Reciprocal Lattice

Reciprocal space
Ernesto Estévez Rams
Instituto de Ciencia y Tecnología de Materiales (IMRE)-Facultad de Física
Universidad de la Habana
IUCr International School on Crystallography, Brazil, 2012.

2 / 26
Definition
While constructing the
stereographic projection
we saw that the angle
between planes could be
also measured as the
angles between the
corresponding normal.

3 / 26
Definition
We can add metric
information to the
normals by asociating
the norm of the vector
somehow to the
interplanar distance.

Definition
We can add metric
information to the
normals by asociating
the norm of the vector
somehow to the
interplanar distance.
The reciprocal vector
⃗ r
∗
hkl
is defined as a vector normal to plane
(hkl) with length 1/d hkl
3 / 26

Definition
⃗ r ∗ hkl || ˆ n hkl
| ⃗ r ∗ hkl | = 1
d hkl
4 / 26

5 / 26
Equation of a plane
(u, 0, 0) (0, v, 0) (0, 0, w)
⃗u = u⃗a
⃗v = v ⃗ b
⃗w = w⃗c

6 / 26
Equation of a plane
⃗n = (⃗u − ⃗v) × (⃗v − ⃗w)
⃗n = (⃗v × ⃗w) + ( ⃗w × ⃗u) + (⃗u × ⃗v)
⃗n = vw( ⃗ b × ⃗c) + uw(⃗c × ⃗a) + uv(⃗a × ⃗ b)

7 / 26
Equation of a plane
⃗n · (⃗r − ⃗u) = 0
⃗n · ⃗r = ⃗n · ⃗u
⃗n = vw( ⃗ b × ⃗c) + uw(⃗c × ⃗a) + uv(⃗a × ⃗ b)
⃗r = x⃗a + y ⃗ b + z⃗c
vwx( ⃗ b × ⃗c) · ⃗a + uwy(⃗c × ⃗a) ·⃗b + uvz(⃗a × ⃗ b) · ⃗c =
uvw( ⃗ b × ⃗c) · ⃗a

8 / 26
Equation of a plane
vwx( ⃗ b × ⃗c) · ⃗a + uwy(⃗c × ⃗a) ·⃗b + uvz(⃗a × ⃗ b) · ⃗c =
uvw( ⃗ b × ⃗c) · ⃗a

8 / 26
Equation of a plane
vwx( ⃗ b × ⃗c) · ⃗a + uwy(⃗c × ⃗a) ·⃗b + uvz(⃗a × ⃗ b) · ⃗c =
uvw( ⃗ b × ⃗c) · ⃗a
[ ] [ ] [
x ⃗b × ⃗c
u ⃗a · ( ⃗ · ⃗a + y ⃗c × ⃗a
b × ⃗c) v ⃗a · ( ⃗ ·⃗b + z ⃗a × ⃗ ]
b
b × ⃗c) w ⃗a · ( ⃗ · ⃗c = 1
b × ⃗c)

Equation of a plane
vwx( ⃗ b × ⃗c) · ⃗a + uwy(⃗c × ⃗a) ·⃗b + uvz(⃗a × ⃗ b) · ⃗c =
uvw( ⃗ b × ⃗c) · ⃗a
[ ] [ ] [
x ⃗b × ⃗c
u ⃗a · ( ⃗ · ⃗a + y ⃗c × ⃗a
b × ⃗c) v ⃗a · ( ⃗ ·⃗b + z ⃗a × ⃗ ]
b
b × ⃗c) w ⃗a · ( ⃗ · ⃗c = 1
b × ⃗c)
⃗a ∗ =
⃗ b × ⃗c
⃗a · ( ⃗ b × ⃗c)
⃗b ∗ ⃗c × ⃗a
=
⃗a · ( ⃗ b × ⃗c)
⃗c ∗ =
⃗a ×⃗ b
⃗a · ( ⃗ b × ⃗c)
8 / 26

9 / 26
Equation of a plane
⃗a ∗ =
⃗ b × ⃗c
⃗a · ( ⃗ b × ⃗c)
⃗b ∗ ⃗c × ⃗a
=
⃗a · ( ⃗ b × ⃗c)
⃗c ∗ =
⃗a ×⃗ b
⃗a · ( ⃗ b × ⃗c)
⃗a · ⃗a ∗ = 1
⃗a · ⃗b ∗ = 0
⃗a · ⃗c ∗ = 0
⃗ b · ⃗a ∗ = 0
⃗ b · ⃗ b ∗ = 1
⃗ b · ⃗ b ∗ = 0
⃗c · ⃗a ∗ = 0
⃗c · ⃗b ∗ = 0
⃗c · ⃗b ∗ = 1

