The reciprocal
space
• Space of the wave vectors
• Fourier space
• Inverse
• Orthogonal
Reciprocal Space:
Geometrical definition
• Definition of basis vectors
• Introduced by Bravais
• Then used by Ewald (1917)
𝒃∧𝒄 ∗
𝒄∧𝒂 ∗
𝒂∧𝒃
𝒂 = 2𝜋
; 𝒃 = 2𝜋
; 𝒄 = 2𝜋
𝑣
𝑣
𝑣
∗
DL
RL
b
b*
a*
a
• with 𝑣 = (𝑎, 𝑏, 𝑐) volume of the cell
• Equivalent definition (2D, 3D...)
𝒂∗ ⋅ 𝒂 = 2𝜋
𝒃∗ ⋅ 𝒂 = 0
𝒄∗ ⋅ 𝒂 = 0
𝒂∗ ⋅ 𝒃 = 0
𝒃∗ ⋅ 𝒃 = 2𝜋
𝒄∗ ⋅ 𝒃 = 0
𝒂∗ ⋅ 𝒄 = 0
𝒃∗ ⋅ 𝒄 = 0
𝒄∗ ⋅ 𝒄 = 2𝜋
• 𝒂∗ is orthogonal to 𝒃 et 𝒄 but NOT in gal to 𝑎
𝒗∗ = (𝒂∗ , 𝒃∗ , 𝒄∗ ) = (2𝜋)3/𝑣
• Reciprocal space: vector space basis 𝒂∗ , 𝒃∗ , 𝒄∗
• Reciprocal lattice: set of points
𝑸ℎ𝑘𝑙 = ℎ𝒂∗ + 𝑘𝒃∗ + 𝑙𝒄∗
ℎ, 𝑘, 𝑙 integers
Definition by plane waves
• 𝑸 belongs to RS iff:
∀𝑹𝑢𝑣𝑤 𝑒 𝑖𝑸∙𝑹𝑢𝑣𝑤 = 1 ⟺ ∀𝑹𝑢𝑣𝑤 𝑸 ∙ 𝑹𝑢𝑣𝑤 = 2𝜋𝑚
• Si
• Si
on pose
entiers.
• Reciprocal space
𝑸ℎ𝑘𝑙 = ℎ𝒂∗ + 𝑘𝒃∗ + 𝑙𝒄∗
• Set of wave vectors 𝒒 of the plane waves 𝑒 𝑖𝒒∙𝒓 with the direct space periodicity
𝒒
𝒒
Properties of the RS
• Symmetry
The reciprocal space
the same point symmetry
as the direct lattice
Let 𝑶 be a symmetry operator of the RL and
𝑹𝑢𝑣𝑤 a point of the RL.
𝑶(𝑸ℎ𝑘𝑙 ) ∙ 𝑹𝑢𝑣𝑤 = 𝑶−1 (𝑶(𝑸ℎ𝑘𝑙 )) ∙ 𝑶−1 (𝑹𝑢𝑣𝑤 )
= 𝑸ℎ𝑘𝑙 ∙ 𝑹𝒖′ 𝒗′ 𝒘′ = 2𝜋𝑚
Thus 𝑶(𝑸ℎ𝑘𝑙 ) belongs to the RL
DL
b
a
b*
a*
• Duality
• The reciprocal lattice of the RL is the direct lattice:
• RL of the RL consists in points 𝑹 such that:
∀𝑸ℎ𝑘𝑙
𝑸ℎ𝑘𝑙 ∙ 𝑹 = 2𝜋𝑚
• If 𝑹 = 𝑹𝑢𝑣𝑤 the relation is verified
• Conversely if 𝑹 = 𝑥𝒂 + 𝑦𝒃 + 𝑧𝒄, it satisfies 𝑥𝑢 + 𝑦𝑣 + 𝑧𝑤 = 𝑚,
thus 𝑥, 𝑦 and 𝑧 are integers and 𝑹 belongs to the RL
RL
• The nodes of a lattice
are regrouped in equally spaced
planes:
The lattice planes
Lattice planes, rows
• Family of planes
[001]
[010]
[100]
