Questions: Miller Indices for Planes and Directions
2 questions to test your understanding
Score: 0 / 2
Question 1 Short Answer
A plane intercepts the crystallographic axes at a = 2, b = 3, c = ∞. What are its Miller indices?
Think about your answer, then reveal below.
Model answer: (3 2 0). Take reciprocals: 1/2, 1/3, 0. Clear fractions by multiplying by 6: 3, 2, 0. The Miller index is (320).
The reciprocal of ∞ is 0, handling the case where the plane is parallel to the c-axis. After taking reciprocals (1/2, 1/3, 0), multiply through by the LCM (6) to get integers: 3, 2, 0.
Question 2 Short Answer
Why do Miller indices use reciprocals of intercepts rather than the intercepts themselves?
Think about your answer, then reveal below.
Model answer: Using reciprocals converts parallel axes (intercept = ∞) into zeros, making the notation finite and tractable. It also aligns with the mathematical relationship between planes and their normal vectors in reciprocal lattice space, which is central to diffraction theory.
The practical motivation is the ∞ problem: a plane parallel to an axis would require ∞ as an index, which is useless. The reciprocal turns that into a clean 0. The deeper reason is that Miller indices directly correspond to the reciprocal lattice, making them naturally suited for Bragg diffraction calculations where plane spacings scale as 1/|hkl|.