10 / 26
Equation of a plane
[ ] [ ] [
x ⃗b × ⃗c
u ⃗a · ( ⃗ · ⃗a + y ⃗c × ⃗a
b × ⃗c) v ⃗a · ( ⃗ ·⃗b + z ⃗a × ⃗ ]
b
b × ⃗c) w ⃗a · ( ⃗ · ⃗c = 1
b × ⃗c)

10 / 26
Equation of a plane
[ ] [ ] [
x ⃗b × ⃗c
u ⃗a · ( ⃗ · ⃗a + y ⃗c × ⃗a
b × ⃗c) v ⃗a · ( ⃗ ·⃗b + z ⃗a × ⃗ ]
b
b × ⃗c) w ⃗a · ( ⃗ · ⃗c = 1
b × ⃗c)
(x⃗a) · (h ⃗a ∗ ) + (y ⃗ b) · (k ⃗ b ∗ ) + (z⃗c) · (l ⃗c ∗ ) = 1
Equation of the plane
hx + ky + lz = 1

Some history
Gibbs (1839-1903)
Gibbs introduced the reciprocal base { a ⃗∗ j
} from the
condition ⃗a i · ⃗a ∗ j =K δ ij
11 / 26

Reciprocal vector
(x⃗a) · (h ⃗a ∗ ) + (y ⃗ b) · (k ⃗ b ∗ ) + (z⃗c) · (l ⃗c ∗ ) = 1
can be written as
where
hx + ky + lz = 1
r xyz ⃗ · r ⃗∗ hkl
= 1
⃗ r ∗ hkl = h ⃗a ∗ + k ⃗ b ∗ + l ⃗c ∗
if
⃗ r ∗ hkl || ˆ n hkl
| ⃗ r ∗ hkl | = 1
d hkl
12 / 26

Reciprocal vector
r⃗
hkl ∗ · (⃗r − p r ⃗ hkl ∗ ) = 0
p| ⃗ r ∗ hkl |2 = ⃗r · ⃗ r
∗
hkl
= 1
Now by construction
p = 1
| ⃗ r ∗ hkl |2
p| ⃗ r ∗ hkl | = d hkl ⇒ | ⃗ r ∗ hkl | = 1
d hkl
13 / 26

Coordinates transformations
⃗r = (⃗a ⃗ b⃗c) · ⎝
from where
⎛
x
y
z
⎞
⎛
⎝
Basis transformation
x ′ ⎞
y ′ ⎠ = F ·
z ′
⎛
⎠ = ( a ⃗′ b ⃗′ c ⃗′ ) · ⎝
⎛
⎝
x
y
z
x ′
y ′
z ′ ⎞
⎞
⎠
⎛
⎠ = ( a ⃗′ b ⃗′ c ⃗′ ) · F · ⎝
x
y
z
⎞
⎠
( ⃗ a ′ ⃗ b ′ ⃗ c ′ ) = (⃗a ⃗ b⃗c) · F −1 14 / 26

Metric tensor transformations
⎛
⃗r · ⃗r = (x y z) · G · ⎝
x
y
z
⎞
from where calling M = (F T ) −1
Metric tensor transformation
⎛
⎠ = (x ′ y ′ z ′ ) · G ′ · ⎝
(x y z) · F T · G ′ · F · ⎝
x ′ ⎞
y ′ ⎠ =
z ′
⎛
x
y
z
⎞
⎠
G ′ = M · G · M T 15 / 26

16 / 26
Metric tensor transformations
Consider the transformation
⎛
⎝
⃗a ∗
⃗b ∗
⃗c ∗
⎞
⎛
⎠ = M · ⎝
⃗a
⃗ b
⃗c
⎞
⎠

16 / 26
Metric tensor transformations
Consider the transformation
⎛
⎝
⃗a ∗
⃗b ∗
⃗c ∗
⎞
⎛
⎠ = M · ⎝
⃗a
⃗ b
⃗c
⎞
⎠
from where
⎛
⎝
⃗a ∗
⃗b ∗
⃗c ∗
⎞
⎛
⎠ · (⃗a ⃗ b⃗c) = M · ⎝
⃗a
⃗ b
⃗c
⎞
⎠ · (⃗a ⃗ b⃗c)

