Planes zones Lattice planes Useful concept for crystallography

Planes & zones

Lattice planes Useful concept for crystallography & diffraction Think of sets of planes in lattice - each plane in set parallel to all others in set. All planes in set equidistant from one another Infinite number of sets of planes in lattice d dinterplanar spacing

Lattice planes Keep track of sets of planes by giving them names - Miller indices (hkl)

Miller � indices (hkl) Choose cell, cell origin, cell axes: origin b a

Miller � indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest: origin b a

Miller � indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin: origin b a

Miller � indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin Find intercepts on cell axes: origin 1, 1, ∞ b 1 a 1

Miller � indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin Find intercepts on cell axes origin 1, 1, ∞ Invert these to get (hkl) (110) b 1 a 1

Lattice planes Exercises

Lattice planes Exercises

Lattice planes Exercises

Lattice planes Exercises

Lattice planes Exercises

Lattice planes Exercises

Lattice planes Exercises

Lattice planes Two things characterize a set of lattice planes: interplanar spacing (d) orientation (defined by normal)

Strange indices For hexagonal lattices - sometimes see 4 -index notation for planes (hkil) where i = - h - k a 3 a 2 a 1 (1120) (110)

Zones 2 intersecting lattice planes form a zone axis [uvw] is ui + vj + wk i j k h 1 k 1 l 1 h 2 k 2 l 2 plane (hkl) belongs to zone [uvw] if hu + kv + lw = 0 if (h 1 k 1 l 1) and (h 2 k 2 l 2 ) in same zone, then (h 1+h 2 k 1+k 2 l 1+l 2 ) also in same zone.

Zones zone axis [uvw] is ui + vj + wk Example: zone axis for (111) & (100) - [011] i j k 1 1 0 0 (100) (111) i j k h 1 k 1 l 1 h 2 k 2 l 2 [011] (011) in same zone? hu + kv + lw = 0 0· 0 + 1· 1 - 1· 1 = 0 if (h 1 k 1 l 1) and (h 2 k 2 l 2 ) in same zone, then (h 1+h 2 k 1+k 2 l 1+l 2 ) also in same zone.

The Wulff net

The Wulff net Can plot plane normals - called poles Can only measure angles along great circles

Give crystal system for each From H. K. D. Bhadeshia All planes in same zone lie on a great circle

Standard projections for cubic http: //www. jcrystal. com/steffenweber/