Crystal Planes & Indices

1. Crystal Planes & Indices • Crystals often have polyhedral shapes bounded by flat faces; this is a consequence of the periodicity of its internal arrangement. • Take a 2-D distribution of lattice points; choose any two (e.g. A & B) and pass a line through those points; can pass parallel lines through every other lattice point (N.B. in 3-D need 3 points not on the same line). • Have generated a set of equivalent equidistant planes. • Because these planes pass equally through the lattice points, what is generated corresponds to the stacking of layers. • The faces of crystals arrive from those planes which most favor the growth of the crystal (i.e. molecules add more easily on some faces than on others).

2. Crystal Planes & Indices • The orientation of a set of parallel planes can be specified by means of intercepts through the axes of the coordinate system (i.e. the unit cell edges). • It is customary to specify the orientations by means of indices, which are proportional to the reciprocals of the intercepts. • Here the intercept, using fractional coordinates, along the a axis is at 1, and at ½ along the b axis. • So, the index would be: 1 2. b a

3. Crystal Planes & Indicies b a • The figure to the right shows a unit cell with a set of parallel planes. • plane I intercepts at ²/3 ¹/2 ∞ (i.e. the plane is parallel to the c axis); the reciprocals are ³/2 2 0. • plane II intercepts at ¹/3 ¹/4 ∞ ; the reciprocals are 3 4 0. • The orientation of the planes is of interest to us. Notice that we can multiply the first set of reciprocals by a common factor (x2) to obtain integers (3 4 0), which are identical to the second plane; that is, these two planes are parallel and part of the same set. • (3 4 0): These 3 numbers (h k l) are called the Miller indices. • Note that they reveal the number of planes that pass across each axis. • Law of Rational Indices: the indices of the faces of a crystal are usually small integers, seldom greater than 3 (Hauy, 1784). ¹/2 b II ²/3 a I

4. Two Methods for Determining Miller Indices • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Method 1: Single Plane. • choose plane of interest. • plane intercepts a at 22/3 • intercepts b at 4 • intercepts c at ∞ • invert: 3/8 ¼ 0 • multiply by common denominator (x8) • Miller Index: (3 2 0) b a origin 4 b 22/3 a

5. Two Methods for Determining Miller Indices • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Method 2: Parallel Planes. • draw unit cell and all parallel planes going through the cell. • a axis cut into 3 equal parts. • b axis cut into 2 equal parts. • c axis not cut at all. • Miller Index: (3 2 0) b a origin

6. Negative Miller Indices • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • I • Line I. • intercepts a at -11/3 • intercepts b at 2. • intercepts c at ∞ • invert: 3/41/2 0 • multiply by common denominator (x9) • Miller Index: (3 2 0) • Line II. • intercepts a at 1 • intercepts b at -11/2 • intercepts c at ∞ • invert: 1 2/3 0 • multiply by common denominator (x3) • Miller Index: (3 2 0) b a II origin - _ - _ _ _ compare: (h k l) ║ (h k l) _

7. Miller Indices • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Although there are an almost infinite number of planes that could be drawn, the only ones that are important in crystals are those whose indices are small whole numbers (empirical observation). • Law of Rational Indices. b a origin

8. Inter-planar Spacings • The inter-planar spacing (d) for a set of parallel planes will be important in crystallography. This value is related to the Miller indices and the unit cell dimensions. • For orthorhombic, tetragonal and cubic unit cells (the axes are all mutually perpendicular), the inter-planar spacing is given by: • as derived by geometry. • For other lattice types, use: h, k, l = Miller indices a, b, c = unit cell dimensions