Advanced Solid State Physics

matrix for everykin the first Brillouin zone:
(
~^2 k^2
2 m

−E

)

Ck+

∑

G

UGCk−G= 0 (49)

By solving these equations we get both, the coefficients C and the energy. This is done for everykto
get the band structure. Before starting to calculate, a potential must be chosen.

Example: FCC-Crystal
We make an example for an fcc crystal. We need to express this periodic potential (U(r) =

∑
GeiGr)
mathematically. For example with a Heaviside stepfunctionΘand a delta function:

Θ(1−

|r|

b

)·

∑

j=fcc

δ(r−rj)

Now we have a mathematical expression for this hard sphere potential. The Fourier transform of this
function is equal to the right side of eqn. (50) because the Fourier transform of the convolution of two
functions is the product of the Fourier transforms of the two functions. The Fourier transformation of
the fcc lattice is a bcc lattice. The Fourier transform of this stepfunction is a Bessel function, which is
zero most of the time, the other time it is the amplitude of the Bessel function. The horizontal lines
in fig. 27 of the Bessel function correspond to the coefficients(UG).

̂

Θ

(

1 −

|r|

b

) ∑

j=fcc

δ(r−rj) =

̂

Θ

(

1 −

|r|

b

) ∑

j=bcc

δ(r−rj) (50)

Figure 27: Bessel function with coefficientsUG

Now to the relationship between real space lattices and reciprocal space lattices. For a bcc in real you
get an fcc in reciprocal space. That’s the same for fcc in real space, there you get a bcc in reciprocal
space. For simple cubic, tetragonal and orthorhombic lattices in real space you get the same in recip-
rocal space.(for a tetragonal lattice:a,a,bin real space the reciprocal lattice is:^2 aπ,^2 aπ,^2 bπ- which is
also tetragonal). An hcp in real space is an hcp rotated by 30 degrees in reciprocal space.

It turns out that hard spheres have these nasty Bessel function - behaviour. So we want to try cubic
atoms, because the Fourier transform is much easier. But an even nicer form to use is a gaussian. The
Fourier transform of a gaussian is a gaussian. It does not have oscillations, and it also looks more like
an atom, for example it could be thes-wave function of an atom.