Explanation:
Miller indices are rationalized reciprocal of fractional intercepts taken along the crystallographic directions.
Calculation:
Given:
Plane intercepts are \(x = \frac{2}{3},y = \frac{1}{3}and\;z = \frac{1}{2}\)
Plane ABCD

x 
y 
Z 
Intercept

\(x = \frac{2}{3}\) 
\(x = \frac{1}{3}\) 
\(x = \frac{1}{2}\) 
Reciprocal 
\(\frac{3}{{2}}\) 
\(\frac{3}{1 }\) 
\(\frac{2}{1 }\) 
Rationalization 
\(x = \frac{3}{2}\times2=3\) 
\(x = \frac{3}{1}\times2=6\) 
\(x = \frac{2}{1}\times2=4\) 
Indices 
3 
6 
4 
∴ Miller index of this plane will be 3,6,4
In the figure shown below, Miller indices [0 2 1] have the direction of
Concept:
Miller Indices:
Miller Indices are used to specify directions and planes.
These directions and planes could be in a lattice or in crystals.
If two points M (x1, y_{1}, z_{1}) and N (x2, y2, z_{2}) then the direction is given by [(x2  x1) (y2  y1) (z2  z1)]
Calculation:
Given:
A1 (1, 0, 0) and A2 (0, 0,1)
Direction of A [(0  1) (0  0) (1  0)] ⇒ [1 0 1]
\(∴ A \;[{\bar{1}}\;0\;1]\)
\({B_1}\left( {\frac{1}{2},1,0} \right)\;and\;{B_2}\;\left( {1,0,1} \right)\)
\({\rm{Direction\;of}}\;B\;\left[ {\left( {1  \frac{1}{2}} \right)\left( {0  1} \right)\left( {1  0} \right)} \right] \Rightarrow \left[ {\frac{1}{2}  1\;1} \right]\)
\(∴ B\left[ {\frac{1}{2}\;\bar 1\;1} \right]\Rightarrow B\;[{1}\;\bar 2\;2]\)
\({C_1}\left( {0,\frac{3}{4},1} \right)\;and\;{C_2}\left( {1,\;0,\;0} \right)\)
\({\rm{Direction\;of}}\;C\left[ {\left( {1  0} \right)\left( {0  \frac{3}{4}} \right)\left( {0  1} \right)} \right] \Rightarrow \left[ {1  \frac{3}{4}\;  1} \right]\)
\(∴ C\left[ {1\;\frac{{\bar 3}}{4}\;\bar 1} \right] \Rightarrow \left[ {4\;\bar 3\;\bar 1} \right]\)
\(\;{D_1}\;\left( {0,\;0,\;0} \right)\;and\;{D_2}\;\left( {0,\;1,\;\frac{1}{2}} \right)\)
\({\rm{Direction\;of}}\;D\left[ {\left( {0  0} \right)\left( {1  0} \right)\left( {\frac{1}{2}  0} \right)} \right] \Rightarrow \left[ {0\;1\;\frac{1}{2}} \right]\)
∴ D [0 2 1]
Concept:
\(h=\frac{1}{p},~k=\frac{1}{q},~l=\frac{1}{r}\)
Where p = intercept of the plane on the xaxis, q = intercept of the plane on the yaxis, and r = intercept of the plane on the zaxis.
Calculation:
Given:
Miller indices of a plane = (632)
Since, Miller indices are obtained by reciprocal of intercept p, q, and r made by the plane on the three rectangular axes x, y, and z respectively. Hence the reciprocal of miller indices will give the intercepts.
Distances from the origin to points at which the plane intersects = \(\frac{{1}}{{6}}, \frac{{1}}{{3}}~and~\frac{{1}}{{2}}\) units
CONCEPT:
\(h=\frac{1}{p},~k=\frac{1}{q},~l=\frac{1}{r}\)
Where p = intercept of the plane on the xaxis, q = intercept of the plane on the yaxis, and r = intercept of the plane on the zaxis.
EXPLANATION:
1) Likewise, the yellow plane can be designated as (∞,1,∞)
And the green plane can be written as (∞,∞,1)
Miller Indices are the reciprocals of the parameters of each crystal face. Thus:
Procedure for Miller indices
Step 1: locate the origin ‘O’, and axis x, y, z
Step 2: find the plane fractions (plane dimensions) = P, Q, R
Step 3: Calculate the reciprocals of plane fractions = \(\frac{1}{P},\frac{1}{Q},\frac{1}{R}\)
[h, k, l] is the miller indices.
For parallel planes
The distance will be the same for all directions from the origin. So, reciprocals of those also will be the same.
For parallel planes, miller indices will be the same
Example:
Here plane is intersecting x and y axes at a and a respectively and extending along the zdirection.
Plane fractions [P, Q, R] = [a, a, ∞]
Plane fractions [P, Q, R] = [1, 1, ∞]
Reciprocals of plane fractions:
\(\left[ {\frac{1}{P},\frac{1}{Q},\frac{1}{R} = \frac{1}{1},\frac{1}{1},\frac{1}{\infty }} \right]\)
[h, k, l] = [1, 1, 0]
