6 Some Crystal Structures

Prof. P. N. Kotru

6.1 Miller Indices of a Crystal Face.

The orientation of any face or plane of a crystal is labelled by a set of three numbers called Miller indices. We first define a plane by taking the intercepts that it cuts on the three axes. Choose a lattice point O as origin and let us take OX, OY and OZ as positive directions along the three edges of the unit cell. We give the intercepts in terms of the dimensions of the unit cell which are the unit distances along the three edges, i.e., in terms of the primitive translations along the three axes. The procedure of determining Miller indices may be described as follows:

1. Take any atom of the crystal as the origin and choose co-ordinate axes from this atom in the direction of the basis vectors.
2. Next find the intercepts of the plane under consideration belonging to the system. Express these intercepts as integral multiples of the basis vectors along the three crystal axes.
3. Take the reciprocals of these numbers.
4. Clear off the fractions by multiplying each by the L.C.M. of the denominators. This will reduce it to three smallest integers having the same ratio. Suppose these integers are, for example, h,k,l. The quantities (h k l) are then the Miller Indices of that system of planes.

Let us illustrate the method by taking the example of a plane whose intercepts are at the lattice distance ‘a’ ,twice the lattice distance ‘b’ (i.e., 2b ) and three times the lattice distance ‘c’ (i.e.,3c), as shown in figure 6.1

Figure 6.1: Plane with miller indices (6 3 2)

Now , take the reciprocals of these intercepts which in this particular example are 1/1 , ½ and 1/3.Multiplying each of the reciprocals by the L.C.M. of the denominator 1 , 2 ,and 3 which comes out to be 6. The numbers that we get are 6, 3, and 2. Place these three numbers in parenthesis as (6 3 2). These are the Miller Indices of the plane. The plane, we may say, is a member of the (6 3 2) family of planes.

Let us consider some more examples and further explain the procedure of assigning Miller indices to crystal faces or planes. Suppose a plane cuts the x-axis at unit distance but is parallel to the Y and Z axes.

So, the plane cuts intercepts 1 a, ∞ and ∞ along X, Y and Z axes respectively. Taking the reciprocals of these numbers which are 1 a, 0 and 0. The Miller indices of this plane, therefore, are (1 0 0). This may also be called as YZ plane or bc plane. Similarly, indices of the planes that are parallel to X and Z axes ( a c plane ) and X and Y axes ( a b plane ) are , therefore , ( 0 1 0 ) and ( 0 0 1 ) respectively. Taking unit cell of a simple cubic crystal , planes (1 0 0) , (0 1 0), (0 0 1) and (1 1 1 ) are shown as shaded planes in schematic diagrams of figures 6.2 (a,b,c,d)

Figure 6.2: (100), (010), (001) and (111) planes of a simple cubic crystal

Six faces of a cubic crystal are assigned the indices as (100), (100), (010), (010), (001), and (001) as shown in the schematic diagrams. {100} denotes the six faces of the cubic form.

Figure 6.3: Faces of a cubic crystal with Miller indices (001), (010), (001), (100) and (010). {100} denote the six faces of the cubic form

In crystallography it is conventional to write the minus signs above and not before the Miller Indices. The digits of six faces of a cubic crystal are 1 or 1 and two zeroes. Likewise, there may be number of systems of planes whose Miller indices differ by permutation of numbers or of minus sign yet they are all crystallographically equivalent planes in respect of density of atoms and interplanar spacing. In a cubic lattice , all the planes identified by permutations of three Miller indices among themselves., as for example, ( hkl) ,(khl),(lhk) and so on as well as those obtained by taking various combinations of minus sign are all crystallographically equivalent. In the language of crystallography, we say the six faces of a cubic crystal belong to the same form. To show the crystallographic equivalence of the six faces belonging to the same form, of which (hkl) is a member, it is customary to enclose the Miller indices in curly brackets i.e., {hkl}. In the case of the six faces of the cubic form described here the form is represented by {100}. For any crystal structure, a set of axes can be found for which the Miller indices reduce to simple integers or zero. This is known as Law of Rational Indices.

Miller indices of a hexagonal crystal are satisfied by choosing four axial directions a1, a2, a3, and c. In this system, a plane is specified by four numbers as shown in figure 6.4

Figure 6.4: Miller indices for hexagonal system

6.2    Indices of Crystallographic Direction

Considering three-dimensional space lattice, we take a point at the origin of co-ordinates and move it in the given direction until it passes through the first lattice point whose co-ordinates are ha, kb, lc. The indices of the direction are then hkl and it is customary to write it within square brackets [h k l]. This operation is equivalent to saying that the desired movement can be accomplished by moving along the x-axis a distance h times unit distance a , along the y-axis a distance k times the unit distance b and along the z-axis a distance l times the unit distance c .So, square brackets enclosing three indices specify direction indices. The normals to crystallographically equivalent planes are crystallographically equivalent directions. The complete set of crystallographically equivalent directions of which [hkl] are a member is denoted by enclosing the direction indices in angle brackets such as <hkl>. In case of cubic crystals, as for an example, a direction represented by direction indices [hkl] is normal to a plane whose Miller indices are (hkl).

6.3 Some Important Crystallographic Formulae

6.3.1 Interplanar spacing.

The interplanar spacing d between adjacent planes holding Miller indices (hkl) are given by using the following equations relevant to the crystal system to which they belong. The axial lengths of unit cell are represented by a, b and c whereas their axial angles are represented by α, β and γ.

TRICLINIC SYSTEM: 1/d² = 1/v² (W₁₁ h² + W₂₂ k² + W₃₃ l² + 2 W12hk +2 W₂₃ kl + 2 W₃₁ hl)

MONOCLINIC SYSTEM: 1/d2 = 1/sin2β {h2/a2 + (k2sin2β)/b2 + l2/c2 ─ (2hlcosβ)/ac}

HEXAGONAL SYSTEM : 1/d2 = 4 ( h2 + hk + k2 ) +    l2

TETRAGONAL SYSTEM: 1/d²= h²+ k² + ^l2

6.3.2 Interplanar angles

Let us consider planes with Miller indices h1 k1 l1 with interplanar spacing d1 and the plane with Miller indices h2 k2 l2 with interplanar spacing d2. If these two set of planes make an angle θ with each other, its value can be found by application of the following formulae: