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Chapter 2 Reciprocal Lattice
An important concept for analyzing periodic structures
• Theory of crystal diffraction of x-rays, neutrons, and electrons.
Where are the diffraction maximum? What is the intensity?
• Abstract study of functions with the periodicity of a Bravais latticeFourier transformation. Reciprocal lattice is closely related to Fourier transformation
• Consideration of how momentum conservation will be modified in periodic potential
Reasons for introducing reciprocal lattice
Basic knowledge: rkie
rr⋅ is a plane wave
( ) ( )rkirkerki rrrrrr
⋅+⋅=⋅sincos
Wave vector nk ˆ2
λ
π=
r
a periodic function
2
One dimensional reciprocal lattice
)()( xfaxf =+
)(xf
∑
∑
∑
>
−
>
−−
>
−+++=
−
++
+=
++=
0
22
0
0
2222
0
0
0
)22
()22
(
22
)2
sin()2
cos()(
n
a
nxi
nna
nxi
nn
n
a
nxi
a
nxi
n
a
nxi
a
nxi
n
n
nn
ei
sce
i
scf
i
ees
eecf
a
nxs
a
nxcfxf
ππ
ππππ
ππ
One dimensional periodic function
na na−0aRedefine coefficients
∑∑ ==n
iGx
n
n
a
nxi
n eaeaxf
π2
)(
We have
n runs through all integers
*
nn aa −= Constraint that ensures f(x) is a real function
a
nG
π2= is one dimensional reciprocal lattice
Expand f(x) into Fourier series
The set of discrete points
f(x) is expanded into a series of plane waves, with a set of wave vectors G’s.
Each plane wave preserve the periodicity of f(x).iGxe
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• Reciprocal lattice points of a Bravais lattice indicate the allowed terms in the Fourier series of a function
with the same periodicity as the Bravais lattice
•But reciprocal lattice does not indicate the magnitude of the each Fourier term an, which depends on the exact
form of f(x).
• Reciprocal lattice depends on the Bravais lattice, but does not depend on the particular form of f(x), as long as
f(x) has the periodicity of the Bravais lattice
∫∫−
−
==a
iGxa
a
nxi
n exfdxa
exfdxa
a00
2
)(1
)(1
π
• Reciprocal space is Fourier space
real space
reciprocal space
Expand to three dimensional case:
Three dimensional periodic function )(rfr
that satisfies )()( Rrfrfrrr
+=
Fourier expansion: ∑ ⋅=G
rGi
Gearf
r
rr
rr)( ∫
⋅−=cell
rGi
c
GerfdV
Va
rr
rr)(
1
Volume of a primitive cell
0a1+a1−a
2+a2−a
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General definition of reciprocal lattice
Rr
rGie
rr⋅
Gv
Bravais lattice
a plane wave
The set of all wave vectors that yield plane waves with
the periodicity of a given Bravais lattice is known as its
reciprocal lattice
rGie
rr⋅
Rr
Analytically, the definition is expressed as rGiRrGiee
rrrrr⋅+⋅ =)(
The set of satisfyingGv
1=⋅RGie
rr
, for all in a Bravais latticeRr
• A reciprocal lattice is defined with reference to a particular Bravais lattice
• A set of vectors satisfying is called a reciprocal lattice only if the set of is a Bravais lattice
•Reciprocal lattice is a Bravais lattice (proven in the following)
Gv
1=⋅RGie
rr
Rr
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Construction of a reciprocal lattice
and Proof that is a Bravais lattice
)(2
321
32
1aaa
aab rrr
rrr
×⋅
×= π )(
2321
13
2aaa
aab rrr
rrr
×⋅
×= π
)(2
321
21
3aaa
aab rrr
rrr
×⋅
×= π
1ar
2ar
3ar
Are primitive vectors of a Bravais lattice
The reciprocal lattice can be generated by the primitive vectors1
br
2br
3br
ijji ab πδ2=⋅rr
Apparently
332211bvbvbvGrrrr
++=Reciprocal lattice vector
An arbitrary vector in reciprocal space can be written as a linear combination of
332211bgbgbggrrrr
++=
332211 anananRrrrr
++=
ig
To qualify for a reciprocal lattice, 1=⋅Rgie
rr
For all Rr
has to be integers
vi is a integer
Apparently, it is also a Bravais lattice with as its primitive vectors
)( 321 aaaVrrr
×⋅=
volume of primitive cell
ibr
Gr
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The reciprocal of the reciprocal lattice
The reciprocal lattice is a Bravais lattice, one can construct its reciprocal lattice, which turns out to
be nothing but the original direct lattice.
