Crystal Analysis: Miller Indices of a Crystal | Materials | Engineering

In this article we will discuss about:- 1. Miller Indices of Direction 2. Miller Indices of Planes 3. Miller-Bravais Indices  4. Important Features.

Miller Indices of Direction:

In a crystal, there exist directions and planes which contain a large concentration of atoms. Therefore, it is necessary to locate these direction and planes for crystal analysis and to use some conventions to specify them in a crystal. A crystallographic direction is defined as a line between two points or a vector. In Fig. 2.52, the vector r⃗, passing through the origin O to a lattice point, can be expressed in terms of the fundamental translation vectors a, b and c, which form the crystal axes.

r = n₁a + n₂b + n₃c

where, n₁, n₂ and n₃ are integers. The C-axis is not shown in the Figure, as r is assumed to lie on the ab plane. The components of r along the three axes are- n₁ = 4, n₂ = 4 and n₃ = 0.

Then the crystal direction denoted by ‘r’ is written as [440] in Miller notation, with square brackets enclosing the indices. If we clear the common factor, we can represent the same direction by [110] in Miller notation, which shows that the coordinates of the first lattice point lying on the line OP are utilized to denote the direction of the line.

Suppose we have to represent a direction [uvw] with u = n₁a, v = n₂b, w = n₃c, where n₁, n₂, n₃ are integers. To show this direction we will require n₁ unit cells packed along the x-axis, n₂ along the y-axis and n₃ along the z-axis. We can do so in a single unit cell by dividing the u, v, w by the integer which is highest in n₁, n₂, and n₃. Say n₁ is the highest integer, then we have by dividing-

For example [234] can be written as and can easily be plotted in a single unit cell. A few directions are shown in Fig. 2.53.

The following steps are utilized to determine the Miller indices of a direction:

(1) A vector of convenient length is positioned such that it passes through the origin of the co-ordinate system.

(2) The length of the vector projection, each on the three axes is determined; these are measured in terms of the unit cell dimensions a, b and c.

(3) These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values.

(4) The three indices are enclosed in square brackets (not separated by commas), thus [hkl]. The h, k and l integers correspond to the reduced projections along the x, y and z axes respectively.

(5) The negative indices are also possible, which are designated by a bar over the appropriate index, the [1 1̅ 1] direction would have a component in the -y direction.

Families of Directions:

In the cubic crystal, the following directions are identical except for our arbitrary choice of the x, y and z labels on the axes:

Any directional property (modulus of elasticity, refractive index etc.) will be identical for these four opposing pairs. Therefore, it is convenient to identify this family of directions as = < 111 > rather than writing the eight separate indices.

Miller Indices of Planes:

The crystal lattice may be regarded as made up of an aggregate of a set of parallel equidistant planes passing through the lattice points which are known as lattice planes. For a given lattice, the lattice planes can be chosen in different ways as shown in Fig. 2.54. The problem is that how to designate these planes in a crystal. Miller evolved a method to designate a plane in a crystal by three numbers (h k l) known as Miller indices and the lattice plane is known as Miller plane.

The steps in the determination of Miller indices of a plane are illustrated with the aid of Fig. 2.55, where a plane intercepts at 2a, 3b and c respectively along x, y, and z-axes.

(i) Determine the coordinates of the intercepts made by the plane along the three crystallographic axes (x, y, z) axes)

(ii) Express the intercepts as multiples of the unit cell dimensions, or lattice parameters along the axes, i.e.,

(iii) Determine the reciprocals of these numbers-

(iv) Reduce these reciprocals to the smallest set of integral numbers and enclose them in brackets-

In general it is denoted by (h, k, I). We also notice that-

Thus, Miller indices may be defined as the reciprocals, of the intercepts made by the plane of the crystallographic axes when reduced to smallest integers.

Families of Planes:

Depending on the crystal system, two or more planes may belong to the same family of planes. In the cubic system, an example of multiple planes includes the following, which constitute a family of planes;

The collective notation for a family of planes is [hkl], each plane is identical except for the consequences of our arbitrary choice of axis labels and directions. It has been verified that {111} family includes eight planes and the {110} family includes 12 planes in a cubic system.

Miller-Bravais Indices for Hexagonal Crystal:

An alternate indexing system, which has four numbers in each set of indices, is often used for hexagonal crystal. The indices are called Miller-Bravais indices. Four numbers are used in order to make the relationship between the indices and the symmetry of the hexagonal lattice more obvious. The hexagonal unit cell, is described with reference to four axes, one along the axis of the hexagonal prism and three (a₁, a₂, a₃) in the base, 120° apart (see Fig. 2.57b).

