A crystallographic plane intercepts the a-axis at 1, the b-axis at 2, and is parallel to the c-axis (intercept at ∞). What are the Miller indices of this plane?
A(1 2 0) — the direct intercept values reduced to integers
B(2 1 0) — take reciprocals (1/1=1, 1/2, 1/∞=0) and scale to smallest integers: 2, 1, 0
C(1 2 ∞) — the parallel axis is written as ∞ in Miller index notation
D(0 0 1) — planes parallel to two axes are indexed only by the axis they intercept
The procedure for Miller indices is: (1) record intercepts in units of lattice parameters: 1, 2, ∞; (2) take reciprocals: 1/1 = 1, 1/2, 1/∞ = 0; (3) multiply through by the smallest common denominator to get integers: multiply by 2 gives 2, 1, 0; (4) write as (210). Option A is the most tempting wrong answer — it uses the raw intercepts rather than their reciprocals. Option C shows the misconception that ∞ is written directly. The reciprocal convention elegantly converts ∞ (parallel to axis) to 0, which is why reciprocals are taken.
Question 2 Multiple Choice
FCC metals like copper preferentially slip on {111} planes in <110> directions. What is the crystallographic reason for this preference?
A{111} planes are the most widely spaced in FCC, minimizing the energy needed to separate them
B{111} planes in FCC have the highest atomic packing density, meaning atoms can slide past each other with less resistance (lowest Peierls stress)
C{111} is chosen because it is the only plane family with exactly 4 members in a cubic crystal
DSlip occurs on {111} because these planes are perpendicular to the applied stress in most loading geometries
Slip occurs on the most densely packed planes because atoms on these planes are arranged most efficiently — the corrugation between planes is minimized, and the interplanar spacing is maximized, both of which reduce the resistance to dislocation glide (the Peierls-Nabarro stress). In FCC structures, the {111} planes contain the densest arrangement of atoms, and the <110> directions within these planes are the most densely packed directions. Together, these give the lowest resistance to dislocation motion. Option A (widest spacing) partially contributes but is not the primary reason; the highest packing density of atoms within the plane is the key factor for the slip direction.
Question 3 True / False
The Miller indices (hkl) of a crystallographic plane represent the direct intercepts that the plane makes on the a, b, and c crystallographic axes.
TTrue
FFalse
Answer: False
Miller indices are the reciprocals of the intercepts, not the intercepts themselves. To find (hkl): intercept the plane with the three axes (in units of lattice parameters), take the reciprocal of each intercept, and reduce to the smallest integers. The reciprocal convention has a practical motivation: a plane parallel to an axis never intersects it (intercept = ∞), and the reciprocal of ∞ is conveniently 0 — a finite integer. If intercepts were recorded directly, parallel planes would require ∞ as an index, making notation impossible. Students who skip the reciprocal step systematically assign wrong indices to all planes.
Question 4 True / False
In a cubic crystal system, the direction [hkl] is perpendicular to the plane (hkl) with the same integer indices.
TTrue
FFalse
Answer: True
In a cubic crystal, the lattice vectors are orthogonal and of equal length, so the crystallographic axes coincide with Cartesian axes. In this special geometry, the direction vector [hkl] (with components h, k, l along a, b, c) is perpendicular to the plane (hkl) by the rules of vector geometry — the normal to a plane with intercepts at 1/h, 1/k, 1/l is parallel to (h, k, l). This convenient property does NOT hold for non-cubic crystal systems (e.g., hexagonal or monoclinic), where the lattice parameters and angles differ. In those systems, the direction [hkl] and plane (hkl) with the same indices are generally not perpendicular.
Question 5 Short Answer
Why are reciprocals of intercepts used to define Miller indices for planes, rather than recording the intercept values directly?
Think about your answer, then reveal below.
Model answer: The reciprocal convention solves two problems simultaneously. First, a plane parallel to a crystallographic axis never intersects it — its intercept is infinity. Taking the reciprocal converts ∞ to 0, producing a finite, usable integer. Without this convention, parallel axes would require ∞ as an index, making the notation mathematically unusable. Second, the reciprocal representation has a natural crystallographic interpretation: planes with small Miller indices (like (100) or (111)) intercept the axes at large unit-cell-sized intervals, meaning they are widely spaced and prominent in diffraction patterns. High-index planes (like (531)) make closely spaced cuts and are associated with large reciprocal lattice vectors and small interplanar spacings. The reciprocal lattice — the mathematical structure underlying X-ray diffraction — is built directly from Miller indices, so the reciprocal convention makes crystallographic calculations natural.
Another way to see it: Miller indices are coordinates in reciprocal space, not in direct space. The convention is not arbitrary — it is the natural language of the reciprocal lattice that governs diffraction.