17 / 26
Metric tensor transformations
⎛
⎜
⎝
⃗a ∗ · ⃗a ⃗a ∗ ·⃗b ⃗a ∗ · ⃗c
⃗b ∗ · ⃗a b∗ ⃗ ·⃗b b∗ ⃗ · ⃗c
⃗c ∗ · ⃗a ⃗c ∗ ·⃗b ⃗c ∗ · ⃗c
⎞
⎛
⎟ ⎜
⎠ = M · ⎝
⃗a · ⃗a ⃗a ·⃗b ⃗a · ⃗c
⃗ b · ⃗a ⃗ b ·⃗ b ⃗ b · ⃗c
⃗c · ⃗a ⃗c ·⃗b ⃗c · ⃗c
⎞
⎟
⎠

17 / 26
Metric tensor transformations
⎛
⎜
⎝
⃗a ∗ · ⃗a ⃗a ∗ ·⃗b ⃗a ∗ · ⃗c
⃗b ∗ · ⃗a b∗ ⃗ ·⃗b b∗ ⃗ · ⃗c
⃗c ∗ · ⃗a ⃗c ∗ ·⃗b ⃗c ∗ · ⃗c
⎞
⎛
⎟ ⎜
⎠ = M · ⎝
⃗a · ⃗a ⃗a ·⃗b ⃗a · ⃗c
⃗ b · ⃗a ⃗ b ·⃗ b ⃗ b · ⃗c
⃗c · ⃗a ⃗c ·⃗b ⃗c · ⃗c
⎞
⎟
⎠
⎛
⎝
1 0 0
0 1 0
0 0 1
⎞
⎛
⎠ ⎜
= M · ⎝
⃗a · ⃗a ⃗a ·⃗b ⃗a · ⃗c
⃗ b · ⃗a ⃗ b ·⃗ b ⃗ b · ⃗c
⃗c · ⃗a ⃗c ·⃗b ⃗c · ⃗c
⎞
⎟
⎠ = M · G
M = G −1
F = G

18 / 26
Metric tensor transformations
⎛
⎝
⃗a ∗
⃗b ∗
⃗c ∗
⎞
⎛
⎠ = G −1 · ⎝
⃗a
⃗ b
⃗c
⎞
⎠

18 / 26
Metric tensor transformations
⎛
⎝
⃗a ∗
⃗b ∗
⃗c ∗
⎞
⎛
⎠ = G −1 · ⎝
⃗a
⃗ b
⃗c
⎞
⎠
and
G = F T · G ∗ · F = G T · G ∗ · G
G −1 · G = G −1 · G · G ∗ · G
I = G ∗ · G

Metric tensor transformations
⎛
⎝
⃗a ∗
⃗b ∗
⃗c ∗
⎞
⎛
⎠ = G −1 · ⎝
⃗a
⃗ b
⃗c
⎞
⎠
and
G = F T · G ∗ · F = G T · G ∗ · G
G −1 · G = G −1 · G · G ∗ · G
I = G ∗ · G
Reciprocal tensor
G ∗ = G −1 18 / 26

19 / 26
Transformation in reciprocal space
⎛
⃗r ∗ = (h k l) · ⎝
⎛
(h k l) · G −1 · ⎝
⃗a
⃗ b
⃗c
⃗a ∗
⃗b ∗
⃗c ∗
⎞
⎛
⎠ = (h ′ k ′ l ′ ⎜
) · ⎝
⎞
⎛
⎠ = (h k l) · G −1 · F T ⎜
⎝
⃗a ∗′
⃗b ∗′
⃗c ∗′
⃗a ′
⃗b ′
⃗c ′
⎛
(h k l) · G −1 · F T · G ′ ⎜
⎝
⎞
⎟
⎠ =
⎞
⎟
⎠ =
⃗a ∗′
⃗b ∗′
⃗c ∗′
⎞
⎟
⎠

Transformation in reciprocal space
⎛
⃗r ∗ = (h k l) · ⎝
⃗a ∗
⃗b ∗
⃗c ∗
⎞
⎛
⎠ = (h k l) · G −1 · F T · G ′ ⎜
⎝
⃗a ∗′
⃗b ∗′
⃗c ∗′
⎞
⎟
⎠
Now
G ′ = M · G · M T = (F T ) −1 · G · F −1
and finally
Reciprocal coordinates
(h ′ k ′ l ′ ) = (h k l) · M T = (h k l) · F −1 20 / 26

Point symmetry transformation in
reciprocal space
Let R be a point symmetry operation acting over the
coordinates of the atoms, then according to the transformation
rules already derived
(h e k e l e ) = (h k l) · R −1
and (h e k e l e ) is a plane symmetry related to (h k l). Therefore:
Point symmetry in reciprocal space
R ∗ = R −1 21 / 26

