Diffraction
1 slit
-λL/a
λL/a
0
λL/d
2 slits
-λL/a
λL/a
0
λL/d
6 slits
-λL/a
0
λL/a
Distance
R. Ubic
Diffraction in Nature
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Diffraction Geometry
incident waves
1
2
specimen
θ
θ
dsinθ
dsinθ
θθ
The path difference
between diffracted waves
1 and 2 is 2dsinθ.
d
θθ
crystal planes
If 2dsinθ = nλ then the
waves are in-phase and
constructive interference
occurs.
∴Condition for diffraction:
transmitted
waves
diffracted
waves
2dsinθ
θ = λ
Bragg’s Law
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Diffraction in the TEM
Sample
(d = planar spacing)
2dsinθ = λ
sinθ ≈ θ
2θ
tan(2θ) = R/L
tan(2θ) ≈ 2θ
∴θ ≈ R/2L
L
∴Rd = Lλ
T
D
R
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Spacing “R” in diffraction pattern
is inversely proportional
to real crystallographic
interplanar spacing “d ”.
Reciprocal Space
b*
b
In reciprocal space,
everyone can hear you whisper.
010
a*
a
c
c*
a = 4Å
100
Where would 001 be?
c*
010
c
b
100
b*
a
Reciprocal lattice
vector is always
perpendicular to the
corresponding real space
plane.
a*
The real and reciprocal
lattice vectors are only
parallel in orthogonal
axis systems (cubic,
tetragonal, &
orthorhombic).
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Diffraction in 2D
Effect of Centering
Centered
rectangular
(square) unit cell
Primitive square
unit cell
hk :
h + k = 2n (bcc)
hk : no restrictions
h,k = 2n or
h,k = 2n+1 (fcc)
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Diffraction in 2D
Effect of Basis
p 4mm
Both patterns have same lattice
but different basis.
Positions of diffracted spots are
identical, but intensity distribution is
different.
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Diffraction in 2D
Effect of lattice size
Primitive square
Primitive square
Small lattice
Large lattice
Large spacing
between
diffraction spots
Small spacing
between
diffraction spots
Rd = Lλ
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Diffraction in 2D
pg
rectangular
oblique
vertical glide plane
hexagonal
{0k}: k = 2n
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Diffraction from Crystals
Nd2Hf2O7
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Reciprocal Basis Vectors
Orthogonal cells
a*=bXc
ghkl = 1/dhkl
a * = 1/a
V
Real Space
b*=cXa
b * = 1/b
V
c*=aXb
c * = 1/c
V
Reciprocal Space
Reciprocal basis vectors
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Reciprocal Lattice
Any point on the reciprocal lattice can be specified by
a vector ghkl = h a* + k b* + l c* (hkl are integers)
[001]
[111]
(002)
(022)
(202)
(222)
(111)
000
[010]
(020)
(200)
[100]
[110]
a
fcc
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(220)
1/a
bcc