The reciprocal

space• Space of the wave vectors

• Fourier space

• Inverse• Orthogonal

Reciprocal Space:Geometrical definition

• Introduced by Bravais• Then used by Ewald (1917)

• Definition of basis vectors

• with volume of the cell

• Equivalent definition (2D, 3D...)

• is orthogonal to et but NOT in gal to

• Reciprocal space: vector space basis • Reciprocal lattice: set of points

DL

RL

ab

b* a*

integers

𝒂∗=2𝜋𝒃∧𝒄𝑣

;𝒃∗=2𝜋𝒄∧𝒂𝑣

;𝒄∗=2𝜋𝒂∧𝒃𝑣

𝑸h𝑘𝑙=h𝒂∗+𝑘𝒃∗+𝑙𝒄∗

Definition by plane waves

• belongs to RS iff:

• Reciprocal space

• Set of wave vectors of the plane waves with the direct space periodicity

• Si mwlvkuhuvwhkl 2)(2.RQ

• Si on pose

entiers.

muvwuvw 2.RQR *** cbaQ zyxhkl

*** cbaQ lkhhkl

zy,x,zyx, hklhklhkl 2.,2.2. cQbQaQ

𝒒

𝑸h𝑘𝑙=h𝒂∗+𝑘𝒃∗+𝑙𝒄∗

𝒒

∀𝑹𝑢𝑣𝑤𝑒𝑖𝑸 ∙𝑹𝑢𝑣𝑤=1⟺∀𝑹𝑢𝑣𝑤𝑸 ∙𝑹𝑢𝑣𝑤=2𝜋𝑚

Properties of the RS • Symmetry

The reciprocal spacethe same point symmetry

as the direct lattice

Let be a symmetry operator of the RL and a point of the RL.

Thus belongs to the RL

• Duality• The reciprocal lattice of the RL is the direct lattice:

• RL of the RL consists in points such that:

• If the relation is verified• Conversely if , it satisfies ,

thus , and are integers and belongs to the RL

b* a*DL RLab

∀𝑸h𝑘𝑙𝑸h𝑘𝑙 ∙𝑹=2𝜋𝑚

• The nodes of a lattice are regrouped in equally spaced planes:

The lattice planes• Family of planes

Lattice planes, rows

[100]

[001]

[010]

<100>

• Row : series of nodes in the direction Ruvw • Notation [uvw], u, v, w relatively prime

• Symmetry equivalent directions are noted: <uvw>

Lattice planes

c

1/3

1/4 1/2b

a

h, k, l Miller indices• Family of lattice planes • Famillies of planes equivalent by symmetry

dhkl

• Distance between planes

• 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

The lattice plane closest to the origin, intersects the cell axes in:

(0,0,1) (3,2,4)

𝒂h

,𝒃𝑘

,𝒄𝑙

Lattice planes and RS

Q010=d*

Q020

d010=2p/Q0102p/Q020

• The lattice plane closest to the origin satisfies:

• It intersects the axes in: h, k, l Miller indices (mutually prime)

To each family of lattice planes of period corresponds

A reciprocal space row of period • This row is orthogonal to the lattice planes

• The smallest vetor of this row has a magnitude

𝒉𝒖+𝒌𝒗+𝒍𝒘=𝟏𝒂h

,𝒃𝑘

,𝒄𝑙

𝒅=𝟐𝝅 /𝑸

𝑹𝑢𝑣𝑤

is a RL vector

cannot be shorter, it is the row period

Miller indices:

𝑑

?

