Banha University

L lies anywhere on the surface of a cone that makes an angle θ with the z axis such
that
The fourth quantum number is the spin magnetic quantum number ms. There are
only two directions to exist for the electron spin. Either the spin is up or down
relative to the z-axis. The values of the spin magnetic quantum number are
ms = - ½, + ½ .
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Solution to Question 2 (18 points)
a) The Fermi energy for copper is 7.0 eV. (i) Find the probability of an energy level
at 7.15 eV is being occupied by an electron at 300K if the electrons in copper follow
the Femi-Dirac distribution function. (ii) Find the Fermi level at 0K if the density of
electrons 8.75x10 22 cm -3 .
Solution
(i) The probability f(E) = ( E EF ) / kT
1
e
For E = 7.15 eV, EF = 7 eV, kT = 0.026 eV
f(E = 7.15 eV) = ( 7.
15 7)
/ 0.
026
1
e
(ii) The electron density n = g ( E)
f ( E)
dE
0
The density of states g(E) =
1
8
2m
3
h
3/
2
1
E
1/
2
The probability at T = 0K: f(E) = 1 for E < EF
= 0 for E > EF
8
Then n =
3/
2 EF
2m
1/
2 2 8
E dE
3
h
3
3/
2
2m
3/
2
E 3 F
h
0
2 / 3
= 3.11x10 -3 = 0.311 %
2
h 3n
The Fermi level EF(0) =
2m
8
34
2
22 6
2
( 6.
63x10
) 3(
8.
75x10
x10
)
31
2(
9.
1 10 )
8
x
1.
154 x10 -18 = 7.213 eV
J
2 / 3