Miller Indices of a Plane: Feature and Spacing | Engineering

In this article we will discuss about:- 1. Concept of Miller Indices 2. Important Features of Miller Indices 3. Spacing of Planes 4. Relation between Interplanar Spacing ‘d’ and Cube Edge ‘a’. 

Concept of Miller Indices:

Miller indices is a system of notation of planes within a crystal of space lattice. They are based on the intercepts of plane with the three crystal axes, i.e., edges of the unit cell. The intercepts are measured in terms of the edge lengths or dimensions of the unit cell which are unit distances from the origin along three axes.

Procedure for finding miller indices:

The Miller indices of a crystal plane are determined as follows: (Refer to Fig. 25)

Step 1:

Find the intercepts of the plane along the axes x, y, z (The intercepts are measured as multiples of the fundamental vector). …4, 2, 3.

Step 2:

Take reciprocals of the intercepts. 1/4, 1/2, 1/3

Step 3:

Convert into smallest integers in the same ration. …3 6 4

Step 4:

Enclose in parentheses. … (3 6 4)

The factor that results in converting the reciprocals of integers may be indicated outside the brackets, but it is usually omitted.

Important Note:

The directions in space are represented by square brackets [ ]. The commas inside the square brackets are used separately and not combined. Thus [1 1 0] is read as “One-one-zero” and not “one hundred ten”. Negative indices are represented by putting a bar over digit, e.g., [1 1 0].

The general way of representing the indices of a direction of a line is [x y z]. The indices of a plane are represented by a small bracket, (h, k I). Sometimes the notations < > and ( ) or { } are also used for representing planes and directions _x respectively.

The following procedure is adopted for sketching any direction:

1. First of all sketch the plane with the given Miller indices.

2. Now through the origin, draw a line normal to the sketched plane, which will give the required direction.

Important Features of Miller Indices:

Some of the important features of Miller indices (particularly for the cubic system) are detailed below:

1. A plane which is parallel to any one of the co-ordinate axes has an intercept of infinity (∞) and therefore, the Miller index for that axis is zero.

2. All equally spaced parallel planes with a particular orientation have same index number (h k I).

3. Miller indices do not only define particular plane but a set of parallel planes.

4. It is the ratio of indices which is only of importance. The planes (211) and (422) are the same.

5. A plane passing through the origin is defined in terms of a parallel plane having non­zero intercepts.

6. All the parallel equidistant planes have the same Miller indices. Thus the Miller indices define a set of parallel planes.

7. A plane parallel to one of the coordinate axes has an intercept of infinity.

8. 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.

9. 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.

10. 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, 12).

11. 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.

12. 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.

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

14. The distance d between adjacent planes of a set of parallel planes of the indices (h k I) is given by-

Where a is the edge of the cube.

Normally the planes with low index numbers have wide interplanar spacing compared with those having high index numbers. Moreover, low index planes have a higher density of atoms per unit area than the high index plane. In fact, it is the low index planes which play an important role in determining the physical and chemical properties of solids.

15. The angle between the normals to the two planes (h₁ k₁ l₁) and (h₂ k₂ l₂) is-

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

17. Miller indices are proportional to the direction consines of the normal to all corresponding plane.

18. 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.

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

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

21. Two planes (h₁ k₁ l₁) and (h₂ k₂ Z₂) both contain line [uvw] if u = k₁ l₂ – 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.

22. 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 I = v₁ w₂ – v₂ w₁.

23. 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.

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

Given Miller Indices How to Draw the Plane:

For the given Miller indices, the plane can be drawn as follows:

Step 1:

Find the reciprocal of the given Miller indices. These reciprocals give the intercepts made by the plane on X, Y and Z axes respectively.

Step 2:

Draw the cube and select a proper origin and show X, Y and Z axes respectively.

Step 3:

With respect to origin mark these intercepts and join through straight lines. The plane obtained is the required plane.

Following points are worth noting:

(i) Take lattice constant as one unit.

(ii) If the intercept for an axis is infinity then proceed parallel to that axis till you reach the next lattice point.

(iii) Try to get two points and join them first.

Fig. 26 (a) and (b) shows important planes of cube. Thick lines with arrows indicate the directions.

Spacing of Planes:

In order to identify different types of crystals it is essential to have knowledge of spacing of planes. It is so because for each crystal there exists a definite ratio between the spacing of planes which are rich in atoms. Refer to Fig. 27. (a).

Bragg by carrying out experiments on different crystals with X-rays not only verified the above ratio but also employed them to determine whether the crystal was simple cubic or B.C.C. type.

Relation between Interplanar Spacing ‘d’ and Cube Edge ‘a’:

Let us assume that the plane shown in Fig. 28 belongs to a family of planes whose Miller indices are <h k l>. The perpendicular ON from origin to the plane represents the interplanar spacing d of this family of planes.

Let the direction cosines of ON be cos α’, cos β’ and cos γ’.

The intercepts of the plane on the three axes are: