Lattice Planes and Miller Indices

Let us start with a few definitions. A lattice plane of a given 3D BL contains at least three noncollinear lattice points and this plane forms a 2D BL. A family of lattice planes of a 3D BL is a set of parallel equally-spaced lattice planes separated by the minimum distance d between planes and this set contains all the points of the BL. The resolution of a given 3D BL into a family of lattice planes is not unique, but for any family of lattice planes of a direct BL, there are vectors of the reciprocal lattice that are perpendicular to the direct lattice planes. Inversely, for any reciprocal lattice vector G, there is a family of planes of the direct lattice normal to G and separated by a distance d, where 2jt/d is the length of the shortest reciprocal lattice vector parallel to G. A proof of these two assertions can be found in Ashcroft and Mermin [1]. 

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. 

In fee and bcc lattices, there are no cubic primitive cells whereas in simple cubic system, the reciprocal lattice is also simple cubic and the Miller indices of a family of lattice planes represent the coordinates of a vector normal to the planes in the usual Cartesian coordinates. As the lattice planes of a fee cubic lattice or a bcc cubic lattice are parallel to those of a sc lattice, it has then been fixed as a rule to define the lattice planes of the fee and bcc cubic lattices as if they were sc lattices with orthogonal primitive vectors of the reciprocal lattice. 

As described in Chapter 1, the facets of a well-formed crystal or internal planes through a crystal structure are specified in terms of Miller Indices, h, k and l, written in round brackets, (,hkl). The same terminology is used to specify planes in a lattice. 

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Electron-density waves hkl lie perpendicular to the sets of crystal lattice planes with Miller indices hkl. The wavelength of the electron-density wave is the spacing of hkl crystal lattice planes, i.e., The amplitude and the relative phase of the electron-... 

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. 

The reciprocal lattice of the fee lattice (see eq. (12)) is bee, and the fourteen planes which bisect the shortest vectors to near-neighbor reciprocal lattice points have Miller indices... 

Figure 1.4 Examples of lattice planes and their Miller indices. (After Lalena and Cleary, 2005. Copyright John Wiley Sons, Inc. Reproduced with permission.)...

Figure 1.10. Examples of lattice planes and their Miller indices.

The angle between two sets of planes in any type of direct-space lattice is equal to the angle between the corresponding reciprocal-space lattice vectors, which are the plane normals. In the cubic system, the [h k /] direction is always perpendicular to the (h k 1) plane with numerically identical indices. For a cubic direct-space lattice, therefore, one merely substimtes the [h k 1] values for [u v w] in Eq. 10.57 to determine the angle between crystal planes with Miller indices h k l ) and (h2 h)- With all other lattice types, this simple... 

In the Bragg formulation of diffraction we thus refer to reflections from lattice planes and can ignore the positions of the atoms. The Laue formulation of diffraction, on the other hand, considers only diffraction from atoms but can be shown to be equivalent to the Bragg formulation. The two formulations are compared in Fig. 2B for planes with Miller indices (110). What is important in diffraction is the difference in path length between x-rays scattered from two atoms. The distance si + s2 in the Laue formulation is the same as the distance 2s shown for the Bragg formulation. The Laue approach is by far the more useful one for complicated problems and leads to the concept of the reciprocal lattice (Blaurock, 1982 Warren, 1969) and the reciprocal lattice vector S = Q 14n that makes it possible to create a representation of the crystal lattice in reciprocal space. 

As described in Chapter 1, for a perfect, infinite crystal the reciprocal lattice is made of points, each representing a set of planes with Miller indices hkl). The diffraction condition in reciprocal space is then defined in terms of a geometrical relation diffraction takes place when incident and diffracted beam are such that the scattering vector = (v — Vq) connects the origin with an hkl) point ... 

The diffraction pattern (Fignre 2.1) then simply results from the interference of the reflections from sets of parallel planes within the crystal. The spacing of the lattice planes is determined by the lattice geometry, that is, it is a function of the unit cell parameters. The orientation of the plane with respect to the axes of the unit cell is defined by three integers, h, k, and / (Miller indices) that denoted the points where the plane intersects the three unit cell axes. Miller indices are defined as h = a X, k = blY, and I = dZ, where X, Y, and Z are the points where the plane intersects the a, b, and c nnit cell axes. Thus, the plane intersecting the unit cell at all, bll, and c/2 would have indices 222. 

