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Miller indices of planes and directions

Short answer

For planes, take reciprocals of intercepts in lattice constants and clear fractions: (a) (321), (b) (210). For directions, subtract tail from head coordinates and reduce to smallest integers: [120] and [̅1̅11].

Explain the procedure for finding Miller indices of a crystal plane. Find the Miller indices of (a) a plane making intercepts of 2a, 3b and 6c on the three axes of a cubic cell, and (b) a plane making intercepts a/2, b and parallel to the c-axis. Also find the Miller indices of the direction joining the origin to the point (1/2, 1, 0), and of the direction from point (1, 0, 0) to point (0, 1, 1).

Miller indices (hkl)(hkl) are a set of three smallest integers that describe the orientation of a crystal plane.

Procedure for planes

  1. Find the intercepts of the plane on the x,y,zx, y, z axes in terms of lattice constants a,b,ca, b, c (use ∞\infty if parallel).
  2. Take the reciprocals of the intercepts.
  3. Clear fractions by multiplying by the smallest common factor.
  4. Write the result in round brackets without commas, (hkl)(hkl); a negative index has a bar over it.

(a) Intercepts 2, 3, 6

Stepxyz
Intercepts236
Reciprocals1/21/31/6
Multiply by 6321

Plane is (321)(321).

(b) Intercepts 1/2, 1, ∞\infty

Stepxyz
Intercepts1/21∞\infty
Reciprocals210

Plane is (210)(210).

Directions

Procedure: subtract the coordinates of the tail from the head, clear fractions to the smallest integers, and write in square brackets [uvw][uvw].

  • Origin to (12,1,0)(\tfrac12, 1, 0): components (12,1,0)(\tfrac12, 1, 0); multiply by 2, giving [120][120].
  • (1,0,0)(1,0,0) to (0,1,1)(0,1,1): components (0−1, 1−0, 1−0)=(−1,1,1)(0-1,\ 1-0,\ 1-0) = (-1, 1, 1), written [1ˉ11][\bar{1}11].

Answer: (a) (321); (b) (210); directions [120] and [1ˉ11][\bar{1}11].

✓ Answer checked by Claude Opus 5.5 (high effort)

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