<100>
• Row : series of nodes in the direction Ruvw
• Notation [uvw], u, v, w relatively prime
• Symmetry equivalent directions are noted: <uvw>
Lattice planes
(3,2,4)
(0,0,1)
c
b
1/4
dhkl
1/2
1/3
a
The lattice plane closest to the origin, intersects the cell axes in:
𝒂 𝒃 𝒄
, ,
ℎ 𝑘 𝑙
h, k, l Miller indices
• Family of lattice planes (ℎ, 𝑘, 𝑙)
• Famillies of planes equivalent by symmetry {ℎ, 𝑘, 𝑙}
• Distance between planes 𝑑ℎ𝑘𝑙
• If N(hkl) is the planar node density, N(hkl)/dhkl is the volumic node density
• The most dense planes are the more distant
• Crystals facets are planes with small indices
Lattice planes and RS
To each family of lattice planes of period 𝑑
corresponds
A reciprocal space row of period 2𝜋/𝑑
• This row is orthogonal to the lattice planes
• The smallest vetor of this row has a magnitude 2𝜋/𝑑
Q020
Q010=d*
2p/Q020
d010=2p/Q010
• The lattice plane closest to the origin satisfies: 𝒉𝒖 + 𝒌𝒗 + 𝒍𝒘 = 𝟏
• It intersects the axes in:
𝒂 𝒃 𝒄
, ,
ℎ 𝑘 𝑙
h, k, l Miller indices
(mutually prime)
𝒅 = 𝟐𝝅/𝑸
𝒏
𝑹𝑢𝑣𝑤
𝑑
𝑹𝑢𝑣𝑤 ∙ 𝒏 = 𝑚𝑑
𝑹𝑢𝑣𝑤 ∙ 2𝜋𝒏/𝑑 = 2𝜋𝑚
𝑸=
? 𝑸
2𝜋𝒏
𝑑
is a RL vector
𝑹𝑢𝑣𝑤 ∙ 𝑸 = 2𝜋
𝑸 cannot be shorter,
it is the row period
𝑸𝑸ℎ𝑘𝑙 ℎ, 𝑘, 𝑙 Miller indices: 𝑑ℎ𝑘𝑙
Distance between lattice planes dhkl
• dhkl distance between planes (hkl)
𝑑ℎ𝑘𝑙
2𝜋
=
𝑄ℎ𝑘𝑙
Qhkl smallest vextor of the row
• General case
𝑑ℎ𝑘𝑙 =
2𝜋
ℎ2 𝑎∗2 + 𝑘 2 𝑏 ∗2 + 𝑙2 𝑐 ∗2 + 2ℎ𝑘𝑎∗ 𝑏 ∗ cos 𝛾 ∗ + 2𝑘𝑙𝑏𝑐 ∗ cos 𝛼 ∗ + 2ℎ𝑙𝑎∗ 𝑐 ∗ cos 𝛽 ∗
• Hexagonal system:
4𝜋
𝑎∗ = 𝑏 ∗ = √3𝑎 , 𝑐 ∗ =
2𝜋
∗
,
𝛾
𝑐
= 60°
𝑑ℎ𝑘𝑙 =
• Cubic system :
𝑎∗ = 𝑏∗ = 𝑐 ∗ =
2𝜋
∗
,
𝛼
𝑎
𝑎
4 2
𝑎
(ℎ + 𝑘 2 + ℎ𝑘) +𝑙2 ( )2
3
𝑐
= 𝛽 ∗ = 𝛾 ∗ = 90°
𝑑ℎ𝑘𝑙 =
𝑎
ℎ2 + 𝑘 2 +𝑙2
Multiple unit cells