The Miller indices of planes in hexagonal lattices can be ambiguous. For example, three sets of planes lying parallel to the c-axis, which is imagined to be perpendicular to the plane of the diagram, are shown in (Figure 2.15). These planes have Miller indices A, (110), B, (120) and C, (210). Although these Miller indices seem to refer... 

Index the lattice planes drawn in the figure below using Miller - Bravais (hkil) and Miller indices (hid). The lattice is hexagonal with the c-axis is normal to the plane of the page and hence the index l is 0 in all cases. 

The important characteristics of the reciprocal lattice that should be noted are (1) that the vector r kl is normal to the crystallographic plane whose Miller indices are (hkl), and (2) that the length r%kl of the vector is equal to the reciprocal of the interplanar spacing dh]d. 

Many of the physical properties of crystals, as well as the geometry of the three-dimensional patterns of radiation diffracted by crystals, are most easily described by using the reciprocal lattice. Each reciprocal lattice point is associated with a set of crystal planes with Miller indices (hkl) and has coordinates hkl. The position of the hkl spot in the reciprocal lattice is closely related to the orientation of the (hkl) planes and to the spacing between these planes, dhki, called the interplanar spacing. Crystal structures and Bravais lattices, sometimes... 

Fig. 5 A lattice plane and definition of its Miller indices. The plane shown intercepts halfway along each axis. It is described by the Miller indices according to all, bl2, dl) = aih, bid c/l). This is. therefore, the (222) plane.

Figure 8.63 (a) A portion of a 3D crystal lattice. The unit cell, or basic repeating unit, of the lattice is shown in heavy outline. The black dots represent the atoms or ions or molecules that make up the crystal, (b) A cubic unit cell, with the comers of the cell located at 1 unit from the origin (O). The triangular plane drawn within the unit cell intersects the x-axis at 1/2, the y-axis at 1/2, and the z-axis at 1. This plane has Miller Indices of (221). (c) A family of planes shown in a 2D lattice. 

It therefore follows that the total intensity of a reflection depends on how far apart the various lattices are, i.e. on the positions and identities of the atoms in the cell. The square root of the total intensity of a reflection from a particular set of planes with Miller indices h, k and I is called a structure amplitude or structure factor, Ffj i. It depends not only on the fractional coordinates of the atoms x, y and z and their scattering powers,/, but also on parameters that describe the vibrational movement of the atoms in the lattice. Every vibrational movement leads to partial destructive interference and therefore a decrease of intensity of the reflected beam. Assuming that each atom i moves isotropically, with a mean-square amplitude of vibration (/, we can define quantities A and B by Eqs 10.8 and 10.9. 

Unlike plane lattice where the lines joining lattice points are lines, here in space lattice it generates a plane known as lattice plane. As it is important to know these lattice planes and as they preserve their individuality in much respect, each such lattice planes are identified by indices known as Miller indices, hkl. These indices are given to a plane by the following procedures [3,4] ... 

In Fig. 6.8, the miit translational vectors of the direct lattice and a plane having Miller indices hkl are shown to derive the further relations between the plane and the vectors in the corresponding new imaginary lattice. ... 

The orientation of a plane is defined by its Miller indices, h,k, . They are generated by taking the reciprocals of the intercepts of the plane with the three lattice vectors measured in lattice vector units, multiplying them by a factor that produces the smallest set of whole numbers with no common denominator, and writing them in parentheses as three numbers with no commas between them. For example, a plane intercepting a at 1 /2, b at 1 /3, and c at 2/3 in Figure 4.5 would have Miller indices given by (2/l,3/l,3/2) x 2 = (463) and would be referred to as the "four-six-three plane. The Miller indices do not refer to a... 

Thus we have constructed so that it is perpendicular to the plane with Miller indices h, k, t and it can be shown (proof left to the student) that the distance from the origin to lattice plane hkt) is given by... 

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. 

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. 

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