Lattices Miller indices

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Figure Bl.21.2. Atomic hard-ball models of stepped and kinked high-Miller-index bulk-temiinated surfaces of simple metals with fee lattices, compared with anfcc(l 11) surface fcc(755) is stepped, while fee...

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. 

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. 

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 spread of the reflections around the reciprocal lattice points due to the finite particle dimensions and to cumulative lattice disorders of the "ideally paracrystalline" type is calculated according to the formula given by Hosemann and Wilke (7). They showed that for a one-dimensional crystal the integral breadth (8) varies with the Miller index (h) of the reflection and can be approximated by the expression... 

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... 

Relations between Interplanar Spacing, Miller Indexes, and Lattice Parameters... 

This approximation is sufficient for broad lines and particle diameters <5 nm. The particle size is determined in the direction normal to the lattice plane indicated by the Miller index of the line used for profile determination. The formula featuring a constant accounting for the fact that the breadth is measured in radians (as is 20) is only valid for spherical particles. 

Besides, the authors discovered an intercalation effect during activation. Intercalation involves the insertion of calcium ions into aluminium hydroxide lattice which possesses a layered structure. Intercalation takes place between hydroxide layers over the planes with Miller index 002. A maximum increase of the distance is observed in Al(OH)3+CaO mixture (from 0.4850 to 0.4875 nm) while in Al(OH)3+ Ca(OH)2 mixture, it changes not so significantly (to 0.4867 nm). 

In some structures, several planes and directions may be equivalent by symmetry. For example, this is the case for the (100), (010), (001), (100), (010), and (OOl) planes in the diamond cubic structure. Equivalent directions are denoted concisely as a group by using angular brackets. Thus, the (100) directions in a diamond cubic lattice include all of the directions that are perpendicular to the six planes noted above. The Miller index notation thus provides a concise designation for describing the surfaces of semiconductor crystals. 

In this crystal lattice system, all surfaces with Miller indices, (hkl), satisfying the conditions h x k x 1 and h k l h are chiral [11]. Although such high Miller index surfaces have been studied for decades, it was not until recently that McFadden et al. specifically pointed out and demonstrated that their low synunetry structures render them chiral and, therefore, that they might have enantiospecific interactions with chiral adsorbates [12]. There has been a growing interest in the enantiospecific properties of naturally chiral metal surfaces and in the possibility of using such surfaces for enantioselective chemical processes. 

Similar to structure solution from first principles, the ab initio indexing implies that no prior knowledge about symmetry and approximate unit cell dimensions of the crystal lattice exists. Indexing from first principles, therefore, usually means that Miller indices are assigned based strictly on the relationships between the observed Bragg angles. 

FIGURE 3.17 In (a) and (b) are two different lattices. In each lattice, the points have been organized as families of planes. The planes, like the points, are periodic throughout the crystal and can be precisely specified by their interplanar spacing and a plane normal (a vector perpendicular to the planes). The families are designated by three integers hkl called Miller indexes, and these specify the slopes of the planes in a family. 

For every family of planes having integral Miller indexes hkl, a vector can be drawn from a common origin having the direction of the plane normal and a length 1 /d, where d is the perpendicular distance between the planes. The coordinate space in which these vectors are gathered, as in Figure 3.19, is called reciprocal space, and the end points of the vectors for all of the families of planes form a lattice that is termed the reciprocal lattice. 

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. 

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. 

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. 

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. 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-terminated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (hep) and body-centred cubic (bcc) lattices (a) fee... 

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. 

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