Miller , crystal faces

The temperature behavior of low446,491,503 558 as well as high Miller index crystal faces of Au447,448 has been examined in 0.01 M perchloric acid solutions. For all gold surfaces studied, C, was found to decrease and Ea=Q moved to less negative values with increasing t 446-448 491503-558... 

Metal (with the Miller indices of the crystal face) 4>e(ar) (eV) Method... 

The anodic dissolution of metals on surfaces without defects occurs in the half-crystal positions. Similarly to nucleation, the dissolution of metals involves the formation of empty nuclei (atomic vacancies). Screw dislocations have the same significance. Dissolution often leads to the formation of continuous crystal faces with lower Miller indices on the metal. This process, termed facetting, forms the basis of metallographic etching. 

A crystal is a polyhedral solid, bounded by a number of planar faces. The arrangement of these faces is termed the habit of the crystal, with the crystal faces being identified using Miller indices. The crystal is built up through the repetition of a fundamental building block, known as the unit cell. The molecules... 

The faces of crystals, both when they grow and when they are formed by cleavage, tend to be parallel either to the sides of the unit cell or to planes in the crystal that contain a high density of atoms, it is useful to be able to refer to both crystal faces and to the planes in the crystal in some way—to give them a name—and this is usually done by using Miller indices. 

The indices of crystal faces are traditionally expressed by the points at which the lattice plane parallel to the face cuts the a-, h-, and c-axes, respectively. Indices are always expressed as integers. A particular face is expressed by parentheses (), and crystallographically equivalent faces are denoted by curly brackets. Miller indices use the reciprocal number of the unit length at which the respective axes are cut, and are widely adopted. 

In many cases ordering is no longer observable in the presence of steps. Ordered carbonaceous layers form on the Ir(l 11) crystal face, for example, while ordering is absent on the stepped iridium surface. Ordering is absent on stepped Pt surfaces for most molecules that would order on the low Miller-Index (111) or (100) surfaces. 

Low Miller index surfaces of metallic single crystals are the most commonly used substrates in LEED investigations. The reasons for their widespread use are that they have the lowest surface free energy and therefore are the most stable, have the highest rotational symmetry and are the most densely packed. Also, in the case of transition metals and semiconductors they are chemically less reactive than the higher Miller index crystal faces. 

The various methods of determining the real surface area usually agree only to about 25%. One of the advantages of single crystals with a well-defined face exposed is that the real area is a known property of the exposed crystal face (see Section 7.8.2 on Miller indices). 

MILLER INDICES. In mineral crystallography the identity of a crystal face consists of a series of whole numbers which arc the products of their parameters relating to that face by their inversion, and where required lire clearing of fractionol values. A parameter is the relative intercept of a crystallographic axis on a given crystal face... 

Assuming parameter values on a given crystal face to be let. l/r, < would on inversion yield I. I. parameters I a. I h. 2 e would on inversion vield I, I. 1 and parameters of 3Miller indices of (112). (2211 and (2.311 respectively. 

Crystallographic identification of facial planes on tl crystal become possible through assignment of numerical values to a faec which represents its relationship to the crystal axes. Parameters, or relative intercepts, aie obtained hy plotting coordinates of crystal laces with respect to their crystallographic axes. The actual distances of axial intercepts of the crystal face are determined and expressed as a unit of measurement. The product of these values is known as Miller indices. 

A Miller indices faec on the from face of a cube would be (100). signifying that face intersects the o axis at I unit length from the center of the crystal, and is parallel to axes a2 and or intersects those axes at infinity. In this system, zero (0) is the numerical substitute for infinity. A (I I I) Miller indices face identifies that facial plane as intersecting each of the three crystallographic axes of that form at I unit length from the crystal center. 

The Miller Indices identify the orientation of a face in relationship to its axes of reference regardless of its sree and position on the crystal. Hatiy first proposed this basic law of crystallography— crystal faces make simple... 

The adsorption and ordering characteristics of the various hydrocarbon molecules on the low Miller index platinum surfaces are discussed in great detail elsewhere. These two surfaces appear to be excellent substrates for ordered chemisorption of hydrocarbons, which permit one to study the surface crystallography of these important organic molecules. The conspicuous absence of C-H and C-C bond breaking during the chemisorption of hydrocarbons below 500 K and at low adsorbate pressures (10 9-10-6 Torr) clearly indicates that these crystal faces are poor catalysts and lack the active sites that can break the important C-C and C-H chemical bonds with near zero activation energy. 

