Miller indices defined

The repeat unit of a stack of planes is introduced when analyzing the distribution of the bulk structure atoms over the atomic planes in the direction z normal to the surface. This distribution depends on the oxide bulk structure and Miller index, defining the surface. The three different possible stacking sequences define three types of surfaces. 

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). 

If the vector defining the surface normal requires a negative sign, that component of the Miller index is denoted with an overbar. Using this notation, the surface defined by looking up at the bottom face of the cube in Fig. 4.4 is the (001) surface. You should confirm that the other four faces of the cube in Fig. 4.4 are the (100), (010), (100), and (010) surfaces. For many simple materials it is not necessary to distinguish between these six surfaces, which all have the same structure because of symmetry of the bulk crystal. ... 

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 convention of Miller indices is used to describe the planes within a unit cell. Miller indices are defined as the reciprocals of the intercept, which the plane makes with each of the three crystal axes. Each plane is denoted by three parameters h k and Planes which are parallel to a crystal axis are given the Miller index of 0 while planes formed in the negative direction are written with a bar over the number in the Miller index. The Miller index of a single specific plane is written within parentheses (hkl) whereas the Miller indices describing a whole family of faces are written with braces hkl. The faces that exist and define the crystal morphology are termed morphologically important and are commonly identified by the Miller indices of the planes represented by those faces. 

These difficulties have stimulated the development of defined model catalysts better suited for fundamental studies (Fig. 15.2). Single crystals are the most well-defined model systems, and studies of their structure and interaction with gas molecules have explained the elementary steps of catalytic reactions, including surface relaxation/reconstruction, adsorbate bonding, structure sensitivity, defect reactivity, surface dynamics, etc. [2, 5-7]. Single crystals were also modified by overlayers of oxides ( inverse catalysts ) [8], metals, alkali, and carbon (Fig. 15.2). However, macroscopic (cm size) single crystals cannot mimic catalyst properties that are related to nanosized metal particles. The structural difference between a single-crystal surface and supported metal nanoparticles ( 1-10 nm in diameter) is typically referred to as a materials gap. Provided that nanoparticles exhibit only low Miller index facets (such as the cuboctahedral particles in Fig. 15.1 and 15.2), and assuming that the support material is inert, one could assume that the catalytic properties of a... 

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. 

As one generally uses a vector normal to a lattice plane to specify its orientation, one can as well use a reciprocal lattice vector. This allows to define the Miller indices of a lattice plane as the coordinates of the shortest reciprocal lattice vector normal to that plane, with respect to a specified set of direct lattice vectors. These indices are integers with no common factor other than 1. A plane with Miller indices h, k, l is thus normal to the reciprocal lattice vector G = hb + kb > + lb->. and it is contained in a continuous plane G.r = constant. This plane intersects the primitive vectors a of the direct lattice at the points of coordinates xiai, X2a2 and X3a3, where the Xi must satisfy separately G.Xjai = constant. Since G.ai, G.a2 and G.as are equal to h, k and /, respectively, the Xi are inversely proportional to the Miller indices of the plane. When the plane is parallel to a given axis, the corresponding x value is taken for infinity and the corresponding Miller index taken equal to zero. 

It has recently been proposed that kink sites of high Miller index metal surfaces should be considered as chiral when the step lengths on either side of the kink are unequal [22], Two such surfaces, which are not superimposable, can be defined - by analogy with the Cahn-Ingold-Prelog rules - as, e. g., Ag(643) and Ag(643). Theoretical calculations predicted that adsorption of chiral molecules should be stereospecific on such surfaces, but the only experimental evidence yet available is the electro-oxidation of d- and L-glucose on Pt(643) and Pt(531) surfaces [23]. It was speculated that with the polycrystalline metal catalyst, which contains equal numbers of (/ )- and (S)-type kink sites, preferential adsorption of a chiral modifier on one type of kink site would leave the other type of site free for catalysis. 

The representation of planes in a lattice makes use of a convention known as Miller indices. In this convention, each plane is represented by three parameters (hkl), which are defined as the reciprocals of the intercepts the plane makes with three crystal axes. If a plane is parallel to a given axis, its Miller index is zero. Negative indices are written with bars over them. Miller indices refer not only to one plane but a whole set of planes parallel to the plane specified. If we wish to specify all planes that are equivalent, we put the indices in braces. For example, 100 represents all the cube faces. Examples of Miller indices in the cubic system are shown in Figure 2.5. 

A plane occupied by atoms or ions within a crystal, or at its surface, is defined by its Miller index, which consists of integers that are the reciprocals of the intersections of that plane with the system of axes appropriate to the crystal... 

If a plane is parallel to a crystallographic axis, the spacing in that direction will be oo, and therefore the Miller index will be 0. The crystallographic planes parallel to one axis are defined by [Okl), (hOl), and (hkO), respectively. Planes parallel to faces A, B, and C of the unit cell will be (/ OO), OkO), and (00/), respectively. 

Finally, catalysis is not confined to well-defined. Miller-indexed metal surfaces. One area of recent interest is in the use of certain minerals to catalyze reactions. Some aluminosilicates—minerals with mixed alumina and silica structures—have pores in which molecules can enter and react catalytically [Figure 22.25(a)]. One type of aluminosilicate is called zeolite, shown in Figure 22.25(b). Thanks to the pores in zeolites (which can vary in size and geometry depending on the exact type of zeolite), the properly sized reactant molecules can enter the structure and a particular reaction can be promoted. In fact, it is thought that such minerals are the future of designed catalysts that can be used to promote a preferred chemical reaction—if the pore size is just right. 

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