Solids 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). 

Fig. 2. Stereographic triangle indicating various crystallographic orientations of fee solid surfaces using Miller index notations.

Platinum crystal surfaces that were prepared in the zones indicated by the arrows at the sides of the triangle are thermally unstable. These surfaces, on heating, will rearrange to yield the two surfaces that appear at the end of the arrows. There is reason to believe that the thermal stability exhibited by various low and high Miller index platinum surfaces is the same for other fee metals. There are, of course, differences expected for surfaces of bcc solids or for surfaces of solids with other crystal structures. 

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). 

One of the most exciting observations of LEED studies of adsorbed monolayers on low Miller index crystal surfaces is the predominance of ordering within these layers (18). These studies have detected a large number of surface structures formed upon adsorption of different atoms and molecules on a variety of solid surfaces. Conditions range from low temperature, inert gas physisorption to the chemisorption of reactive diatomic gas molecules and hydrocarbons at room temperature and above. A listing of over 200 adsorbed surface structures, mostly of small molecules, adsorbed on low Miller index surfaces can be found in a recent review (/). 

The theories of surface structure and bonding have been reviewed. It should be clear to the reader that surface structural chemistry is indeed a frontier area for both theorists and experimental researchers. From an experimentalists viewpoint the data base of atomic and molecular surface structures is very small at present. Most investigations have been carried out on flat, low Miller index surfaces of monatomic solids, either clean or with atomic or small molecules as adsorbates. 

Figure 22.14 shows our point First, Figure 22.14a shows Miller index planes inside a solid crystal. The designation (110) is consistent with our understanding of Miller... 

The clean siuface of solids sustains not only surface relaxation but also surface reconstruction in which the displacement of surface atoms produces a two-dimensional superlattice overlapped with, but different from, the interior lattice structure. While the lattice planes in crystals are conventionally expressed in terms of Miller indices (e.g. (100) and (110) for low index planes in the face centered cubic lattice), but for the surface of solid crystals, we use an index of the form (1 X 1) to describe a two-dimensional surface lattice which is exactly the same as the interior lattice. An index (5 x 20) is used to express a surface plane in which a surface atom exactly overlaps an interior lattice atom at every five atomic distances in the x direction and at twenty atomic distances in the y direction. 

A better solution is to use a four-axis, four-index system for hep solids. A few examples are depicted in Fig. 4.10. The Miller indices are found as before by taking the reciprocals of the intercepts of the plane with the four axes. Using this system, the equivalent planes discussed above become (1100) and (1010). Now the six equivalent planes resulting from the sixfold axis of symmetry can be identified as the 1100 family of planes. Using four axes, the [hkil] direction is normal to the (hkil) plane, in the same way it was for cubic solids using the three-axis system. ... 

There is a long history of calculations of adsorption potentials for simple gases adsorbed on the exposed low index Miller planes of ionic crystals, especially alkali halides (see the review [26] and references therein). The total interaction potential energy between an adsorbed molecule and the surface of a solid is generally expressed as a sum of dispersion, repulsion, induction, and electrostatic contributions (see, e.g.. Ref. [27]) ... 

FIGURE 5 A typical tornadic thunderstorm sounding of the atmosphere plotted on a log pressure-skew temperature adiabatic chart. The thick solid lines represent environmental temperature and dew-point temperature measurements. The dashed line Is the temperature of a parcel lifted from the earth s surface. Plus and minus signs Indicated regions of positive and negative buoyancy experienced by the parcel. Since the parcel is 6°C warmer than the environment at 500 mbar, the lifted index (stability indicator) Is defined to be -6°C. [Adapted from Fawbush, E. J., and Miller, R. C. (1953). Bull. Am. Meteorol. Soc. 34, 235-244.]... 

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,... 

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