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September 21th, 2010
Class 4
Fourier series
Given a function of time (let’s say a wave), its composition in frequencies can be obtained via
the Fourier series


f ( t )  an cos n t   bn sin n t  or f (t )   cn e int  n=2f
n
n
The coefficients in the series an and bn (with n>0) are real and determine how much each
particular mode participate in the signal. If the exponential expansion is chosen, the cn (with -∞<
n<∞) can be complex but must meet the condition c * n  c n what ensures that f(t) is real. The
coefficients can be obtained as:
1
cn 
f ( t )e in t dt

2
Thus, if the signal (real space function) is known, the spectral composition can be obtained by
anti-transforming the function what indicates the importance of each particular mode. If instead
of discrete frequencies we consider a continuous of frequencies, then the summation above
becomes an integral in  and the anti-transformation renders c()
Since waves are also function of the coordinates, a transformation is also possible on the
coordinates, where now, in the Fourier space we have wavenumbers k (instead of frequencies).
Notice that as ω=2π/T (T the time period)
k=2π/λ (“λ“ space period)
Scattered Wave Amplitude
Let’s consider scattering (i.e. specular reflection) from the arrangement of atoms below
(alternating large and small atoms).
Any atomic property (electron density, charge concentration, magnetic moment, etc) when
mapped into a crystal will show the same periodicity as the atomic arrangement. For instance
Such an arrangement would result in the periodic electron number density given below.
a
a
a
1
Mathematically, this electron number density must have the same periodicity of the lattice, thus
it must satisfy the expression
n(x + T) = n(x) or in 3-D n(r + T) = n(r).
Translations into Fourier Space (or Reciprocal Space)
Problems with such periodic systems are much easier to treat in Fourier space. In 1D, the
Fourier transform for the electron density is
n( x )  n o

 C
p 0
p

 2px 
 2px 
 2ipx 
cos
n p exp
  S p sin 
 or n( x) 


 a 
 a 
 a 
 p  

where p are integers, Cp and Sp are the Fourier Coefficients (Real values). np can be complex
numbers with the condition n* p  n p what ensures that n(x) is real.
For each point x in real space, there is a reciprocal lattice point in Fourier space defined as
The reciprocal lattice points are
2 4 6
,
,
,…
a a a
2p
.
a
3D Translation into Fourier Space
The extension to 3-D is straight forward
n( x )   nG expiG  r 
G
where G are reciprocal lattice vectors in 3D.
The Reciprocal Lattice a Bravais Lattice
Given a real space Bravais lattice with translation vectors a1, a2, and a3, there always exist a
reciprocal lattice associated with it with translation vectors, that in terms of the real lattice
translation vectors, are given by:
 

a xa
b1  2 2 3
Vcell
 

a xa
b2  2 3 1
Vcell
 

a xa
b3  2 1 2
Vcell
The electronic density must have the same periodicity of the lattice and thus
n(r + T) = n(r)
Where T is a translation in real space. By writing the electronic density in terms of the fourier
series


nr    nG exp iG.r

 
G
The condition of periodicity imposed to the electronic density implies that
2

 
  

  


n r  T   nG exp iG  r  T   nG exp iG  r . exp iG  T  nr 
G
G
The above condition is warranted if G is written as a linear combination of the reciprocal vectors
as defined above where the coefficients are all integer.
In other words if G is a vector in the reciprocal space then
G.T=(v1b1+v2b2+v3b3). (u1a1+u2a2+u3a3)=2(u1v1+u2v2+u3v3)
Thus to ensure that the crystal properties have appropriate periodicity, all vi must be integers and
thus the reciprocal lattice must be a Bravais lattice with bi a set of translation vectors in the
reciprocal space.
The condition above can also be understood as that a wave with wave number G has the same
periodicity of the lattice, as such, the reciprocal space can be defined as the collection of
wavenumbers with the same periodicity of the lattice.
Every crystal has two lattices associated with it, a real space lattice and a reciprocal space lattice.
The diffraction pattern of a crystal is the map of the reciprocal lattice space for the crystal. The
image by an optical microscope (if it could be resolved to the atomic level), is a map of the direct
lattice.
By using the relationship between reciprocal and direct space vectors, the diffraction patterns
can be translated into real lattice vectors what allows to understand where the atoms are
positioned. Particularly the distance between different planes dhkl can be obtained. Using the
structure factors, a map for the electron density can be obtained.
Activity 3: reciprocal lattice vectors
Diffraction Condition in reciprocal space (the Laue condition)
The Bragg’s condition requires the existence of two planes what involved more than two
diffraction centers. This can be generalized however, considering just two points from which the
wave is scattered, the incoming and the diffracted waves form an arbitrary angle with respect to
each other. If k and k’ are respectively the wavenumbers of the incoming and diffracted waves. A
phase factor can be defined as exp i k  k'   r where
k=2π/λ with λ the wavelength.
Notice that the phase difference at O between the
waves reaching O and those reaching O’ is
O’
2r cos   
r cos=r.n
r
 k  r while the phase difference between

O
scattered waved by O compared to those scattered at O’
 
r cos’=-r.n’
is  k  r
The scattering amplitude, assuming it is proportional to
the electron density in the volume element, is then given by

3


 

  
 
F   dV n( r ) exp i k  k '  r   dV n( r ) exp  ik  r

with
k=k’-k. The phase factor is
added to account for the interference of the two waves, when the difference in path is a multiple
m of the wavelength, then the phase factor is equal to 2πm and constructive interference occurs.
Using that n( x )  nG expiG  r 
G

 
 

F   dV nG exp i G  k  r
For G-k=0, the exponential is 1 and the integral gives VnG. As k becomes different from G,
the oscillatory nature of the complex exponential will make the integral decreases rapidly. Thus
the Laue condition for diffraction is that the vector difference between the incoming wave and
the outgoing wave must be one of the reciprocal lattice vectors.
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