The Miller-Bravais indices of a plane are denoted by h, k, i and l enclosed in parentheses, (h k i I). These indices are the reciprocals of intercepts on the a₁, a₂, a₃ and c-axes, respectively. As with Miller indices, the reciprocals are usually divided by the largest common factor. Since only three non-coplanar axes are necessary to specify a plane in space, the four indices cannot be independent. The additional condition which their values must satisfy is-

h + k = -i … (v)

In Fig. 2.57 (b) the basal plane (0001)—and the prism plane (101̅0) are indicated.

The Miller-Bravais indices of a direction are the vector components of the direction, resolved along each of the four coordinate axes and reduced to smallest integers. For consistency, the same restriction (Equation (v)) which applies to planes is applied to directions. The three coplanar components may be made to satisfy Equation (v) by drawing them on the triangular net shown on the base of the prism in Fig. 2.57 (a).

Many crystallographers prefer to express directions in a hexagonal lattice with reference to the three co-ordinate system a₁, a₂ and c and not the four co-ordinate system a₁, a₂, a₃, and c. We give the transformation relation between the indices of a direction [uvw] and [hkil] when referred to the system of axes a₁, a₂, and c and a₁, a₂, a₃ and c respectively follows-

Let us consider a lattice vector in three index system,

Since in hexagonal system a = b ≠ c and we have considered coplanar vectors, so we can have-

The above vector can also be written in forth index notation as-

Now on comparing the coefficients of a₁⃗, a₂⃗ and c⃗ of equations (I) and (II), we have-

(2h + k) = u, (2k + h) = v and w = I

On solving these equations, we get,

h = (2u – v/3), k = (2v – u/3), l = w

Now we will convert [110] to four index system by using the above relations, we have-

h = 1/3, k = 1/3, i = -2/3 and l = 0

or [hkil] = [112̅0].

Important Features of Miller Indices:

(i) All the parallel equidistant planes have the same Miller indices. Thus the Miller indices define a set of parallel planes.

(ii) A plane parallel to one of the coordinate axes has an intercept at infinity.

(iii) If the Miller indices of two planes have the same ratio (i.e. 844 and 422 or 211), then the planes are parallel to each other.

(iv) If (h k I) are the Miller indices of a plane, then the plane cuts the axes into a/h, b/k and c/l equal segments respectively.

(v) When the integers used in the Miller indices contain more than one digit, the indices must be separated by commas for clarity e.g. (3, 11, and 12).

(vi) The crystal directions of a family are not necessarily parallel to one another. Similarly, not all members of a family of planes are parallel to one another.

(vii) By changing the signs of all the indices of a crystal direction, we obtain the antiparallel or opposite direction. By changing the signs of all the indices of a plane, we obtain a plane located at the same distance on the other side of the origin.

(viii) The normal to the plane with indices (hkl) is the direction [hkl].

(ix) The angle between the normal to the two planes (h₁ k₁ l₁) and (h₂ k₂ l₂) is –

for orthorhombic system.

for cubic system.

(xi) A negative Miller index shows that the plane (h̅kl) cuts the x-axis on the negative side of the origin.

(xii) Miller indices are proportional to the direction cosines of the normal to the corresponding plane.

(xiii) The purpose of taking reciprocals in the present scheme is to bring all the planes inside a single unit cell so that we can discuss all crystal planes in terms of the planes passing through a single unit cell.

(xiv) Most planes which are important in determining the physical and chemical properties of solids are those with low index numbers.

(xv) The plane (hkl) is parallel to the line [uvw] if hu + kv + Iw = 0.

(xvi) Two planes (h₁ k₁ l₁) and (h₂ k₂ l₂) both contain line [uvw] if u = k₁ l₁ – k₂ l₁, v = l₁ h₂ – l₂ h₁, and w = h₁ k₂ – h₂ k₁

Then both the planes are parallel to the line [uvw] and therefore their intersection is parallel to [uvw] which defines the zone axis.

(xvii) The plane [hkl] belongs to two zones [u₁ v₁ w₁] and [u₂ v₂ w₂] if h = v₁ w₂ – v₂ w₁, k = v₁ w₂ – v₂ w₁ and l = v₁ w₂ – v₂ w₁.

(xviii) The plane (h₃ k₃ l₃) will be among those belonging to the same zone as (h₁ k₁ l₁) and (h₂ k₂ l₂) if,

h₃ = h₁ ± h₂, k₃ = k₁ ± k₂, and l₃ = l₁ ± l₂.

(xix) The angle between the two directions [u₁ v₁ w₁] and [u₂ v₂ w₂] for orthorhombic system is-