𝒏

Distance between lattice planes dhkl

• General case

• Hexagonal system:

• Cubic system :

• dhkl distance between planes (hkl)

Qhkl smallest vextor of the row

𝑑h𝑘𝑙=2𝜋

√h2𝑎∗ 2+𝑘2𝑏∗2+𝑙2𝑐∗ 2+2 h𝑘𝑎∗𝑏∗ cos𝛾∗+2𝑘𝑙𝑏𝑐∗cos𝛼∗+2 h𝑙𝑎∗𝑐∗ cos 𝛽∗

𝑑h𝑘𝑙=2𝜋𝑄h𝑘𝑙

𝑑h𝑘𝑙=𝑎

√ 43

(h¿¿2+𝑘2+h𝑘)+ 𝑙2(𝑎𝑐

)2

¿

𝑑h𝑘𝑙=𝑎

√h2+𝑘2+ 𝑙2

Multiple unit cells

• Body centered cell

• The condition implies

1) integers (Reciprocal space of lattice 2)

• Reflection conditions

cbaR

cbaR

0.5)w0.5)v0.5)u

wvu

uvw

uvw

(((I

F

PIFA

PFIA

Conditions

abA

B

b* a*

A*

B*

a

a*

• Hexagonal lattice• A = a-b; B=a+b; C=c

*)*(2

1

2

)(2

22*

*)*(2

1

2

)(2

22*

abbacAC

B

bacbaCB

A

vv

vv

2nkh

h+𝑘+ 𝑙=2𝑛

∀𝑹𝑢𝑣𝑤𝑸 ∙𝑹𝑢𝑣𝑤=2𝜋𝑛

same parity

• Definition• Fonction or distribution

• Direct lattice described by a ’’node density’’ function:

Fourier transform of the RS

• The Fourier transform of direct lattice is the reciprocal lattice

‘‘node density’’ of RL

• The reciprocal space is the FT of the Direct space

𝑆 (𝒓 )=∑𝑢𝑣𝑤

𝛿(𝒓−𝑹𝑢𝑣𝑤)

𝑇𝐹 (𝑆 (𝒓 ) )=𝐹 (𝒒 )=∫∑𝑢𝑣𝑤

𝛿(𝒓 −𝑹𝑢𝑣𝑤)𝑒−𝑖𝒒 ∙𝒓 𝑑3𝒓

𝐹 (𝒒 )=𝑣∗∑h𝑘𝑙

𝛿(𝒒−𝑸h𝑘𝑙)

¿ ∑𝑢𝑣𝑤

𝑒−𝑖𝒒 ∙𝑹𝑢𝑣𝑤=∑𝑢

𝑒− 2𝑖 𝜋 𝑞𝑥𝑢∑𝑣

𝑒−2 𝑖 𝜋 𝑞𝑦 𝑣∑𝑤

𝑒− 2𝑖 𝜋 𝑞𝑧𝑤

¿∑h

𝛿 (𝑞𝑥− h )∑ 𝛿 (𝑞𝑦 −𝑘 )∑𝑙

𝛿 (𝑞𝑧− 𝑙)

∑h

𝛿 (𝑞−h𝑇 )= 1𝑇 ∑

𝑛=− ∞

+∞

𝑒− 2𝑖 𝜋 𝑛 𝑞

𝑇

Série de Fourier duPeigne de Dirac

Properties of the FT

uvwuvw

hklhkl vTF )())((1 RrQq

• 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

)(')'('))'((

)()())((

3'.3'.

3)(.3).( 1

qFrrrrO

rrrrq

rqrq

rOqrq

∫∫∫∫

deSdeS

deSdeSOF

ii

iiO

• Convolution• Convolution of f and g is f * g

∫ udurur 3)()())(( gfgf

)()()2()(

)()()(3 gTFfTFπfgTF

gTFfTFgfTF

Application to low dimenbsion objects

2p/a

a

a

• 1D : chain )()u()(S

u// rarr

• 2D : planes )()()( // rbarr

uvw

vuS

hk

yx kqhqF )()()(q

h

x hqF )()(q

** baq yx qq

*aq xq

Set of parallel plane

Lattice of lines

a*

b b*

a*

Relation with diffraction• Bragg’s law

• Diffraction on lattice planes, spacing

Vecteur de diffusion• q normal to the lattice planes

Diffraction

belongs to RS(to the direction planes)

ki kdqq

d

𝒒=2𝜋𝑑

𝒏 𝒒=22𝜋𝑑𝒏

2𝑑 sin𝜃=𝑚𝜆 𝒒=𝒌𝑑−𝒌𝑖

𝒒=2𝑘 sin𝜃=4𝜋𝜆

sin𝜃=2𝜋𝑑𝑚