Plane Miller indices

It is sometimes useful to be able to calculate the perpendicular distance dhu between parallel planes (Miller indices hkl). When the axes are at right angles to one another (orthogonal) the geometry is simple and for an orthorhombic system where a b i c and... 

Empirically, however, the results are reasonably accurate, and the approach is a very useful one. An application of it to various Miller index planes is given by... 

Figure Bl.21.1 shows a number of other clean umeconstnicted low-Miller-index surfaces. Most surfaces studied in surface science have low Miller indices, like (111), (110) and (100). These planes correspond to relatively close-packed surfaces that are atomically rather smooth. With fee materials, the (111) surface is the densest and smoothest, followed by the (100) surface the (110) surface is somewhat more open , in the sense that an additional atom with the same or smaller diameter can bond directly to an atom in the second substrate layer. For the hexagonal close-packed (licp) materials, the (0001) surface is very similar to the fee (111) surface the difference only occurs deeper into the surface, namely in the fashion of stacking of the hexagonal close-packed monolayers onto each other (ABABAB.. . versus ABCABC.. ., in the convenient layerstacking notation). The hep (1010) surface resembles the fee (110) surface to some extent, in that it also...

The electrochemically induced creation of the Pt(lll)-(12xl2)-Na adlayer, manifest by STM at low Na coverages, is strongly corroborated by the corresponding catalyst potential Uwr and work function O response to galvanostatic transients in electrochemical promotion experiments utilizing polycrystalline Pt films exposed to air and deposited on (T -AbCb. 3637 Early exploratory STM studies had shown that the surface of these films is largely composed of low Miller index Pt(lll) planes.5... 

Structure Sensitivity over Pe. Table II presents the rates of ammonia synthesis over each of the low Miller index planes of Pe. 

Figure 1,2 Atomic arrangement on various clean metal surfaces. In each of the sketches (a) to (h) the upper and lower diagrams represent top and side views, respectively. Atoms drawn with dashed lines lie behind the plane of those drawn with thick lines, Atoms in unrelaxed positions (i.e. in the positions they occupy in the bulk) are shown as dotted lines. From G.A. Somorjai, Chemistry in Two Dimensions, Cornell University Press, London, 1981, p. 133, For the Miller index convention in hexagonal close-packed structures, see also G.A. Somorjai loc. cit, Used by permission of Cornell University Press,...

A.R. Siedle, 3M Central Research Laboratory If one of the extended structures described by Professor McCarley were truncated through a low Miller index plane, can one, following the appproach of Solomon, predict what metal orbitals would protrude from the surface so generated Have ultraviolet photoelectron spectra been obtained on single crystals of any of these materials ... 

For an fee lattice a particularly simple surface structure is obtained by cutting the lattice parallel to the sides of a cube that forms a unit cell (see Fig. 4.6a). The resulting surface plane is perpendicular to the vector (1,0,0) so this is called a (100) surface, and one speaks of Ag(100), Au(100), etc., surfaces, and (100) is called the Miller index. Obviously, (100), (010), (001) surfaces have the same structure, a simple square lattice (see Fig. 4.7a), whose lattice constant is a/ /2. Adsorption of particles often takes place at particular surface sites, and some of them are indicated in the figure The position on top of a lattice site is the atop position, fourfold hollow sites are in the center between the surface atoms, and bridge sites (or twofold hollow sites) are in the center of a line joining two neighboring surface atoms. 

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). 

To decide which of these many vectors to use, it is usual to specify the points at which the plane intersects the three axes of the material s primitive cell or the conventional cell (either may be used). The reciprocals of these intercepts are then multiplied by a scaling factor that makes each reciprocal an integer and also makes each integer as small as possible. The resulting set of numbers is called the Miller index of the surface. For the example in Fig. 4.4, the plane intersects the z axis of the conventional cell at 1 (in units of the lattice constant) and does not intersect the x and y axes at all. The reciprocals of these intercepts are(l/oo,l/oo,l/l), and thus the surface is denoted (001). No scaling is needed for this set of indices, so the surface shown in the figure is called the (001) surface. 

The Miller Indices for planes in three-dimensional lattices are given by hkl, where 7 Is now the index for the z-axis. The principles are the same. Thus a plane is indexed hkl b ... 

The metal substrates used in the LEED experiments have either face centered cubic (fee), body centered cubic (bcc) or hexagonal closed packed (hep) crystal structures. For the cubic metals the (111), (100) and (110) planes are the low Miller index surfaces and they have threefold, fourfold and twofold rotational symmetry, respectively. 

