Crystallography lv.

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Presentation transcript:

crystallography lv

Useful concept for crystallography & diffraction 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 d - interplanar spacing

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

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: 1,1,∞ origin 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 1,1,∞ Invert these to get (hkl) (110) origin 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 Exercises

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)

For hexagonal lattices - sometimes see 4-index Strange indices For hexagonal lattices - sometimes see 4-index notation for planes (hkil) where i = - h - k a3 a1 a2 (110) (1120)

2 intersecting lattice planes form a zone Zones 2 intersecting lattice planes form a zone zone axis zone axis zone axis [uvw] is ui + vj + wk i j k h1 k1 l1 h2 k2 l2 plane (hkl) belongs to zone [uvw] if hu + kv + lw = 0 if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then (h1+h2 k1+k2 l1+l2 ) also in same zone.

zone axis [uvw] is ui + vj + wk Zones Example: zone axis for (111) & (100) - [011] zone axis [uvw] is ui + vj + wk i j k h1 k1 l1 h2 k2 l2 i j k 1 1 1 1 0 0 (100) (111) [011] (011) in same zone? hu + kv + lw = 0 0·0 + 1·1 - 1·1 = 0 if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then (h1+h2 k1+k2 l1+l2 ) also in same zone.

Reciprocal lattice Real space lattice

Real space lattice - basis vectors Reciprocal lattice Real space lattice - basis vectors a a

Real space lattice - choose set of planes Reciprocal lattice Real space lattice - choose set of planes (100) planes n100

Real space lattice - interplanar spacing d Reciprocal lattice Real space lattice - interplanar spacing d (100) planes d100 1/d100 n100

Real space lattice ––> the (100) reciprocal lattice pt planes d100 n100 (100)

Reciprocal lattice The (010) recip lattice pt n010 (010) planes d010 (100)

The (020) reciprocal lattice point planes d020 (010) (020) (100)

More reciprocal lattice points (010) (020) (100)

The (110) reciprocal lattice point planes n110 d110 (010) (020) (100) (110)

Still more reciprocal lattice points (010) (020) (100) the reciprocal lattice (230)

Reciprocal lattice Reciprocal lattice notation

Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

Reciprocal lattice