Reciprocal lattice

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Reciprocal lattice
How to construct reciprocal lattice Meaning of
reciprocal lattice Relation between reciprocal
lattice and diffraction Geometrical relation
between reciprocal lattice and original lattice
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? How to construct a reciprocal lattice from a
crystal (1) Pick a set of planes in a crystal
parallel
Plane set 2
d2
d1
Plane set 1
Does it really form a lattice? Draw it to
convince yourself!
Plane set 3
d3
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Example a monoclinic crystal
Reciprocal lattice (a and c) on the plane
containing a and c vectors.
(b is out of the plane)
c
b
a
(-100)
(100)
c
(001)
(102)
c
(001)
(002)
O
(002)
?
O
a
O
a
(00-2)
c
(00-2)
(002)
O
a
c
(001)
c
c
(101)
(002)
?
?
(002)
O
?
a
a
(10-1)
(00-1)
a
(00-2)
2D ?form a 3-D reciprocal lattice
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Lattice point in reciprocal space
Integer
Lattice points in real space
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Reciprocal lattice cells for cubic crystals
Simple cubic
z
y
x
(002)?
z
z
(010)
(100)
y
y
a
a
x
x
a
(110)
(001)
Simple cubic
y
y
x
x
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Base centered cubic (BCC)
z
y
x
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(111)
(111)
O
O
(111)
FCC
corner
Up and down
O
F and B
L and R
(222)
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You should do the same for a FCC and show it
forms a BCC lattice! (Homework!)
Vector dot and cross product
v
v.u vucos?
?
Projection of v onto u and times each other
(scaler)!
u
vcos?
v?u vusin? w
vsin?
?
vusin? is the area of the parallelogram. w ?
v and u
u
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Relationships between a, b, c and a, b,
c Monoclinic plane ? y-axis (b)
? c ? c.
c
?
c
Similarly,
d001
b
a
c?c cccos?, c 1/d001 ? ccos?
d001 ? c?c 1
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Similarly, a?a 1 and b?b 1.
c //a?b, Define c k (a?b), k a
constant. c?c 1 ? c?k(a?b) 1 ? k
1/c?(a?b)1/V.
V volume of the unit cell
Similarly, one gets
?
? The Weiss zone law or zone equation
A plane (hkl) lies in a zone uvw (the plane
contains the direction uvw). dhkl ?
(hkl) ? dhkl ?ruvw 0 ?
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nth plane
ruvw
Define the unit vector in the dhkl direction i,
r2
dhkl
uvw
r1
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? Reciprocal Lattice Fourier transform of the
spatial wavefunction of the original
lattice ? wave process (e. g.
electromagnetic) in the crystal
Crystal periodic
Physical properties function of a crystal
Crystal translation vector
Periodic function ? Exponential Fourier Series
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u, v, w integer
Translation vectors of the original crystal
lattice
for all T
h, k, l integer
If
Vectors of the reciprocal lattice
always integer
If k ? (reciprocal) lattice T ? original
lattice!
Vice versa!
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In crystallography
In SSP
k (in general) momentum space vector G
reciprocal lattice points
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Proof the reciprocal lattice of BCC is FCC
Use primitive translation vectors only
BCC
FCC
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BCC
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The vector set is the same as the FCC
primitive translation vector.
Unit of the reciprocal lattice is 1/length.
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Mathematics of Interference
Sum of two waves
assume A1 A2 A
?R
new amplitude AR
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Geometrical analysis of Interference
term ? rotation vector
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Complex Wave Representation of Interference
assume A1 A2 A
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Diffraction conditions and reciprocal lattices
Theorem The set of reciprocal lattice
vectors G determines the possible X-ray
reflections.
k
r
r
k
http//en.wikipedia.org/wiki/Plane_wave
k
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Complex exponential form
Complex number
What happen to the time dependent term?
X-ray wavelength 0.1 nm ? ? 3x1018 1/s
Detectors get the average intensity! Detectors
measured the intensity only!
A lot of time, examining is enough!
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Path difference
Similarly,
Phase angle
dV
Phase angle (2?/?)rsin? k?r
r
O
?
Fourier expansion n(r)
? 0 for G ?k otherwise, 0
The diffraction condition is G ?k. ? k
G k
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G reciprocal lattice, -G reciprocal lattice?
____
Bragg condition?
k?
G
(hkl) plane
?
?k
or
dhkl
k
Bragg law
G ?k.
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?More geometric relation between direct lattice
and reciprocal lattice e1, e2, e3
contravariant basis vector of R3 covariant
basis vectors e1, e2, e3 (reciprocal lattice)
ei and ei are not normal, but mutually
orthonormal
For any vector v
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v can be expressed in two (reciprocal) ways
Einsteins summation convention, omitting ?
No proof here, but you can check whether these
relation is correct or not?
Use BCC or FCC lattice as examples, next page!
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Use the BCC lattice as examples
Assume
You can check the other way around. (homework)
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where
Similarly,
Prove
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gij metric tensor in direct lattice a, b, c and
?, ?, ? direct lattice parameters (standard
definition of the Bravais lattice)
detgij V2.
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Matrix inversion
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Inverting the matrix? gij.
gij metric tensor in reciprocal lattice a, b,
c and ?, ?, ? reciprocal lattice parameters
These two are the same!
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One gets relation like
Similarly,
..
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d-spacing of (hkl) plane for any crystal system
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Interplanar spacing
Cubic
Tetragonal
Orthorombic
Hexagonal
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Get the metric tensor! Perform the inversion of
the matrix! Comparing the inversion of the
metric tensor in direct lattice with the metric
tensor in reciprocal lattice ? Geometrical
relation between reciprocal lattice and
direct lattice can be obtained!