Rr
Gr
1=⋅RGie
rr
Direct lattice
Reciprocal lattice
Look for the reciprocal of the reciprocal lattice: Kr
which satisfies 1=⋅GKie
rr
The set of vectors is a subset of the set of vectors Rr
Kr
KRrr
⊆
For any vector that does not belong to the direct lattice R:
332211axaxaxrrrrr
++= At least one xi is non-integer
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≠===⋅⋅⋅ iiiii xibaixbriGri eeee
πrrrrrr
For ibGrr
= we have
So r does not belong to K
So the set of vectors R is identical to the set of vectors K
for all Rr
Or, every satisfies for allRr GRi
err
⋅Gr
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Examples of reciprocal lattice
xaa ˆ1 =r
3
3214
1)( aaaaV =×⋅=
rrr
yaa ˆ2 =r
zaa ˆ3 =r
Simple cubic Bravais lattice
xa
b ˆ2
1
π=
r ya
b ˆ2
2
π=
r
za
b ˆ2
3
π=
r
Reciprocal lattice
Also a simple cubic lattice
Face centered cubic
)ˆˆ(2
1zy
aa +=r )ˆˆ(
22 xz
aa +=r
)ˆˆ(2
3 yxa
a +=r
Primitive vectors of Reciprocal lattice
)ˆˆˆ(2
1zyx
ab ++−=
πr)ˆˆˆ(
22 zyx
ab +−=
πr
)ˆˆˆ(2
3 zyxa
b −+=πr
They are the primitive vectors of a body centered cubic lattice
Cubic unit cell with an edge of a
π2
Size of the cubic cell: 3)
4(
a
π
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Body centered cubic
)ˆˆˆ(2
1 zyxa
a ++−=r
)ˆˆˆ(2
2 zyxa
a +−=r )ˆˆˆ(
23 zyx
aa −+=r
Primitive vectors
Primitive vectors for reciprocal lattice
)ˆˆ(2
1zy
ab +=
πr
)ˆˆ(2
2zx
ab +=
πr)ˆˆ(
23 yx
ab +=
πr
They are the primitive vectors of a fcc lattice
Fcc and bcc are reciprocal to each other
Brillouin Zones
The Wigner Seitz primitive cell of the reciprocal lattice is known as the first Brillouin zone.
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It is a regular 12 faced solid
The first Brillouin zone is enclosed by
perpendicular bisectors between the central lattice
point and its 12 nearest neighbors
First Brillouin zone for bcc structure
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First Brillouin zone of fcc structure
Primitive vectors of fcc structureBrillouin zones of fcc structure
The reciprocal lattice is bcc lattice
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Lattice planes and reciprocal lattice vectors
Lattice planes: any plane that contains at least three noncollinear Bravais latice points
Due to trnaslational symmetry, any lattice plane will contain infinitely many lattice points, which form a
two dimensional Bravais lattice
Family of lattice planes:
A set of parallel, equally spaced lattice planes which together contains all the points of the three
dimensional Bravais lattice.
