The reciprocal space Space of the wave vectors Fourier space Inverse Orthogonal

Reciprocal Space: Geometrical definition Introduced by Bravais Then used by Ewald (1917) DL RL ab b* a*

Definition by plane waves Si Si on pose entiers.

Properties of the RS b* a* DLRL ab

The nodes of a lattice are regrouped in equally spaced planes: The lattice planes Family of planes Lattice planes, rows [100] [001] [010] Row : series of nodes in the direction R uvw Notation [uvw], u, v, w relatively prime Symmetry equivalent directions are noted:

Lattice planes c 1/3 1/4 1/2 b a d hkl The lattice plane closest to the origin, intersects the cell axes in: (0,0,1) (3,2,4)

Lattice planes and RS Q 010 =d* Q 020 d 010 =2  /Q  /Q 020 The lattice plane closest to the origin satisfies: It intersects the axes in: h, k, l Miller indices (mutually prime)

Distance between lattice planes d hkl General case Hexagonal system: Cubic system : d hkl distance between planes (hkl) Q hkl smallest vextor of the row

Multiple unit cells I F PIFAPIFA PFIAPFIA Conditions a b A B b* a* A* B* a a* Hexagonal lattice A = a-b; B=a+b; C=c

Fourier transform of the RS The Fourier transform of direct lattice is the reciprocal lattice The reciprocal space is the FT of the Direct space Série de Fourier du Peigne de Dirac

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 2  /a a a 1D : chain 2D : planes Set of parallel plane Lattice of lines a* b b* a*

Relation with diffraction Vecteur de diffusion q normal to the lattice planes kiki kdkd q  d