Part 5. Reciprocal lattice

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Part 5. Reciprocal lattice (From Chapter 6 of
Textbook 1, part of chapter 2 of ref. 1)
?Introduction ? The reciprocal lattice
vectors define a vector space that enables
many useful geometric calculations to be
performed in crystallography. Particularly useful
in finding the relations for the
interplanar angles, spacings, and cell
volumes for the non-cubic systems. ? Physical
meaning is the k-space to the real crystal
(like frequency and time), is the real to
Fourier variables. ? Lets start first with
the less elegant approach. One has to have
basic knowledge of vectors and their rules.
?Reciprocal lattice vectors ? Consider a
family of planes in a crystal, the planes can
be specified by two quantities (1) orientation
in the
2
crystal and (2) their d-spacings. The
direction of the plane is defined by their
normals. ? reciprocal lattice vector with
direction plane normal and magnitude ?
1/(d-spacing). See Fig. 6.1.
Plane set 2
d2
d1
Plane set 1
k proportional constant, taken to be a value
with physical meaning, such as in
diffraction, wavelength ? is usually
assigned. 2dsin? ? ? ?/d 2sin?.
Longer vector ? smaller spacing ? larger ?.
Does it really form a lattice? Draw it to
convince yourself!
Plane set 3
d3
3
?Reciprocal lattice unit cells ? Use a
monoclinic crystal as an example. Exam the
reciprocal lattice vectors in a section
perpendicular to the y-axis, i.e.
reciprocal lattice (a and c) on the plane
containing a and c vectors.
(-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)
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? The process has constructed the reciprocal
lattice points (do form a lattice), which
also shows the reciprocal lattice unit cell
for this section outlined by a and c. One
can extend this section to other sections , see
Fig. 6.5, to form a 3D reciprocal lattice with
in reciprocal space
in real space
?Reciprocal lattice cells for cubic crystals
? The reciprocal lattice unit cell of a simple
cubic is a simple cubic. What is the
reciprocal lattice of a non- primitive unit
cell? For example, BCC and FCC? See Fig.
6.7 as an example. Look at the reciprocal
lattice of a BCC crystal on the x-y plane.
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(010)
O
y
In BCC crystal, the first plane encountered in
the x-axis is (200) instead of (100). The same
for y-axis.
(200)
(100)
(110)
1/2
x
(020)
O
? Get a reciprocal lattice with a centered
atom on the surface. The same for each surface.
Exam the center point. In BCC, the first plane
encountered in the (111) direction is (222). ?
FCC unit cell
6
022
222
002
? Reciprocal lattice of BCC crystal is
a FCC cell.
No 111
011
101
220
110
000
200
In FCC the first plane encountered in the x-axis
is (200) instead of (100). The same for y-axis.
But, the first plane encountered in the diagonal
direction is (220) instead of (110). Centered
point disappear
1/2
y
(220)
(110)
x
In FCC, the first plane encountered in the
111 direction is (111).
(111)
7
022
222
002
202
? Reciprocal lattice of FCC crystal is
a BCC cell.
111
020
220
000
200
? Another way to look at the reciprocal
relation is the inverse axial angles
(rhombohedral axes).
FCC SC BCC Real
60o 90o
109.47o. Reciprocal 109.47o
90o 60o
? In real space, one can defined the
environments around lattice points In terms
of Voronoi polyhedra (or Wigner -Seitz
cells (Section 3.4 of textbook 1, see Fig. 3.6).
The same definition for the environments
around reciprocal lattice points ?
Brillouin zones. (useful in SSP)
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a) Select a lattice point and draw
construction lines to the nearest
neighboring points.
b) Draw lines that perpendicularly bisect
the construction lines
c) The smallest enclosed area represents the
Wigner-Seitz cell. Here shown in
orange.
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Real Space
Reciprocal Space
BCC WS cell
BCC BZ
FCC WS cell
FCC BZ
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?Proofs of some geometrical relationships using
reciprocal lattice vectors ? Relationships
between a, b, c and a, b, c See Fig. 6.9.
Plane of a monoclinic unit cell ? to y-axis.
? angle between c and c.
c
?
c
d001
a
Similarly,
Consider the scalar product c?c
cccos?, since c 1/d001 by definition and
ccos? d001 ? c?c 1
Similarly, a?a 1 and b?b 1.
Since c //a?b, one can define a proportional
constant k, so that c k (a?b). Now, c?c 1 ?
c?k(a?b) 1 ? k 1/c?(a?b) 1/V. V volume of
the unit cell
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Similarly, one gets
? The addition rule the addition of
reciprocal lattice vectors
? The Weiss zone law or zone equation
A plane (hkl) lies in a zone uvw ? the plane
contains the direction uvw. Since the
reciprocal vectors dhkl ? the plane ?
dhkl ?ruvw 0 ?
uvw lies on the plane through the origin
When a lattice point uvw lies on the n-th plane
from the origin, what is the relation? See, Fig.
6.10. See the text or the following simple
explanation.
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ruvw
Define the unit vector in the dhkl direction i,
dhkl
uvw
r1
? d-spacing of lattice planes (hkl)
? The rest angle between plane normals, zone
axis at intersection of planes, and a plane
containing two directions. See text or part
four.
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?Reciprocal lattice in Physics ? In order to
describe physical processes in crystals more
easily, in particular wave phenomena, the
crystal lattice constructed with unit
vectors in real space is associated with
some periodic structure called the reciprocal
lattice. Note that the reciprocal lattice
vectors have dimensions of inverse length.
The space where the reciprocal lattice
exists is called reciprocal space. The
question arises what are the points that make a
reciprocal space? Or in other words what
vector connects two arbitrary points of
reciprocal space? ? Consider a wave process
associated with the propagation of some
field (e. g., electromagnetic) to be observed in
the crystal. Any spatial distribution of the
field is, generally, represented by the
superposition of plane
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waves such as
The concept of a reciprocal lattice is used
because all physical properties of an ideal
crystal are described by functions whose
periodicity is the same as that of this lattice.
If f(r) is such a function (the charge density,
the electric potential, etc.), then obviously,
We expand the function f(r) as a three
dimensional Fourier series
This series of k (some uses G) defined the
reciprocal lattice which corresponds to the
real space lattice. R is the translational
symmetry of the crystal.
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Thus, any function describing a physical
property of an ideal crystal can be expanded
as a Fourier series where the vector G runs
over all points of the reciprocal lattice
What is the meaning of this equation?
is the phase of a wave exp(ik?R)1 ?
k?R2?n, some defined the reciprocal lattice as
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Consider an incident wave from the source along
direction ki. The incident wave is
proportional to expiki?(r-rs), where r is
any point in the sample and rs is the location
of the source relative to the origin of
coordinates. The wave scattered into the detector
is then proportional to n(r)expikf?(rd-r) ,
where kf is the propagation direction of the
scattered wave, n(r) is the nuclei density,
and rd is the position of the detector.
Because each of the atoms may scatter wave, the
total amplitude at the detector is
dV
r
rd
rs
detector
source
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a crystal can be expanded in Fourier series G
? 0 for G ki - kf 0 otherwise, 0
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?Another approach to Reciprocal lattice Ref.
C. Kittel, Introduction to Solid State Physics,
John Wiley and Sons, Inc., Singapore, 1986.
? Assume a1, a2, and a3 are primitive vectors of
the crystal lattice and b1, b2, and b3 are
the axis vectors of the reciprocal lattice
The factor 2? are not used by
crystallographers but are convenient in
solid state physics.
Points in the reciprocal lattice are mapped
by the set of vectors
where v1, v2, v3 are integers
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A crystal property n(r), say electron
density, under any crystal translation T
u1a1u2a2u3a3 is invariant under the
Fourier expansion in the vectors G.
1
integer
? Diffraction conditions Theorem The
set of reciprocal lattice vectors G
determines the possible X-ray reflections. The
scattering amplitude of the following
schematics is
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Phase angle 2?rsin?/? k?r
Similarly,
Phase angle -k?r
dV
r
A
k
?
B
O
k
?
Fourier expansion n(r)
? 0 for G ?k otherwise, 0
The diffraction condition is G ?k. ? k
G k
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Since G is reciprocal lattice, -G is also.
Condition for diffraction Bragg condition
k
G
(hkl) plane
dhkl
?
If the factor 2? is not used in the
definition of the reciprocal lattice
vectors. Then, dhkl 1/G
The integer hkl that defined G are not
necessarily identical with the indices of
an actual crystal plane, a common n is used
to define hkl.
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Bragg law
Laue Equations started from ?k G and
express it in another way to get Laue
equations. Take the scalar product of both
?k and G successively with a1, a2, a3.
Geometrical interpretation of the first
equation is that ?k lies on a certain cone
about the direction of a1. At a reflection
?k must satisfy all three equations it
must lie at a common line of intersection of
three cones.
Ewald construction Ewald sphere a
sphere of radius k 2?/?. Diffraction
occurs at the condition of k G k.
k
?
G
k
2?
Reciprocal lattice
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Reciprocal lattice to simple cubic
lattice Primitive translation vectors
of a simple cubic lattice
The volume of the cell is a3. The reciprocal
lattice vectors are