• Body centered cell
I
a
• The condition ∀𝑹𝑢𝑣𝑤 𝑸 ∙ 𝑹𝑢𝑣𝑤 = 2𝜋𝑛
implies
1) ℎ, 𝑘 , 𝑙 integers (Reciprocal space of lattice (𝑎, 𝑏, 𝑐))
2) ℎ + 𝑘 + 𝑙 = 2𝑛
•
Reflection conditions
F
a*
• Hexagonal lattice
• A = a-b; B=a+b; C=c
B
a
b
A
Conditions
P
P
I ℎ + 𝑘 + 𝑙 = 2𝑛 F
F ℎ, 𝑘, 𝑙 same parity I
ℎ + 𝑘 = 2𝑛
A
A
b*
a*
B*
A*
Fourier transform of the RS
• Definition
• Fonction or distribution 𝑆(𝒓)
𝑇𝐹 𝑆 𝒓
=𝐹 𝒒 =
𝑇𝐹 −1 𝐹 𝒒
=𝑆 𝒓 =
𝑆(𝒓)𝑒
1
(2𝜋)3
ℎ
−𝑖𝒒∙𝒓 3
𝑑 𝒓
1
𝛿 𝑞 − ℎ𝑇 =
𝑇
+∞
𝑒
𝑛=−∞
Série de Fourier du
Peigne de Dirac
𝐹 𝒒 𝑒 𝑖𝒒∙𝒓 𝑑 3 𝒒
• Direct lattice described by a ’’node density’’ function:
𝑆 𝒓 =
𝛿(𝒓 − 𝑹𝑢𝑣𝑤 )
𝑢𝑣𝑤
𝑇𝐹 𝑆 𝒓
𝛿(𝒓 − 𝑹𝑢𝑣𝑤 ) 𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
=𝐹 𝒒 =
𝑢𝑣𝑤
𝑒 −𝑖𝒒∙𝑹𝑢𝑣𝑤 =
=
𝑢𝑣𝑤
=
𝑢
𝛿 𝑞𝑥 − ℎ
𝑒 −2𝑖𝜋𝑞𝑦 𝑣
𝑒 −2𝑖𝜋𝑞𝑥 𝑢
𝑣
𝛿 𝑞𝑦 − 𝑘
ℎ
𝑒 −2𝑖𝜋𝑞𝑧 𝑤
𝒒 = 𝑞𝑥 𝒂∗ + 𝑞𝑦 𝒃∗ + 𝑞𝑧 𝒄∗
𝑤
𝛿 𝑞𝑧 − 𝑙
𝑙
𝐹 𝒒 = 𝑣∗
𝛿(𝒒 − 𝑸ℎ𝑘𝑙 )
ℎ𝑘𝑙
−2𝑖𝜋𝑛
‘‘node density’’ of RL
𝑣 ∗ = 𝒂∗ , 𝒃∗ , 𝒄∗ = 2𝜋/𝑣
• The Fourier transform of direct lattice is the reciprocal lattice
• The reciprocal space is the FT of the Direct space
𝑞
𝑇
Properties of the FT
• Duality of RS and DS
• RS and DS have the same point symmetry
• Let O be a symmetry operator of the DS
…then O is a symmetry operator of RS
• Convolution
• Convolution of f and g is f * g
Application to low dimenbsion objects
• 1D : chain
2p/a
a
a*
Set of parallel plane
b
b*
a
a*
• 2D : planes
Lattice of lines
Relation with diffraction
• Bragg’s law
• Diffraction on lattice planes, spacing 𝑑
2𝑑 sin 𝜃 = 𝑚𝜆
ki
q
d
q
kd
𝒒 = 𝒌𝑑 − 𝒌𝑖
Vecteur de diffusion
• q normal to the lattice planes
4𝜋
2𝜋
𝒒 = 2𝑘 sin 𝜃 =
sin 𝜃 =
𝑚
𝜆
𝑑
Diffraction

𝒒 belongs to RS
(to the direction  planes)
𝒒=
2𝜋
𝒏
𝑑
𝒒=2
2𝜋
𝒏
𝑑