264 Problems €4 Sdutioru on Thermodyurm’cs €4 Stati3tical Mechanics
We have D(E) = fi. const.
Then
- leF&&da -
- iEF *
&=
I,” &de
(b) For p >> mc, we have E = pc, and D(E) = E’. const. Therefore,
2092
Derive the density of states D(e) as a function of energy E for a free
electron gas in one-dimension. (Assume periodic boundary conditions or
confine the linear chain to some length L.) Then calculate the Fermi energy
EF at zero temperature for an N electron system.
( wis co ns in)
Solution:
The energy of a particle is E = p2/2m. Thus,
Taking account of the two states of spin, we have
or
a(€) = L (F) ’”/..
At temperature 0 K, the electrons will occupy all the states whose energy
is from 0 to the Fermi energy EF. Hence