
It would be sufficient for most purposes to know the E_{n}(k)
curves – the dispersion relations – along the major directions of the reciprocal lattice (n is
the band index) (see quantum mechanics script as
well). 


This is exactly what is done when real band diagrams of crystals are shown. Directions are
chosen that lead from the center of the WignerSeitz unit cell – or the Brillouin zones in the more generalized picture
– to special symmetry points. These points are labeled according to the following rules: 
 
 Points (and lines) inside the Brillouin zone are denoted with Greek letters.
 Points on the surface of the Brillouin zone with Roman letters.
 The center of the WignerSeitz cell is always denoted by a G



For cubic reciprocal lattices, the points with a high symmetry on the WignerSeitz cell are
the intersections of the Wigner Seitz cell with the lowindexed directions in the cubic elementary cell. 


We use the following nomenclature: ( red for fcc,
blue for bcc): 


The intersection point with the [100]
direction is called X (H); the line G—X is called D.
The intersection point with the [110] direction is called K
(N); the line G—K is called S.
The intersection point with the [111] direction is called L (P);
the line G—L is called L .



The picture above already used this kind of labelling. Since the tracing of the
dispersion curve can be done on different levels – corresponding to the 1st, second, etc. Brillouin zone –
the points are often indexed with the number of the Brillouin zone they use. 


This may look like this:

 



The top pictures show the elementary cell of the diamond lattice or of the ZnS type lattice; the lower left picture the Bravais lattice
of the fcc type and the primitive (noncubic) lattice
which is an equally valid, if less symmetric, representation of the fcc lattice.. 


The lower right picture shows the cubic reciprocal lattice of
the cubic
fcc lattice (which is a bcc lattice) and the WignerSeitz cells (identical with the first Brillouin zone)
which also represent the reciprocal lattice 

We now can draw the band diagrams along all kinds of lines – not only from
G
to some point on the Brillouin zone, but also from point to point, e.g., from L to K or to some other
points not yet labeled. An example for the fcc structure and the free electron gas approximation
is shown below. 





The first Brillouin zone with the proper indexing of the relevant points and some dispersion
parabola along prominent directions are shown. The picture is taken from Hummel's
book. 


The indexing of the various branches is a bit more complicated than in the illustration example for reasons explained below. 

Contemplate this picture a bit and then ask yourself: 
 
 Do I find this picture alarming ? ("Gee, if even the most simple situation produces such a complicated structure,
I'm never going to understand it)
 Do I find this picture exciting? ("Gee, what a wealth of information one can get in a simple diagram if you pick
a smart way of representation").



Yes, it is a bit confusing at first. But do not despair: If you need it, if you work with
it, you will quickly catch on! 