
Electron waves with
wave vectors on or near a BZ are diffracted; all others are not. 


This means simply that electrons
with wave vectors near or at a BZ  let's call them
k_{BZ} electrons  feel the periodic potential of the
crystal while the others do not (in a first
approximation). 


In other words, k_{BZ}
electrons interact with the crystal, and
this must express itself in their energies. 

Contrariwise, we expect that
"normal" electrons not feeling any diffraction, still pretty much
obey the relation for the total energy
E 


E 
= 
E_{kin}^{ } 
= 
p^{2}
2m^{ } 
= 
(k)^{2}
2m^{ } 




For k_{BZ} electrons,
however, we must expect major modifications. 

What we will get in the most general
terms is a splitting of the energy
value if a given k ends exactly at the Brillouin zone,
i.e. for a k_{BZ} electron. Instead of
E(k) = (k)^{2}/2m_{e}
we obtain. 


E(k_{BZ}) 
= 
(k)^{2}
2m 
± 
DE 




In words: Electrons at the BZ can have
two energies for the same wave vector and
thus state. One value is somewhat lower than the free electron gas value, the
other one is somewhat higher. 

Energies between these values are
unobtainable for any electrons  there is
now an energy gap in the E =
E(k) relation for all k vectors ending on a
Brillouin zone. 

The timehonored way to visualize this energy gap
is to look at a onedimension crystal  i.e. a chain
of atoms. 


In this case the electron wave meeting the Bragg condition
will be reflected back on itself. We have 





in this case  and the effect will be a
standing wave. 


The solutions of the Schrödinger equation will be
described by the possible superpositions of the two waves and there are
two possibilities to do that: 


y^{+} 
= 
æ
ç
è 
1
L 
ö
÷
ø 
3/2 
· 
æ
è 
exp (ikr) +
exp –(ikr) 
ö
ø 
y^{–} 
= 
æ
ç
è 
1
L 
ö
÷
ø 
3/2 
· 
æ
è 
exp (ikr) – exp –(ikr) 
ö
ø 



Both solutions are no longer propagating plane
waves with y · y*
= const. throughout the crystal, but standing
waves. 


y · y*, i.e. the probability
density of finding the electron, is no longer the same everywhere in
the crystal, but follows a relation given by 


y · y* 
= 
const. 
· cos^{2} 
æ
è 
px
a 
ö
ø 




With the maxima being at
the coordinates of the atoms for the y^{–} solution and between the atoms for the y^{+}
solution. 


In the first case the potential energy of the electrons is
lowered, in the second case it is raised  there is an energy gap! 

Note that we now have a potential energy, but only
because we now implicitely assumed that the potential is no longer
constant. 