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. 

The slice through a bulk crystal can differ from both the 111 plane and the 100 plane by small angles. This produces a kink in the face of the step. By an extension of the analysis that leads to step characterization, these kinks can also be characterized. For example, a plane with Miller indices 10,8,7 has 111 terraces seven atoms wide, 110 steps one atom high, and kinks of 100 orientation every two atoms. Because of the greater thermodynamic stability of the planes of low Miller index, these surfaces of ordered roughness are stable and can be prepared and studied. Since it is sensitive to periodicity over a domain about 20 nm in diameter, LEED sees the pattern associated with terraces of various widths and may be used to characterize these surfaces. Satisfactory LEED patterns do not require absolute uniformity of terrace width but may be obtained with experimental surfaces that display a distribution of widths. 

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). 

A 3D crystal has its atoms arranged such that many different planes can be drawn through them. It is convenient to be able to describe these planes in a systematic way and Fig. 4 shows how this is done. It illustrates a 2D example, but the same principle applies to the third dimension. The crystal lattice can be defined in terms of vectors a and b, which have a defined length and angle between them (it is c in the third dimension). The box defined by a and b (and c for 3D) is known as the unit cell. The dashed lines in Fig. 4A show one set of lines that can be drawn through the 2D lattice (they would be planes in 3D). It can be seen that these lines chop a into 1 piece and b into 1 piece, so these are called the 11 lines. The lines in B, however, chop a into 2 pieces, but still chop b into 1 piece, so these are the 21 lines. If the lines are parallel to an axis as in C, then they do not chop that axis into any pieces so, in C, the lines chopping a into 1 piece and which are parallel to b are the 10 lines. This is a simple rule. The numbers that are generated are known as the Miller indices of the plane. Note that if the structure in Fig. 6.4 was a 3D crystal viewed down the c axis, the lines would be planes. In these cases, the third Miller index would be zero (i.e., the planes would be the 110 planes in A, the 210 planes in B, and the 100... 

The Miller-Bravais index system for identifying planes and directions in hexagonal crystals is similar to the Miller index system except that it uses four axes rather than three. The advantage of the four-index system is that the symmetry is more apparent. Three of the axes, ai, a2, and a3, he in the hexagonal (basal) plane at 120° to one another and the fourth or c-axis is perpendicular to then, as shown in Figure 3.1. 

One of the strengths of the approach employed here is that we have freedom over the choice of the transition metal for the tip and also the structure of the tip employed. Usually we use Pt and W tips and represent the tip apex as a pyramid-like cluster epitaxed on a substrate that is orientated along some low Miller index crystal plane (for example, 111, 110, or 100 surface planes). Generally we find that the structure of the tip has quite a big impact on the images obtained. Sharp tips, such as those constructed on Pt 100, Pt 111 or W 100 surfaces tend to yield higher resolution images than those obtained with more blunt tips (for example the 111 surface of (bcc) W as shown in Scheme II). However,... 

If now any face on the crystal makes intercepts of a/h,b/k, and cll on the axes OA, OB, and OC, respectively, it is said to have the Miller indices (hid), which have no common divisor. The Miller indices of any face are thus calculated by dividing its intercepts on the axes by a, b, c, respectively, taking the reciprocals, and clearing them of fractions if necessary. If a plane is parallel to an axis, the intercept is at infinity and the corresponding Miller index is zero. The Miller indices of the standard face are (111), and the plane outlined by the dotted lines in Fig. 9.1.2(a) has intercepts a/3, b, c/2, which correspond to the Miller indices (312). In Fig. 9.1.2(b) another plane is drawn parallel to the aforesaid plane, making intercepts a, 3b, 3c/2 it is obvious that both planes outlined by dotted lines in Fig 9.1.2 have the same orientation as described by the same Miller indices (312). 

Once the crack is initiated, the metal surface inside the crack may be quite different from the normal surface of the metal. Thus, in the course of plastic deformation, the metal could have developed slip steps [see Fig. 12.77(c)] which contain crystallographic planes of high Miller index at which the specific dissolution rate (or exchange current density) may be larger than that at the normal metal surface. Anodic current densities of some 104 times those at a passive surface have been shown to appear at a metal surface that is yielding under stress (Despic and Raicheff, 1978). 

These three numbers enclosed in parentheses and not separated with commas, that is, (M/), named the Miller indexes, denote the crystallographic plane. 

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.

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