Indices, Miller

The Miller index is the least whole-numbered multiple of the reciprocals of the intercepts of the plane (hkl) with the axes a, b, and c. 

Basic to the representation of crystals, lattices, and motifs is their symmetry. Group theory, summarized in Appendix 14 with Fig. A. 14.1, is the branch of mathematics dealing with interrelationships between the symmetry elements. Some elementary group theory is needed for the operations, described next. 

The geometric characteristics of a crystal structure are described using a tool called Miller indices. Because a crystal is a three-dimensional object, it must be described in three directions, similar to a 3D graph that uses x, y, and z directions to describe a point in space. In crystallography, or the study of crystals, instead of being called x, y, and z, the directions are called Miller indices, denoted by h, k, and 1. This helps avoid confusion, because when you see h, k, or 1, you know someone is talking specifically about crystals, and not just about 3D objects. 

Recognizing the Miller indices and crystal dimensions is important when doing x-ray diffraction (XRD). This technique measures the size and shape of the crystal, along with the positions of all the atoms in the unit cell. For more information, see Chapter 22. 

Generally speaking crystal structures fall into one of three categories simple, binary, or complex. The number of atoms bonded together, the number of different elements involved, and the size and other characteristics determine whether the crystal formed is simple, binary, or complex. 

For the smaller face, the intercepts are determined relative to the reference face. Thus, the MPO face intercepts the a- and b-axes at the same positions as the reference face and therefore the intercepts are one unit. On the c-axes the intercept is half of the distance to N, which was taken as one unit (notice that the units are relative). Thus, the intercept of MPO on the c-axis is The reciprocal of is 2 and the Miller indices for the face MPO therefore are (112). It is helpful to recognize that the larger the value of the index of the c-axes, the smaller the inclination of the face toward the axis. 

Ionic Compounds, By Claude H. Yoder Copyright 2006 John Wiley Sons, Inc. 

This face intercepts the a-axis at what we will designate as one unit distance and does not intercept (that is, runs parallel to) the b- and c-axes. The Miller indices are therefore 1/1, 1/infinity, and 1/infinity. Since anything divided by infinity is zero, the indices are (100). 

Look at the face labeled a and determine how it would intersect the axes if the face were extended until it was large enough to meet the axes. 

The face would intersect each axis the same distance from the origin. We will arbitrarily call these distances lai, la2, and la3. The Miller indices for this face are (111). 

Since the change in size and shape of a crystal does not alter the angles between the faces (experimental observation) and, moreover, as long as 

FIGURE 2.39 Deteimining the tauto-zonalfaces from a fundamental tetrahedron. 

In this sense, the orientation of a crystallographic plane will be customized by a set of three numbers, called the Miller indices. They are successively written in the order corresponding to the fundamental axes, without 

FIGURE2.40 The coordination of the points in a lattice the points which have a central or intermediate position are noted by the fractional indices, after Chiriac-Putz-Chiriac (2005). 

FIGURE 2.41 The unit cell (left) and the determination of the Miller indices associated to an arbitrary plane from the crystal (right) in relation with the parameters of the unit cell, after Chiriac-Putz-Chiriac (2005). 

For a mathematical description of crystal faces, take any three non-parallel faces (chosen to be mutually orthogonal, if possible) and take their intersections as reference axes, which are labeled OA, OB, and OC with the origin at O, as shown in Fig. 9.1.2(a). Let another face (the standard face or parametral face A B C ) meet these axes at A, B, and C, making intercepts OA = a, OB = b and OC = c, respectively. The ratios a b c are called the axial ratios. 

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). 

In 1784, Haiiy formulated the Law of Rational Indices, which states that all faces of a crystal can be described by Miller indices (hid), and for those faces that commonly occur, h, k, and l are all small integers. The eight faces of an octahedron are (111), (111), (ill), (111), (111), (111), (III), and (III). The form symbol that represents this set of eight faces is 111. The form symbol for the six faces of a cube is 100. Some examples in the cubic system are shown in Figs. 9.1.3. and 9.1.4. 