Theorem:
For any family of lattice planes separated by a distance d, there are reciprocal lattice vectors
perpendicular to the planes, the shortest of which have a length of . Conversely, for any
reciprocal lattice vector , there is a family of lattice planes normal to and separated by a
distance d, where is the length of the shortest reciprocal lattice vector parallel to
d
π2
Gr
Gr
d
π2 Gr
The theorem is a direct consequence of
(1) The definition of reciprocal lattice
(2) the fact that a plane wave has the same value at all points lying in a family of planes that are
perpendicular to its wave vector and separated by an integral number of wavelength.
1=⋅RGie
rr
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Proof of the first part of the theorem:
Given a family of lattice planes with a separation of d , let be a unit vector normal to the planes. Let’s make up a
wave vector:n̂
nd
K ˆ2π
=r
We need to prove that is a reciprocal lattice vector, and it is the shortest one in that directionKr
To qualify for a reciprocal vector, has to be satisfied for all of the Bravais lattice1=⋅RKie
rr
Rr
We know this condition is satisfied for the origin of the Bravais lattice 0=rr
because 10 =⋅Ki
er
Then must be true for all the lattice points in the lattice plane that contains the origin1=⋅rKie
rr
0=rrr
r
is a periodic function with a wavelength in the direction of its wave vector
Because is perpendicular to that plane, and should be a constant in the planeKr
rKrr
⋅
rKie
rr⋅
dK
== rπ
λ2
The wavelength happens to be the separation of the planesd=λ
Therefore for all the lattice points in the family of planes, is satisfiedRr
1=⋅RKie
rr
So is a reciprocal vector, and we can write it asKr
nd
G ˆ2π
=r
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If there is another vector shorter than G′r
Gr
The wavelength of isG′r
dG
>′
=′ rπ
λ2
GGrr
<′
It is impossible for to have the same value on two planes with a distance shorter than the wavelength1=⋅′ rGie
rr
So can not be a reciprocal vectorG′r
Proof of the converse of the theorem
Given a reciprocal lattice vector, let be the shortest parallel reciprocal lattice vectorGr
Let’s construct a set of real space planes (not necessarily lattice planes) on which 1=⋅rGie
rr
Now we need to prove that this set of planes are lattice planes and contains all the Bravais lattice points
These planes must contain . They must be all perpendicular to and separated by a distance 0=rr
Gr
Gd
πλ
2==
For all Bravais lattice vector , must be true for any reciprocal vector1=⋅RGie
rr
Rr
So the set of planes must contain all the Bravais lattice points.
Lastly, we need to prove that each of these planes contain lattice points instead of every nth of them
Suppose only every nth of the planes contain lattice points. According to the first part of the theorem, the shortest
reciprocal vector perpendicular to the planes will be, , which contradict with assmuption that G is the
shortest reciprocal vector in that directionn
G
nd=
π2
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Consider a plane with miller indices h, k, l. is the shortest reciprocal vector normal
to that plane. Let’s prove that
Reciprocal lattice vectors and Miller indices of lattice planes
The miller indices of a lattice plane are the coordinates of the shortest reciprocal lattice vector normal to that plane,
with respect to a specified set of primitive reciprocal lattice vectors. Thus a plane with Miller indices h, k, l, is normal
to the reciprocal lattice vector 321blbkbhGrrrr
++=
1ar
2ar
3ar
11axr
22axr
33axr
Gr
321blbkbhGrrrr
′+′+′=
Proof
llkkhh =′=′=′
For any point on plane,rr
ArG =⋅rr
AaxG
AaxG
AaxG
=⋅
=⋅
=⋅
)(
)(
)(
33
22
11
rr
rr
rr
3
2
1
2
2
2
x
Al
x
Ak
x
Ah
π
π
π
=′
=′
=′
So
Then it follows that321
1:
1:
1::
xxxlkh =′′′
Since G is the shortest reciprocal vector perpendicular to the plane. There should be no common factors
between . These parameters satisfy the original definition of miller indices
Therefore llkkhh =′=′=′
h′ k′ l′