The Miller indices for the plane in the figure shown below are:
Concept:
Crystallographic plane:
Calculation:
Given:
Now, let, 'a' be the side of the given cube and plane intersects in the middle of the x and yaxis.
So, the plane fractions are = \(\frac{a}{2}\), \(\frac{a}{2}\), ∞
Reciprocal of plane fraction = \(\frac{2}{a}\), \(\frac{2}{a}\), 0
To convert reciprocal into least integer, multiply by a
∴ Miller indices = [220]
Concept:
Miller indices are the styles to designate the planes and directions in the unit cells and crystals.
Miller indices (hkl) are expressed as a reciprocal of intercepts p, q, and r made by the plane on the three rectangular axes x, y and z respectively. These are the unit distances from the origin along the three axes. Thus
\(h=\frac{1}{p},~k=\frac{1}{q},~l=\frac{1}{r}\)
where, p = intercept of the plane on the xaxis, q = intercept of the plane on the yaxis, and r = intercept of the plane on the zaxis.
Reciprocal of these intercepts are then converted into whole numbers. This can be done by multiplying each reciprocal by a number obtained after taking LCM of the denominator.
This gives the Miller indices of the required plane. The Miller indices are expressed by three smallest integers.
A unit cell of a crystal is shown in the figure. The miller indices of the direction (arrow) shown in the figure is
Concept:
Procedure To find Miller Indices:
To find miller indices of direction, take the given direction vector as resultant and find its component along the x, y and zaxis.
If components are in fraction, then convert it to an integer.
Calculation:
Given:
Now, From fig. as shown above
Here components are, \(\left[ {\frac{1}{2}1\;0} \right]\)
Now,
Converting them to an integer, we get
Miller indices, [1 2 0]
For calculating miller indices of direction, reciprocal is not taken but for plane reciprocal is taken before converting into the smallest set of integers.
Concept:
Miller Indices:
Calculation:
Given,
Intercepts of plane = (a, 2b, \(  \frac{{3c}}{2}\))
Intercept = (1, 2, \(  \frac{{3}}{2}\))
Reciprocal of intercept = 1, \( \frac{{1}}{2}\), \(  \frac{{2}}{3}\))
LCM = 6
Miller indicates of the plane = LCM × Reciprocal of intercept
∴ Miller indicates of the plane = (6, 3,  4)
Concept:
Miller indices are the styles to designate the planes and directions in the unit cells and crystals.
Miller indices (hkl) are expressed as a reciprocal of intercepts p, q and r made by the plane on the three rectangular axes x, y and z respectively. These are the unit distances from the origin along the three axes. Thus
\(h=\frac{1}{p},~k=\frac{1}{q},~l=\frac{1}{r}\)
where, p = intercept of the plane on the xaxis, q = intercept of the plane on the yaxis, and r = intercept of the plane on the zaxis.
Reciprocal of these intercepts are then converted into whole numbers. This can be done by multiplying each reciprocal by a number obtained after taking LCM of the denominator.
This gives the Miller indices of the required plane. The Miller indices are expressed by three smallest integers.
Calculation
.
Plane AFGD is the diagonal plane.
Intercept on x, y and z axis (1, 1, ∞)
Reciprocal of intercept = 1, 1, 0
Indices = (110)
Explanation:
Miller indices are rationalized reciprocal of fractional intercepts taken along the crystallographic directions.
Calculation:
Given:
Plane intercepts are \(x = \frac{2}{3},y = \frac{1}{3}and\;z = \frac{1}{2}\)
Plane ABCD

x 
y 
Z 
Intercept

\(x = \frac{2}{3}\) 
\(x = \frac{1}{3}\) 
\(x = \frac{1}{2}\) 
Reciprocal 
\(\frac{3}{{2}}\) 
\(\frac{3}{1 }\) 
\(\frac{2}{1 }\) 
Rationalization 
\(x = \frac{3}{2}\times2=3\) 
\(x = \frac{3}{1}\times2=6\) 
\(x = \frac{2}{1}\times2=4\) 
Indices 
3 
6 
4 
∴ Miller index of this plane will be 3,6,4