
The potential for the electron in a crystal
lattice is periodic with the lattice, i.e. V(r + T) =
V(r) with T = translation vector of the
lattice. It is therefore always possible to develop V(r) into a
Fourier
series in reciprocal
space and we obtain 


V(r) 
= 
S_{G}
V_{G} · exp (i · G
·r) 




The V_{G} are the Fourier coefficients
of the potential. 


G is a reciprocal lattice vector; the sum
must be taken over all reciprocal vectors and there are infinitely many.
We will, however, no longer use the underlining
for vectors. 


In simple approximations it will be generally sufficient to
consider only a few vectors of reciprocal space; i.e., most
V_{G} are 0. 

The wave function y(r) can also be Fourier transformed. Without
loss of generality it can be expressed as a sum of the plane waves which are
the solutions of the free electron gas problem (with V = 0): 


y(r) 
= 
S_{k}
C_{k} · exp (ikr) 



The C_{k} are the Fourier
coefficients of the wave function and k denotes the wave vector
as obtained from the simple free electron
gas model (e.g. k_{x} = ± n_{x}
· 2p/L_{x}). 


Since L_{x} must be a multiple
of the lattice constant a, i.e. L_{x} =
N · a (with N = L_{x}/a
= number of elementary cells in L_{x}), all
kvectors can be written as 


k_{x} 
= ± 
n_{x} · 2p
N · a 




We know that 2p/a is
simply the magnitude of the reciprocal lattice vector characterizing the set of
planes perpenduclar to the direction of a (if taken as a unit
vector of the elementary cell) with spacing a; i.e. for the
{100} planes, n_{x} · 2p/a gives the whole set of reciprocal lattice
vectors with the same direction, and 1/N intersperses
N points between the reciprocal lattice points defined by
n_{x}. 


In other words and generalized for three dimensions: The
allowed points for kvectors are points in reciprocal space
interspersed between the lattice points of the reciprocal lattice. 

We have the following picture, with (for reasons
of clarity, very few) blue kpoints
between the red Gpoints: 





The picture makes clear that any arbitrary
wave vector k can be written as a sum of some reciprocal lattice
vector G plus a suitable wave vector k'; i.e., we
can always write k = G + k' and k' can always be confined to the 1.
Brillouin zone (= the elementary cell of the reciprocal lattice).



Alternatively, any reciprocal lattice vector G
can always be written as G = k – k' This is a
relation that should look
familiar; we are going to use it a few lines further down. 

If we now pluck both expressions into the
Schrödinger
equation 


– 
^{2}
2m 
· 
¶^{2}y
¶r^{2} 
+ V(r) 
= 
E · y(r) 




and do the differentiations, we obtain 


S_{k} 
( · k)^{2}
2m^{ } 
· C_{k} · exp (ikr)
+ S_{k'}
S_{G} C_{k'}
· V_{G} · exp (i · [k' + G] ·
r) 
= 
E · S_{k} C_{k} · exp (ik r)





We have written k' in the
double sum to indicate that it is not important how we sum up the components.
That allows us to rename the summation indices and to replace k'
as shown: 





Reshuffling the equation we obtain 


S_{k} exp
(ik · r) 
· 
æ
ç
è 
æ
è 
( ·
k)^{2}
2m 
– E 
ö
ø 
· C_{k} + S_{G} (C_{k – G} ·
V_{G}) 
ö
÷
ø 
= 0 




If this looks a bit like magic, you should
consult the link. 

Since this equation holds for any space vector
r, the expression in the red brackets must be zero by itself and
we obtain 


æ
è 
( ·
k)^{2}
2m 
– E 
ö
ø 
· C_{k} +
S_{G}(C_{k –
G} · V_{G}) 
= 
0 

