**The crystal lattice may be regarded as made**

**up of an infinite set of parallel equidistant**

**planes passing through the lattice points**

**which are known as lattice planes.**

**In simple terms, the planes passing through**

**lattice points are called ‘lattice planes’.**

**For a given lattice, the lattice planes can be**

**chosen in a different number of ways.**

**The orientation of planes or faces in a crystal can be described in terms of their intercepts on the three axes. Miller introduced a system to designate a plane in a crystal.**

### Procedure for finding Miller Indices

Step 1:
Determine the intercepts of
the plane along the axes X,Y and Z in
terms of the lattice constants a,b and
c.

Step 2:
Determine the reciprocals
of these numbers.

Step
3:
Find
the least common denominator (lcd) and multiply each by this lcd.

Step 4:The result is written in paranthesis.This is called the `Miller
Indices’ of the plane in the form (h k l). This is called the `Miller Indices’ of the
plane in the form (h k l).

Plane
ABC has intercepts of 2 units along X-axis,3 units along Y-axis and 2 units along Z-axis

### DETERMINATION OF ‘MILLER INDICES’

Step
1:The
intercepts are 2,3 and 2
on the three axes.

Step
2:The
reciprocals are 1/2, 1/3 and 1/2.

Step
3:The
least common denominator is ‘6’.

Multiplying each reciprocal by lcd,

we get, 3,2 and 3.

we get, 3,2 and 3.

Step
4:Hence
Miller
indices for the plane ABC is (3 2 3)

### IMPORTANT FEATURES OF MILLER INDICES

For
the cubic crystal especially, the important features of Miller indices are, A plane which is parallel to any one of the
co-ordinate axes has an intercept of infinity (¥).
Therefore the Miller index for that axis is zero; i.e. for an intercept at
infinity, the corresponding index is zero.

**Example**
In
the above plane, the intercept along X axis is 1 unit. The plane is parallel to Y and Z axes. So, the
intercepts along Y and Z axes are ‘¥’.

Now the intercepts are 1, ¥ and ¥.

The reciprocals of the intercepts are = 1/1,
1/¥ and 1/¥.

Therefore the Miller indices for the above
plane is (1 0 0).

### IMPORTANT FEATURES OF MILLER INDICES

- A plane passing through the origin is defined in terms of aparallel plane having non zero intercepts.

- All equally spaced parallel planes have same ‘Miller indices’ i.e. The Miller indices do not only define a particular plane but also a set of parallel planes. Thus the planes whose intercepts are 1, 1,1; 2,2,2; -3,-3,-3 etc., are all represented by the same set of Miller indices.

- It is only the ratio of the indices which is important in this notation. The (6 2 2) planes are the same as (3 1 1) planes.

- If a plane cuts an axis on the negative side of the origin,corresponding index is negative. It is represented by a bar, like (1* 0 0). i.e. Miller indices (1* 0 0) indicates that the plane has an intercept in the –ve X –axis.

*NOTE: here due to technical reasons bar above a numerical value is shown as 1**

- If (h k l) is the Miller indices of a crystal plane then the intercepts made by the plane with the crystallographic axes are given as

- A certain crystal has lattice parameters of 4.24, 10 and 3.66 Å on X, Y, Z axes respectively. Determine the Miller indices of a plane having intercepts of 2.12, 10 and 1.83 Å on the X, Y and Z axes.

Lattice
parameters are = 4.24, 10 and 3.66 Å

The
intercepts of the given plane = 2.12, 10 and 1.83 Å

i.e.
The intercepts are, 0.5, 1 and 0.5.

Step
1: The Intercepts are 1/2, 1 and 1/2.

Step
2: The reciprocals are 2, 1 and 2.

Step
3: The least common denominator is 2.

Step
4: Multiplying the lcd by
each reciprocal we get, 4, 2 and 4.

Step
5: By writing them in parenthesis we get
(4 2 4)

Therefore the Miller indices of the
given plane is (4 2 4) or (2 1 2).

2. Calculate
the miller indices for the plane with intercepts 2a, - 3b and 4c the along the crystallographic axes.

The
intercepts are 2, - 3 and 4

Step
1: The intercepts are 2, -3 and 4 along
the 3 axes

Step
2: The reciprocals are

Step 3: The least common denominator is 12.

Multiplying
each reciprocal by lcd, we get
6 -4 and 3 _

Step 4: Hence the Miller indices
for the plane is (6,4,3)

- The angle ‘q’ between any two crystallographic directions [u1 v1 w1] and [u2 v2 w2] can be calculated easily. The angle ‘q’ is given by,

- The direction [h k l] is perpendicular to the plane (h k l)

where h, k and l are the miller indices and a is length of side of cube.

Consider
a cubic crystal
of side ‘a’, and a plane ABC. This plane belongs to a family of planes whose Miller indices are (h k l) because Miller
indices represent a set of planes.

the interplanar
spacing between two adjacent parallel planes of Miller indices
(h k l ) is given by, NM = OM
– ON i.e.Interplanar
spacing

### Example

a =
4.031 Å

(h k l) = (2 1 1)

Interplanar spacing

d
= 1.6456 Å

### Example:

Find the perpendicular distance between the
two planes indicated by the Miller
indices (1 2 1) and (2 1 2) in a unit cell of a cubic lattice with a lattice constant parameter ‘a’.

We know
the perpendicular distance between the origin and the plane is (1 2 1)

and the perpendicular distance between
the origin and the plane (2 1 2),

The perpendicular distance between the
planes (1 2 1) and (2 1 2) are,

d = d1 – d2 =

(or) d = 0.0749 a.