Problems and Solutions on Thermodynamics and Statistical Mechanics

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