The chemisorption of over 25 hydrocarbons has been studied by LEED on four different stepped-crystal faces of platinum (5), the Pt(S)-[9(l 11) x (100)], Pt(S)-[6(l 11) x (100)], Pt(S)-[7(lll) x (310)], and Pt(S)-[4(l 11 x (100)] structures. These surface structures are shown in Fig. 7. The chemisorption of hydrocarbons produces carbonaceous deposits with characteristics that depend on the substrate structure, the type of hydrocarbon chemisorbed, the rate of adsorption, and the surface temperature. Thus, in contrast with the chemisorption behavior on low Miller index surfaces, breaking of C-H and C-C bonds can readily take place at stepped surfaces of platinum even at 300 K and at low adsorbate pressures (10 9-10-6 Torr). Hydrocarbons on the [9(100) x (100)] and [6(111) x (100)] crystal faces form mostly ordered, partially dehydrogenated carbonaceous deposits, while disordered carbonaceous layers are formed on the [7(111) x (310)] surface, which has a high concentration of kinks in the steps. The distinctly different chemisorption characteristics of these stepped-platinum surfaces can be explained by... 

In a series of studies, the dehydrogenation and hydrogenolysis of cyclohexane was studied on both the stepped and low Miller index (111) crystal faces of platinum at a surface temperature of 300°C and a hydrogen to cyclohexane ratio of 20 1. While the rates on the stepped and low Miller index surfaces were not very different for the formation of benzene and hexane, the formation of cyclohexene was very structure sensitive, its rate being 100 times greater on the stepped surface than on the (111) crystal face. In Table III mrnnare the initial turnover numbers for the various reactions at low... 

Crystalline solids consist of periodically repeating arrays of atoms, ions or molecules. Many catalytic metals adopt cubic close-packed (also called face-centred cubic) (Co, Ni, Cu, Pd, Ag, Pt) or hexagonal close-packed (Ti, Co, Zn) structures. Others (e.g. Fe, W) adopt the slightly less efficiently packed body-centred cubic structure. The different crystal faces which are possible are conveniently described in terms of their Miller indices. It is customary to describe the geometry of a crystal in terms of its unit cell. This is a parallelepiped of characteristic shape which generates the crystal lattice when many of them are packed together. 

Miller indices of crystal faces (100), (010), and (001), along with directions of the normals hkl (double arrows) to these faces The normal to (100) is parallel to a the normal to (010) is parallel to 6 the normal to the face (001) is parallel to c. Note that hkl indicates a set of hkl) planes, where the indices are related by the symmetry operators of the crystal these hkl are called "forms of planes." By convention, the normals of a face are enclosed by square brackets For example, [100] indicates the vector a, while [111] indicates the vector a — b + c, a family of symmetry-related vectors in direct space are enclosed in < > for example, if [110], [011], and [110] are symmetry-related, then < 110 > refers to all three. 

Similarly, the Miller indices H0 = (h0k0l0) = haQ + kbQ + lcQ denote the unit normal to an imaginary plane in a crystal, or a real crystal face they transform to Hn = (hnknln) as... 

Miller indices of a crystal face or a single net plane are enclosed in parentheses. (123) or (hkl) is a plane or set of planes that describe crystal faces (h hjh ) is a single net plane. 

Usually the crystal faces are described by the reciprocals of the multiples of the standard intercepts, hence the name the law of rational indices. In Figure 9-8 three lines are adopted as axes which may also be directions of the crystal edges. A reference face ABC makes intercepts a, b, c on these axes. Another face of the crystal, e.g., DEC, can be defined by intercepts alh, blk, dl. Here h, k, l are simple rational numbers or zero. They are called Miller indices. The intercept is infinite if a face is parallel to an axis, and horkorl will be zero. For orthogonal axes the indices of the faces of a cube are (100), (010), and (001). The indices of the face DEC in Figure 9-8 are (231). 

William Whewell (1794-1866) introduces a notation system relating crystal faces to coordinate axes (Miller indices). These were published in a book by Whewell s student William Hallowes Miller (1801-1880) in 1839. 

Absorption correction can be based on an accurate description of the crystal shape in terms of Miller planes (faces), so that the t can be calculated for each incident and diffracted beam. The total absorption is then given by equation (27), where A is the absorption factor. 

It is important to describe each crystal face in a numerical way if data on different crystals or from different laboratories are to be compared. The method used to describe crystal faces is derived from the Law of Rational Indices, proposed by Haiiy and Arnould Carangeot. This Law states that each face of a crystal may be described, by reference to its intercepts on three noncollinear axes, by three small whole numbers (that is, by three rational indices)/ From this law, William Whewell introduced a specific way of designating crystal faces by such indices, and William Hallowes Miller popularized it. The integers that characterize crystal faces are called Miller indices h, k, and 1. When this method is used to describe crystal faces, it is rare to find h, k, or / larger than 6, even in crystals with complicated shapes. An example of the buildup of unit cells to give crystals with different faces is shown in Figure 2.11. 

FIGURE 2,12. The indexing of crystal faces, (a) The derivation of the integers used to describe the crystal faces (Miller indices). The (213) face is marked as an example. It intersects the axes of the unit cell at a/2, 6/1, and c/3. (b) Several parallel planes can be represented by these Miller indices. They intersect at, for example, a, 26, 2c/3 3a/2, 36, c. (c) A crystal with its (100) face (stippled), intersecting the x tixis at x = a and the y and z axes at infinity (that is, parallel to them), (d) A crystal with a (213) face stippled,... 

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