Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of...

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Useful concept for crystallography & diffractionUseful concept for crystallography & diffraction

Lattice planesLattice planes

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

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 set of planes in latticeInfinite number of set of planes in lattice

dd

d - interplanarspacing

d - interplanarspacing

Keep track of sets of planes by giving them names - Miller indices

(hkl)

Keep track of sets of planes by giving them names - Miller indices

(hkl)

Lattice planesLattice planes

Miller indices (hkl)

Choose cell, cell origin, cell axes:

Miller indices (hkl)

Choose cell, cell origin, cell axes:

a

b

originorigin

Miller indices (hkl)

Choose cell, cell origin, cell axesDraw set of planes of interest:

Miller indices (hkl)

Choose cell, cell origin, cell axesDraw set of planes of interest:

a

b

originorigin

Miller indices (hkl)

Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest origin:

Miller indices (hkl)

Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest origin:

a

b

originorigin

Miller indices (hkl)

Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest originFind intercepts on cell axes:

1,1,∞

Miller indices (hkl)

Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest originFind intercepts on cell axes:

1,1,∞a

b

originorigin

1

1

Miller indices (hkl)

Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest originFind intercepts on cell axes

1,1,∞

Invert theseto get (hkl)

(110)

Miller indices (hkl)

Choose cell, cell origin, cell axesDraw set of planes of interestChoose plane nearest originFind intercepts on cell axes

1,1,∞

Invert theseto get (hkl)

(110)

a

b

originorigin

1

1

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Exercises Exercises

Lattice planesLattice planes

Two things characterize a set of lattice planes:

interplanar spacing (d)

orientation (defined by normal)

Two things characterize a set of lattice planes:

interplanar spacing (d)

orientation (defined by normal)

Strange indicesStrange indices

For hexagonal lattices - sometimes see 4-index

notation for planes (hkil)where i = - h - k

For hexagonal lattices - sometimes see 4-index

notation for planes (hkil)where i = - h - k

a1

a2

a3

(110)(1120)

ZonesZones

2 intersecting lattice planes form a zone2 intersecting lattice planes form a zone

zoneaxiszoneaxis

zoneaxiszoneaxis zone axis

[uvw] is ui + vj + wk

zone axis [uvw] is ui + vj + wk

i j k h1 k1 l1 h2 k2 l2

i j k h1 k1 l1 h2 k2 l2

plane (hkl) belongs to zone [uvw] if hu + kv + lw = 0 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.

if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then

(h1+h2 k1+k2 l1+l2 ) also in same zone.

ZonesZones

zone axis [uvw] is ui + vj + wk

zone axis [uvw] is ui + vj + wk

(011) in same zone? hu + kv + lw = 00·0 + 1·1 - 1·1 = 0

(011) in same zone? hu + kv + lw = 00·0 + 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.

if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then

(h1+h2 k1+k2 l1+l2 ) also in same zone.

Example: zone axis for

(111) & (100) - [011]

Example: zone axis for

(111) & (100) - [011] i j k

h1 k1 l1 h2 k2 l2

i j k h1 k1 l1 h2 k2 l2

i j k 1 1 1 1 0 0

i j k 1 1 1 1 0 0

(100)(100)

(111)(111) [011][011]

Reciprocal latticeReciprocal lattice

Real space latticeReal space lattice

a

a

Reciprocal latticeReciprocal lattice

Real space lattice - basis vectorsReal space lattice - basis vectors

(100)planes

n100

Reciprocal latticeReciprocal lattice

Real space lattice - choose set of planesReal space lattice - choose set of planes

(100)planes

n100

d100

1/d100

Reciprocal latticeReciprocal lattice

Real space lattice - interplanar spacing dReal space lattice - interplanar spacing d

(100)planes

n100

d100

(100)

Reciprocal latticeReciprocal lattice

Real space lattice ––> the (100) reciprocal lattice ptReal space lattice ––> the (100) reciprocal lattice pt

(100)planes

n010

d010

(100)

(010)

Reciprocal latticeReciprocal lattice

The (010) recip lattice ptThe (010) recip lattice pt

The (020) reciprocal lattice pointThe (020) reciprocal lattice point

(020)planes

n020

d020

(100)

(010) (020)

Reciprocal latticeReciprocal lattice

More reciprocal lattice pointsMore reciprocal lattice points

(100)

(010) (020)

Reciprocal latticeReciprocal lattice

The (110) reciprocal lattice pointThe (110) reciprocal lattice point

(100)planes

(100)

(010) (020)

n110d110

(110)

Reciprocal latticeReciprocal lattice

Still more reciprocal lattice pointsStill more reciprocal lattice points

(100)planes

(100)

(010) (020)

the reciprocal lattice(230)

Reciprocal latticeReciprocal lattice

Reciprocal lattice notationReciprocal lattice notation

Reciprocal latticeReciprocal lattice

Reciprocal lattice for hexagonal real space latticeReciprocal lattice for hexagonal real space lattice

Reciprocal latticeReciprocal lattice

Reciprocal latticeReciprocal lattice

Reciprocal lattice for hexagonal real space latticeReciprocal lattice for hexagonal real space lattice

Reciprocal latticeReciprocal lattice

Reciprocal lattice for hexagonal real space latticeReciprocal lattice for hexagonal real space lattice

Reciprocal latticeReciprocal lattice

Reciprocal lattice for hexagonal real space latticeReciprocal lattice for hexagonal real space lattice

Reciprocal latticeReciprocal lattice