Reciprocal Lattice

Geometric Definition

The reciprocal lattice is fundamental for all diffraction effects and other processes in a crystal lattice where momentum is transferred.
The reciprocal lattice of any geometrical point lattice has a simple geometric definition:
It can be constructed by drawing the direct lattice, picking three sets of lattice planes (hi, ki, li) (i=1,2,3) that are not coplanar, and by constructing three vectors gh,k,l which are perpendicular to the respective lattice planes and with a length (measured in cm–1) that is given by |g|=2p/dh,k,l, with dh,k,l=distance between the lattice planes (h,k,l).
The three vectors thus obtained, if reduced to the three shortest ones possible (take three lattice planes with largest distance, i.e. lowest values of (h,k,l)) define the reciprocal lattice.
This is, of course, just a complicated way of saying:
Take the (100), (010), and the (001) planes, and use the vectors perpendicular to those planes with a length given by 2p/d for these {100} type planes as the base vectors of the reciprocal lattice.
 

The Reciprocal Lattice as Fourier Transform of the Regular Lattice

The reciprocal lattice, however, is best looked at as the Fourier transform of the regular lattice. We are showing this by constructing the Fourier transform of a real crystal.
It is easier to look at a real crystal (not just a lattice) because otherwise you have to work with d-functions.
A real crystal has atoms. And atoms contain charge densities r(r), or, if we start simple and one-dimensional, r(x).
Now, r(x) must be periodic in x-direction with the lattice constant a:
r(x + na)  =  r(x),   n=0 ±1, ±2, ...
We thus can expand r(x) into a Fourier series, i.e.
r(x)  = 
S
n
rn · exp i · x· n· 2p
a
The three-dimensional case, in analogy, can be written as
r(r)  = 
S
G
rG· exp (i · G· r)
The vector G so far is just a mathematical construct defining the "inverse" space needed for the Fourier transform.
However, since we can always substitute for any r a vector r + T (T= translation vector of the lattice), or written out, r + n1a1 + n2a2 + n3a3 with ni= integers and ai= base vectors of the lattice defining the crystal, the product r · G must not change its value if we substitute r with r + n1a1 + n2a2 + n3a3.
This requires that G · T=2p · m with m= integer.
This is essentially a definition of the vectors G that serve as the Fourier transforms of the vector T, i.e. the lattice in space. These reciprocal lattice vectors, as they are called, can be obtained from the base vectors defining the regular lattice in the following way:
If we write G in components we obtain
G  =  h · g1+ k · g2+ l · g3
With h, k, l= integers.
The vectors g1, g2, and g3 are then the unit vectors of the reciprocal lattice. (yes – they are underlined, you just don't see it with some fonts!)
If we now form the inner product of G · T, e.g., for simplicity, with T= n1 · a1, we obtain
(h · g1+ k· g2 + l · g3) · (n1· a1)  =  2p· m
For an arbitrary n1 this only holds if
g1 · a1  =  2p    
         
g2· a1  =  g3· a1    =  0
In general terms, we have
gi· aj  =  2p· dij
With dij=Kronecker symbol, defined by: dij=0 for ¹ and dij=1 for i=j.
The above equation is satisfied with the following definitions for the unit vectors of the reciprocal lattice:
g1  =  2 p· a2 × a3
a1· a2· a3
       
g2  =  2 p· a3× a1
a1 · a2· a3
       
g3  =  2 p· a1 × a2
a1 · a2· a3

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© H. Föll (Semiconductors - Script)