crystallography l v. useful concept for crystallography & diffraction lattice planes think of...

$: Crystallography l v. Useful concept for crystallography & diffraction Lattice planes Think of sets of planes in lattice - each plane in set parallel to$
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hy lv

crystallograp

hy lv

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)


Miller indices (hkl)

Choose cell, cell origin, cell axes:


Choose cell, cell origin, cell axes:

a

b

originorigin


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


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

a

b

originorigin


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


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

a

b

originorigin


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

1,1,∞


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


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)


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


Exercises Exercises


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.



ZonesZones



(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





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


Real space lattice - basis vectorsReal space lattice - basis vectors

(100)planes

n100


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

(100)planes

n100

d100

1/d100


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

(100)planes

n100

d100

(100)


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

(100)planes

n010

d010

(100)

(010)


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)


More reciprocal lattice pointsMore reciprocal lattice points

(100)

(010) (020)


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

(100)planes

(100)

(010) (020)

n110d110

(110)


Still more reciprocal lattice pointsStill more reciprocal lattice points

(100)planes

(100)

(010) (020)

the reciprocal lattice(230)


Reciprocal lattice notationReciprocal lattice notation


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


crystallography l v. useful concept for crystallography & diffraction lattice planes think of...

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