Reciprocal space (Reziproker Raum) k-space (k-Raum)

k-space is the space of all wave-vectors.

A k-vector points in the direction a wave is propagating.

2   wavelength:    momentum: p  k k kz  G

ky

kx Plane wave:   exp(iGr ) cos( GxGyGzxyz  ) i sin( GxGyGz xyz  ) Reciprocal lattice (Reziprokes Gitter)

Any periodic function can be written as a Fourier series    fr( )  fe iG r  G G Reciprocal lattice vector G Structure factor ky    

G1 b 1   2 b 2   3 b 3

vi integers kx

        a2 a 3 a 3  a 1 a1 a 2 b12   , b 2  2     , b 3  2     a1 a 2 a 3  a 1  a 2 a 3  a 1  a 2 a 3  Plane waves (Ebene Wellen)

    ik r   2 ecos kri   sin kr         k

    expikrr    exp ikr  

Most functions can be expressed in terms of plane waves

    ik r f( r )   F k e dk A k-vector points in the direction a wave is propagating. Determine the structure factors in 1-D

iGx fx( )   feG G

Multiply by e-iG'x and integrate over a period a

iGx' iGGx (')  f( x ) e dx   e dx unit cellunit cell G

  cos((G  G ')) x i sin( G G ')) x dx fG' a G unit cell

 1 iGx fG  f cell ( x ) e dx a -

The structure factor is proportional to the Fourier transform of the pattern that gets repeated on the Bravais lattice, evaluated at that G-vector. Fourier transforms

Most functions can be expressed in terms of plane waves

    ik r f( r )   F k e dk

This can be inverted for F(k)    1 ik  r  F k f( r ) e dr   d 2  

Fourier transform of f(r)

http://lamp.tu-graz.ac.at/~hadley/ss1/crystaldiffraction/ft/ft.php Fourier transforms

a

f x

a / 2 1 ikx sinka / 2 Fourier transform: F k   e dx  2a / 2  k

 sinka / 2 ikx Inverse transform: f x    e dk   k Sine integral

k0 1 1 sinka / 2 ikx Si(kx02 )Si( kx 0  2 ) Transmitted pulse: f x   e dk  k0 k  Notations for Fourier Transforms

d = number of dimensions 1,2,3 a,b = constants Notations for Fourier Transforms

f(r) is built of plane waves Notations for Fourier Transforms

Matlab Notations for Fourier Transforms

Mathematica Notations for Fourier Transforms

Engineering literature, usually on the 1-d case is considered. http://lamp.tu-graz.ac.at/~hadley/ss1/crystaldiffraction/ft/ft.php http://lampx.tugraz.at/~hadley/ss2/linearresponse/dft/dft0.php Properties of Fourier transforms Convolution (Faltung)

      fr()*() gr frgrrdr ()(   ) Reciprocal lattice (Reziprokes Gitter)

Any periodic function can be written as a Fourier series    fr( )  fe iG r  G G Reciprocal lattice vector G Structure factor ky    

G1 b 1   2 b 2   3 b 3

vi integers kx

        a2 a 3 a 3  a 1 a1 a 2 b12   , b 2  2     , b 3  2     a1 a 2 a 3  a 1  a 2 a 3  a 1  a 2 a 3  Bravais lattice and reciprocal lattice in 1-D

a

real x 2 a

k reciprocal

2px  2  cos   cosGx  G  p a  a Reciprocal lattice of an orthorhombic lattice is an orthorhombic lattice

c

b 2 2 a c b 2 a

reciprocal lattice The reciprocal lattice of an fcc lattice is a bcc lattice

a 4 a  a a   a1  xˆ  y ˆ  2 2 a1 a 2 b3  2     a a a a  a a xˆ  zˆ 1 2 3  2 2 2  2  a a ˆ ˆ ˆ a yˆ  zˆ b3  xyz   3 2 2 a The reciprocal lattice is the Fourier transform of the real space lattice

crystal = Bravais_lattice(r) * unit_cell(r)

F(crystal) = F(Bravais_lattice(r))F(unit_cell(r))

2 a

k reciprocal Cubes on a bcc lattice

   fr( )  fe iG r  G G   Multiply by e  iG   r and integrate over a primitive unit cell.    fre( ) iG  r dr3  fV  G unit cell http://lamp.tu-graz.ac.at/~hadley/ss1/crystaldiffraction/fourier.php Cubes on a bcc lattice

   fre( ) iG  r dr3  fV  G unit cell

V is the volume of the primitive unit cell.

 1   3 f  fr( )exp  iGrdr  G V  cell  

fG is the Fourier transform of fcell evaluated at G. fcell is zero outside the primitive unit cell.

a a a  4 4 4 1  3 2C f  fr( )exp  iGrdr exp  iGx exp iGy exp  iGzdxdydz G cell  3  x   y   z  V a  a  a  a 4 4 4 Volume of conventional u.c. a3. Two Bravais points per conventional u.c. Cubes on a bcc lattice

a a a Gx a  4 4 4 2sin   expiGxx  cos  Gxi x  sin  Gx x  4  expiGx x  dx   a iGxa  iG x  a G x 4 4 4

Gx a  Gy a  Gz a  16C sin  sin  sin    4   4   4  fG  3 aGGGx y z

The Fourier series for any rectangular cuboid with dimensions Lx×Ly×Lz repeated on any three-dimensional Bravais lattice is:

Gx L x  Gy L y  Gz L z  8C sin  sin  sin    2  2 2   fr( )  exp iGr     G VGGx y G z Spheres on an fcc lattice

   fr( )  fe iG r  G G   iG  r Multiply by e and integrate over a primitive unit cell.   1  3C  3 f  fr( )exp  iGrdr   exp  iGrdr  . G cell      V V sphere R    C  2 f exp(  iG  r ) r sin drd j d G  V 0 0  R   C 2  cosG r cos   i sin G r cos    r sin j drd d V 0 0  Integrate over j R  2C 2 f cos G r cos  i sin G r cos  r sin drd G       V 0 0 Spheres on an fcc lattice

R  2C 2 f cos G r cos  i sin G r cos  r sin drd G       V 0 0

Both terms are perfect differentials d cosGr cos Gr sin Gr cos  sin and d     d sinGr cos Gr cos Gr cos  sin, d     0  2C R Integrate over : f sin Gr cos  iGr cos cos  dr G      V 0 0 4C R sin G r f    rdr G  V0 G Spheres on any lattice

R 4C sin G r 2 f    r dr G  V0 Gr Integrate over r

4C fG 3 sin GRGR   cos GR  . V G  

The Fourier series for non-overlapping spheres on any three- dimensional Bravais lattice is:

4C sinGR GR cos GR   fr( )    exp iGr  .  3   V G G Molecular orbital potential

 Ze2 1 U( r )     4 0 rj r rj

position of atom j

The Fourier series for any molecular orbital potential is:

   Ze2 exp iG r U( r )     2 V 0 G G

Volume of the primitive unit cell Interference Interference Interference

     r k r k a  b  k k k

elastic scattering

k k     k r k  k  path difference: a b  k    a b r k  k     phase shift: j2  2     rkk     kr  k      i  kr  Amplitude: F F0  Fe 0 Interference   i  kr  3 Fe3    k Fei  kr  1   1 i  kr  2 Fe2

F0  k  r3   r1 r2

0

  i  kr  i Amplitude: Ftot  Fe i i