Reciprocal lattice to BCC lattice
Primitive translation vectors of a BCC lattice
The volume of the cell is a3/2. The reciprocal
lattice vectors are

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Reciprocal lattice to FCC lattice
Primitive translation vectors of a FCC lattice
The volume of the cell is a3/4. The reciprocal
lattice vectors are

Reciprocal lattice of FCC lattice is BCC
lattice and vice versa.
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?More geometric relation between real lattice
and reciprocal lattice ? If e1, e2, e3 are
contravariant basis vector of R3 (not
necessarily orthogonal nor of unit norm ) then
the covariant basis vectors of their
reciprocal system are
Note that even ei and ei are not normal, they are
still by this definition mutually orthonormal
Then, the contravariant coordinates of any vector
v can be obtained by the dot product of v with
the contravariant basis vectors
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Likewise, the covariant coordinates of any vector
v can be obtained from the dot product of v with
the covariant basis vectors
Then v can be expressed in two (reciprocal) ways
Einsteins summation convention, omitting ?
No proof here. But you can check whether they
are correct? (use the BCC or FCC lattice as
examples)
where
Similarly,
One can prove that
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One can easily compute gij, metric tensor. Let a,
b, c and ?, ?, ? be the direct lattice parameters.
detgij V2.
Inverting the matrix, one get gij.
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One can easily compare gij with the metric
tensor calculated in terms of a, b, c and ?,
?, ?, i.e. in reciprocal lattice. Should be
identical
One gets relation like
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Similarly,
Finally, one can get the d-spacing of (hkl) plane
in any crystal
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Use the BCC lattice as examples
Assume
You can check the other way around.