Point symmetry transformation in
reciprocal space
22 / 26
Example:
Three fold axis 3 [0 0 1] :
(x y z) −→ (−y x − y z) −→ (y − x − x z)

Point symmetry transformation in
reciprocal space
22 / 26
Example:
Three fold axis 3 [0 0 1] :
(x y z) −→ (−y x − y z) −→ (y − x − x z)
⎛
R = ⎝
0 ¯1 0
1 ¯1 0
0 0 1
⎞
⎠

Point symmetry transformation in
reciprocal space
22 / 26
Example:
Three fold axis 3 [0 0 1] :
(x y z) −→ (−y x − y z) −→ (y − x − x z)
⎛
R = ⎝
0 ¯1 0
1 ¯1 0
0 0 1
⎞
⎠
⎛
R ∗ = ⎝
¯1 1 0
¯1 0 0
0 0 1
⎞
⎠

Point symmetry transformation in
reciprocal space
22 / 26
Example:
Three fold axis 3 [0 0 1] :
(x y z) −→ (−y x − y z) −→ (y − x − x z)
⎛
R = ⎝
0 ¯1 0
1 ¯1 0
0 0 1
⎞
⎠
⎛
R ∗ = ⎝
¯1 1 0
¯1 0 0
0 0 1
⎞
⎠
(h k l) −→ (−h − k h l) −→ (k − h − k l)

23 / 26
Symmetry group in reciprocal space
Symmetry group in reciprocal space
{R ∗ } = {R}

Fourier transform and reciprocal
vectors
24 / 26
Fourier transform
Γ [f(⃗r)] = ̂f( ⃗r ∗ ) = F ( ⃗r ∗ ) ≡ ∫ ∞
−∞ f(⃗r) exp (−2πi ⃗r ∗ · ⃗r)d⃗r
where
⃗r ∗ = x ∗ ⃗a ∗ + y ∗ ⃗ b ∗ + z ∗ ⃗c ∗
⃗r = x⃗a + y ⃗ b + z⃗c
and therefore
⃗r ∗ · ⃗r = x ∗ x + y ∗ y + z ∗ z

25 / 26
Fourier transform of the lattice
Definition (Dirac comb)
L = ∑ ∞
u,v,w=−∞ δ(⃗r − u⃗a − v⃗ b − w⃗c)
∫
Γ [L(⃗r)] =
∫ ∞
· · ·
=
∞∑
−∞ u,v,w=−∞
∞∑
u=−∞
=
e (−2πiu ⃗r ∗·⃗a)
δ(⃗r − u⃗a − v ⃗ b − w⃗c) exp (−2πi ⃗r ∗ · ⃗r)d⃗r
∞∑
u,v,w=−∞
∞∑
v=−∞
exp (−2πi ⃗r ∗ · (u⃗a + v ⃗ b + w⃗c))
e (−2πiv ⃗r ∗·⃗b)
∞∑
w=−∞
e (−2πiw ⃗r ∗·⃗c)

Fourier transform of the lattice
Dirac comb
1
|a|
∑ ∞
n=−∞ exp (2πinx/a) = ∑ ∞
m=−∞ δ(x − ma) 26 / 26

26 / 26
Fourier transform of the lattice
Dirac comb
1
|a|
∑ ∞
n=−∞ exp (2πinx/a) = ∑ ∞
m=−∞
δ(x − ma)
∞∑
Γ [L(⃗r)] =
e (−2πiu ⃗r ∗·⃗a)
∞∑
e (−2πiv ⃗r ∗·⃗b)
∞∑
u=−∞
v=−∞
w=−∞
e (−2πiw ⃗r ∗·⃗c)

26 / 26
Fourier transform of the lattice
Dirac comb
1
|a|
∑ ∞
n=−∞ exp (2πinx/a) = ∑ ∞
m=−∞
δ(x − ma)
∞∑
Γ [L(⃗r)] =
e (−2πiu ⃗r ∗·⃗a)
∞∑
e (−2πiv ⃗r ∗·⃗b)
∞∑
u=−∞
v=−∞
w=−∞
e (−2πiw ⃗r ∗·⃗c)
∞∑
L ∗ ( ⃗r ∗ ) = δ( ⃗r ∗ · ⃗a − h)δ( ⃗r ∗ ·⃗b − k)δ( ⃗r ∗ · ⃗c − l)
h,k,l=−∞
=
∞∑
h,k,l=−∞
δ( ⃗r ∗ − h ⃗a ∗ )δ( ⃗r ∗ − k ⃗ b ∗ )δ( ⃗r ∗ − l ⃗c ∗ )