparing the measured structure-factor amplitudes IFobsl (which are proportional to (/obs ) /2) with amplitudes IFca(c I from the current model. In calculating the new phases at each stage, we learn what intensities our current model, if correct, would yield. As we converge to the correct structure, the measured Fs and the calculated Fs should also converge. The most widely used measure of convergence is the residual index, or R-factor (Chapter 6, Section V.E). 

There has been no basic formula analogous to the Poisson summation formula, characteristic of translational invariance, on which to base an analysis of quasicrystal diffraction patterns. Here successive values of reciprocal space have geometric ratios instead of the arithmetic spacing of the peaked functions observed with ordinary crystalline diffraction. Fig. 2.15 illustrates a two-dimensional section in reciprocal space of a diffraction pattern. The five-fold symmetry is exact, and typically six indices instead of three are required to index each point, with the choice of origin arbitrary, and for assigiunent of indices, ambiguous. The features of interest are ... 

This process is commonly known as indexing of diffraction data and in three dimensions it usually has a unique and easy solution when both the lengths and directions of reciprocal vectors, d A , are available. On the contrary, when only the lengths, of the vectors in the reciprocal space are known, the task may become extremely complicated, especially if there is no additional information about the crystal structure other than the array of numbers representing the observed Vdhu [= d hkl values. 

The most effective is the reciprocal space approach, in which several low Bragg angle peaks are chosen as a basis set, and then an exhaustive permutation-based assignment of various combinations of hkl triplets to each peak from the basis set is carried out. Index permutation algorithms are more complex in realization than direct space algorithms but the former are many orders of magnitude faster than the latter. This occurs because the indices of... 

For every family of planes having integral Miller indexes hkl, a vector can be drawn from a common origin having the direction of the plane normal and a length 1 /d, where d is the perpendicular distance between the planes. The coordinate space in which these vectors are gathered, as in Figure 3.19, is called reciprocal space, and the end points of the vectors for all of the families of planes form a lattice that is termed the reciprocal lattice. 

Any reciprocal lattice vector, or reciprocal lattice point is uniquely specified by the set of three integers, hkl, which are the Miller indexes of the family of planes it represents in the crystal. Thus there is a one-to-one correspondence between reciprocal lattice points and families of planes in a crystal. It will be seen shortly that the reciprocal lattice is the Fourier transform of the real lattice, and vice versa. This was in fact demonstrated experimentally in Figure 1.7 of Chapter 1 by optical diffraction. As such, reciprocal space is intimately related to the distribution of diffracted rays and the positions at which they can be observed. Reciprocal space, in a sense, is the coordinate system of diffraction space. 

Consider then the plane of reflections in reciprocal space having indexes hOl. We can write their structure factors as... 

In the case of a periodic, three-dimensional function of x, y, z, that is, a crystal, the spectral components are the families of two-dimensional planes, each identifiable by its Miller indexes hkl. Their transforms correspond to lattice points in reciprocal space. In a sense, the planes define electron density waves in the crystal that travel in the directions of their plane normals, with frequencies inversely related to their interplanar spacings. 

Remember further that each reciprocal lattice point represents a vector, which is normal to the particular family of planes hkl (and of length 1 /d u) drawn from the origin of reciprocal space. If we can identify the position in diffraction space of a reciprocal lattice point with respect to our laboratory coordinate system, then we have a defined relationship to its family of planes, and the reciprocal lattice point tells us the orientation of that family. In practice, we usually ignore families of planes during data collection and use the reciprocal lattice to orient, impart motion to, and record the three-dimensional diffraction pattern from a crystal. Note also that if we identify the positions of only three reciprocal lattice points, that is, we can assign hkl indexes to three reflections in diffraction space, then we have defined exactly the orientation of both the reciprocal lattice, and the real space crystal lattice. 

This is known as the reciprocal lattice, which exists in so-called reciprocal space. As we will see, it turns out that the points in the reciprocal lattice are related to the vectors defining the crystallographic planes. There is one point in the reciprocal lattice for each crystallographic plane, hkl. For now, just consider h, k and / to be integers that index a point in the reciprocal lattice. A reciprocal lattice vector hhki is the vector from the origin of reciprocal space to the (hkl) reciprocal lattice point ... 

In some crystalline materials a phase transition on lowering the temperature may produce a modulated structure. This is characterized by the appearance of satellite or superstructure reflections that are adjacent reflections (called fundamental reflections) already observed for the high temperature phase. The superstructure reflections, usually much weaker than fundamental reflections, can in some cases be indexed by a unit cell that is a multiple of the high temperature cell. In such a case the term commensurate modulated structure is commonly used. However, the most general case arises when the additional reflections appear in incommensurate positions in reciprocal space. This diffraction effect is due to a distortion of the high temperature phase normally due to cooperative displacements of atoms, ordering of mixed occupied sites, or both. Let us consider the case of a displacive distortion. 

The label a is a generalized index to indicate the energy of a particular excitation associated with a given configuration. We note that the present discussion emphasizes the local atomic-level relaxations and how they can be handled within the context of the Ising-like model introduced earlier. However, there are additional effects due to long-range strain fields that require further care (they are usually handled in reciprocal space) which are described in Ozoli s et al. (1998). 

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