All the faces of a crystal can be described and numbered in terms of their axial intercepts. The axes referred to here are the crystallographic axes (usually three) which are chosen to fit the symmetry one or more of these axes may be axes of symmetry or parallel to them, but three convenient crystal edges can be used if desired. It is best if the three axes are mutually perpendicular, but this cannot always be arranged. On the other hand, crystals of the hexagonal system are often allotted four axes for indexing purposes. 

for example, three crystallographic axes have been decided upon, a plane that is inclined to all three axes is chosen as the standard or parametral plane. It is sometimes possible to choose one of the crystal faces to act as the parametral plane. The intercepts X, Y and Z of this plane on the axes x, y and z are called parameters a, b and c. The ratios of the parameters a b and b c etvQ called the axial ratios, and by convention the values of the parameters are reduced so that the value of b is unity. 

Miller suggested, in 1839, that each face of a crystal could be represented by the indices h, k and /, defined by 

For the parametral plane, the axial intercepts X, Y and Z are the parameters a, b and c, so the indices h, k and / are aja, bjb and cjc, i.e. 1, 1 and 1. This is usually written (111). The indices for the other faces of the crystal are calculated from the values of their respective intercepts X, Y and Z, and these intercepts can always be represented by ma, nb and pc, where m, n and p are small whole numbers or infinity (Haiiy s Law of Rational Intercepts). 

The generally accepted notation for Miller indices is that Qikl) represents a crystal face or lattice plane, while hkl represents a crystallographic form comprising all faces that can be derived from hkl by symmetry operations of the crystal. 

However inaccurate the depiction, the idea of X-ray diffraction does bring up the concept that planes of atoms are important in an understanding of crystal structures. In the first approximation, individual planes of atoms reflect X rays, and the constructive interference of many reflections from many planes yields refraction of X rays at the correct angle. How do we define a plane of atoms in terms of the unit cell There is another consideration, too At some point, the supposed infinite array of unit cells must, in fact, stop and make the surface of the crystal. This surface is usually considered planar. (In fact, many examples of large crystals are used as examples because they have well-defined planar surfaces. A well-cut diamond, for example, has a very specific shape in terms of the planes that terminate the unit cells.) It becomes clear that we must be able to define planes of corresponding atoms within arrays of unit cells. 

FIGURE 21.18 How to determine Miller indices of parallel planes of atoms. See text for details. 

Unless otherwise noted, all art on this page Is Cengage Learning 2014. 

FIGURE 21.19 Miller indices of some planes in cubic crystals. 

How do we designate these parallel planes in terms of the unit-cell dimensions a, b, and c We use the following steps  

Planes that contain one or more axes or go through the origin can be shifted by unit cells so that their intercepts can be located. 

Just as in the case of lattice directions, families of planes, such as the six face planes (100), (010), (001), (Too), (010), (001) are designated with a special symbol, in this case, 100. However, note that (100) and (TOO), (010) and (010), (001) and (001) are equivalent. 

As stated previously, the Miller indices refer to a family of parallel planes. However, somefimes if is necessary fo refer to a specific plane. In this case, the plane is specified withouf reducing if fo fhe lowest set of whole numbers. For example, a plane intercepting a at 1 /2, b at oo, and c at oo would be written as (200) instead of (100). The reason is that the atoms in that particular plane are of interest and the (200) plane may very well have a different set of atoms than the (100) plane. This notation is sometimes referred to as Laue notation and much use is made of this notation in x-ray crystallography as seen in Chapter 6. 

So far, no assumptior s have been made concerning the properties of the lattice vectors a, b, and c, so all of the notations we have developed thus far apply to any set of lattice vectors. For cubic systems only, the Miller indices of a plane are the same as the direction indices of a vector that is perpendicular to it i.e., the (321) plane is perpendicular to the [321] vector as demonstrated in Section 4.1.5. 

---

In describing a particular surface, the first important parameter is the Miller index that corresponds to the orientation of the sample. Miller indices are used to describe directions with respect to the tluee-dimensional bulk unit cell [2]. The Miller index indicating a particular surface orientation is the one that points m the direction of the surface nonual. For example, a Ni crystal cut perpendicular to the [100] direction would be labelled Ni(lOO). 

Single-crystal surfaces are characterized by a set of Miller indices that indicate tlie particular crystallographic orientation of the surface plane relative to the bulk lattice [5]. Thus, surfaces are labelled in the same way that atomic planes are labelled in bulk x-ray crystallography. For example, a Ni (111) surface has a surface plane... 

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...

