
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
than 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! 