Diffraction Miller indices

Unlike the case of diffraction of light by a ruled grating, the diffraction of x-rays by a crystalline solid leads to the observation that constructive interference (i.e., reflection) occurs only at the critical Bragg angles. When reflection does occur, it is stated that the plane in question is reflecting in the nth order, or that one observes nth order diffraction for that particular crystal plane. Therefore, one will observe an x-ray scattering response for every plane defined by a unique Miller index of (h k l). 

The first term a i. h is the Miller index component h, which is an integer. Thus in the interference fimction it becomes unity since exp(2 ni)=l. This corresponds to strong Bragg diffraction when q=0. The second term when put in the interference fimction becomes... 

Since dhk and L are constant, Equation (100) predicts that A is inversely proportional to dhk, with A the distance of the diffraction spot from the spot produced by the primary. Figure 9.14 shows that dhk is largest for planes of low Miller indices since A varies inversely with dhk, it follows that spots nearest the primary spot are due to low Miller index planes. Likewise, more distant spots are due to planes of higher index. There is a reciprocal relationship between the location of the spot on the photographic plate and the separation of the planes responsible for the spot. 

We call this Pt(100) surface reconstructed. Surface reconstruction is defined as the state of the clean surface when its LEED pattern indicates the presence of a surface unit mesh different from the bulklike (1 x 1) unit mesh that is expected from the projection of the bulk X-ray unit cell. Conversely, an unreconstructed surface has a surface structure and a so-called (1 x 1) diffraction pattern that is expected from the projection of the X-ray unit cell for that particular surface. Such a definition of surface reconstruction does not tell us anything about possible changes in the interlayer distances between the first and the second layers of atoms at the surface. Contraction or expansion in the direction perpendicular to the surface can take place without changing the (1 x 1) two-dimensional surface unit cell size or orientation. Indeed, several low Miller index surfaces of clean monatomic and diatomic solids exhibit unreconstructed surfaces, but the surface structure also exhibits contraction or expansion perpendicular to the surface plane in the first layer of atoms (9b). 

Figure 3. X-ray diffraction patterns for TiN0.26D0.15 (ss) samples annealed at (a) 1270 K and (,b) 900 K. Above peaks Miller indexes of the reflecting planes are indicated.

FIG. 3 X-ray diffraction pattern of a microcrystaliine powder of Cgo- Inset (upper left) is a single-crystal electron diffraction pattern Indexed with Miller indices compatible with the X-ray pattern. The pattern is from a thin platelet such as those in Fig. 1 with the electron beam perpendicular to the flat face. 

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. 

The family of planes within a single unit cell cannot, of course, provide a diffracted ray intense enough to be measured, but the family of planes cutting through all of the unit cells in the crystal can. As noted already, as long as they have integral Miller indexes, then every... 

In summary then, a crystal can be conceived of as an electron density wave in three-dimensional space, which can be resolved into a spectrum of components. The spectral components of the crystal correspond to families of planes having integral, Miller indexes, and these can, as we will see, give rise to diffracted rays. The atoms in the unit cell don t really lie on the planes, but we can adjust for that when we calculate the intensity and phase with which each family of planes scatter X rays. The diffracted ray from a single family of planes (which produces a single diffraction spot on a detector) is the Fourier transform of that family of planes. The set of all diffracted rays scattered by all of the possible families of planes having integral Miller indexes is the Fourier transform of the crystal. Thus the diffraction pattern of a crystal is its Fourier transform, and it is composed of the individual Fourier transforms of each of the families of planes that sample the unit cells. 

Given these unknowns, it might appear that X-ray data collection would be a very difficult process indeed. It is not, in fact. X-ray crystallographers only rarely think about planes in the crystal, or their orientation. They use instead the diffraction pattern to guide them when they orient and manipulate a crystal in the X-ray beam. Recall that the net, or lattice, on which the X-ray diffraction reflections fall is the reciprocal lattice, and that every reciprocal lattice point, or diffraction intensity, arises from a specific family of planes having unique Miller indexes hki. 

There are a few, relatively early, studies of Se and Te adsorption on metals. Selenium is found to adsorb at the high coordination (hollow) sites on the low Miller index surfaces Ni(100) and Ag(lOO). On the most open surface to have been examined, Ni(J10), the bond distance to the Ni atom in the second substrate layer (2.35 A) is slightly shorter than that to the top layer (2.42 A), suggesting the formation of a Ni-Se bond to the second substrate layer. However, it should be noted that the LEED studies of Se adsorption on metals originate before 1975, whilst more recent studies (1982) were by photo-electron diffraction only. Consequently, detailed substrate distortions, of the type seen in more recent studies of O and S adsorption on metals, have not been searched for. 

Measurements of the diffraction peak intensities at various temperatures indicated that the vibrations of the ions in the AIN lattice were anisotropic. The Intensities of different diffraction lines depended differently on the temperature (Fig. 3). Variation of the temperature affected most the lines with large values of the Miller index I and least the lines characterized by I — 0. This indicated that the vibrations of the ions had a larger amplitude along the c axis than in the basal plane. Under these conditions, the atomic scattering factors of the Al and the N ions should also be anisotropic, I.e.,... 

Fig. 4.31 Comparison of the X-ray diffraction patterns between the renatured s-SPG (a) and the s-SPG/poly(C) complex (b). The fiber axis is vertical and the Miller indexes are assigned by referring to lentinan and curdlan.

The crystallographic planes and their representation through Miller indices were rationalized in this section at the level of Ewald sphere of diffraction in the reciprocal space. This complements the structure factors, and implicitly the electronic densities, equally expressed in the previous section through the Miller indexing. However, the electronic localization is made around their attraction centers and the diffraction model should contain also the atomic representation, as basic structural unit. Accordingly, the atomic information should be contained in the diffraction description, and then into the determination of the crystalline structure type, towards the molecular and bio-molecular structural determination by crystallization, as will be eiqiosed in the next section. 

FIGURE 9.6 X-ray diffractograms of DND (a) and CVD diamond film on silicon substrate (b) and diamond particles produced from HPHT diamond by milling method (c). Positions for interplanar distances are marked. The corresponding intensities are given by the lengths of the dashed lines. Positions of diffraction maxima are marked by corresponding Miller indexes. The asterisk identifies an impurity diffraction maximum. The (b, c) curves are received with CoK and CuK radiation. 

Let us first concentrate on the surfaces of flat, low-Miller-index planes. Figure 8 shows the (100) crystal face of platinum. When clean, this surface is reconstructed and its diffraction pattern indicates the presence of a 5 x20 surface structure. When the surface is impure or has a fraction of a monolayer of adsorbates, the square unit cell shown in figure 8 is obtained, which is what one would expect from projection of the bulk unit cell up to the surface. While this 5 x20 surface structure was first detected in our laboratory in 1965 [9], it was actually solved by surface crystallography in 1981 [10]. 

Figure 33 shows the surface structures of several stepped surfaces of high Miller index. These surfaces are stable in a stepped/terrace configuration [26]. On the clean surface of a close-packed metal, the steps are usually of one atom in height, periodically distributed, and are separated by terraces of roughly equal width. Typical diffraction patterns of such surfaces are shown in figure 34. The formation of doublets or triplets indicates the appearance of new periodicities from which the stepped structure of these high Miller index surfaces can be obtained. 

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