A kinked surface, like fee (10,8,7), can also be approximately expressed in this fomi the step plane (h k / ) is a stepped surface itself, and thus has higher Miller indices than tlie terrace plane. However, the step notation does not exactly tell us the relative location of adjacent steps, and it is not entirely clear how the terrace width M should be counted. A more complete microfacet notation is available to describe kinked surfaces generally [5]. 

Figure C2.3.12. Two-dimensional neutron scattering by EOggPO gEOgg (Pluronic F88) micellar solution under shear witli (a) tlie sample shear axis parallel to tlie beam, and (b) tlie sample rotated 35° around tlie vertical axis. Reflections for several of tlie Miller indices expected for a bee lattice are annotated. Reproduced by pennission from figure 4 of [84]-...

We could make scale drawings of the many types of planes that we see in all unit cells but the concept of a unit cell also allows us to describe any plane by a set of numbers called Miller Indices. The two examples given in Fig. 5.5 should enable you to find the... 

Fig. 5.5. Miller indices for identifying crystal planes, showing how the (131) plane and the (110) planes are defined. The lower part of the figure shows the family of 100 and of 110) planes.

The arrangement of lattice points in a 2D lattice can be visualized as sets of parallel rows. The orientation of these rows can be defined by 2D Miller indices (hksee Figure lb). Inter-row distances can be expressed in terms of 2D Miller indices, analogous to the notation for 3D crystals. 

Fig. 1.2 Hard-sphere model of face-centred cubic (f.c.c.) lattice showing various types of sites. Numbers denote Miller indices of atom places and the different shadings correspond to differences in the number of nearest neighbours (courtesy Erlich and Turnbull )...

An account of the use of Miller indices to describe crystal planes and lattice directions is beyond the sco[>e of this article a very adequate treatment of this topic is, however, given in Reference 1. 

Figure 9-7. Elastic electron-diffraction pattern of a highly textured hcxaphenyl film. The Miller indices arc assigned using the intcrplauar spacings calculated in Kef. 11371. Inset Intensity of the f020) peak as a function of the angle between momentum transfer and the Teflon rubbing direction (see text) - taken from Ref. 138. ...

The distance of each reflection from the center of the pattern is a function of the fiber-to-film distance, as well as the unit-cell dimensions. Therefore, by measuring the positions of the reflections, it is possible to determine the unit-cell dimensions and, subsequently, index (or assign Miller indices to) all the reflections. Their intensities are measured with a microdensitometer or digitized with a scanner and then processed.8-10 After applying appropriate geometrical corrections for Lorentz and polarization effects, the observed structure amplitudes are computed. This experimental X-ray data set is crucial for the determination and refinement of molecular and packing models, and also for the adjudication of alternatives. 

It turns out that the method used to decribe the planes given above for the cubic lattice can also be used to define the planes of all of the known lattices, by use of the so-called "Miller Indices", which are represented by ... 

The way that Miller Indices arose CcUi be understood by considering the history of crystal structure work, accomplished by many investigators. 

Returning to the unit-ceU, we can also utilize the vector method to derive the origin of the Miller Indices. The general equation for a plane in the lattice is ... 

Thus, these intercepts are given in terms of the actual unit-cell length found for the specific structure, and not the lattice itself. The Miller Indices are thus the indices of a stack of planes within the lattice. Planes are important in solids because, as we will see, they are used to locate atom positions within the lattice structure. 

It is of tremendous practical importance that at many metal surfaces chemisorption of a large variety of molecules can occur. This can easily be explained from a discussion on a molecular level as follows. At the surface of a crystal the co-ordination number of the metal atoms is lower than in the bulk of the crystal. Fig. 3.4 illustrates this for Ni. In the bulk the coordination number of all Ni atoms is 12, whereas on the three faces in Fig. 3.4 these numbers are only 8, 7 and 9 for the (100), (110) and (111) surfaces, respectively. So-called free valences exist at the surface. The numbers between brackets are the Miller indices of the surfaces. 

Figure 1.2 Miller indices of some important planes.

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

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. 

MILLER INDICES AND BRAGG DISTANCES FOR POLY (TRANS-1, 4 -HEXADIENE )... 

Furthermore, crystals whose structures are not centrosymmetric have different hardnesses on opposite sides of a given crystal even though the Miller indices of the surface planes are the same. For example, the hardness of the (0001) plane of ZnS (zinc blende structure) is not the same as that of the (000-1) plane. 

---