EMS 172
Profs. Y. Takamura and A. Moulé, 2009
UNIVERSITY OF CALIFORNIA, DAVIS
Department of Chemical Engineering and Materials Science
EMS 172: Electronic, Magnetic, and Optical Properties of Materials
Homework 2  Solutions
Due Date: Tuesday, Oct. 13
th
, 2009, inclass
Note: You must show all the steps you used to obtain your answer in order to receive full credit.
1. FermiDirac Distribution
[14 pts]
(a) At what temperature can we expect a 10% probability that electrons in a given material have an energy
which is 2% above the Fermi energy? (Use
E
F
=5.0 eV)
When dealing with probability, we are referring to the FermiDirac distribution.
E=E
F
+ 0.02*E
F
, EE
F
=0.02*E
F
()
()
1
/
10
62
.
8
0
.
5
02
.
0
exp
1
10
.
0
1
exp
1
5
+
⎟
⎠
⎞
⎜
⎝
⎛
×
×
×
=
=
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
−
T
K
eV
eV
T
k
E
E
E
F
B
F
()
()
()
)
9
ln(
/
10
62
.
8
0
.
5
02
.
0
9
/
10
62
.
8
0
.
5
02
.
0
exp
10
.
0
1
1
/
10
62
.
8
0
.
5
02
.
0
exp
5
5
5
=
×
×
×
=
⎟
⎠
⎞
⎜
⎝
⎛
×
×
×
=
+
⎟
⎠
⎞
⎜
⎝
⎛
×
×
×
−
−
−
T
K
eV
eV
T
K
eV
eV
T
K
eV
eV
Solving for
T
= 528 K.
[1 pt for eqn, 1 pt for value, 1 pt for trying]
(b) We stated that the FermiDirac distribution function can be approximated by classical Boltzmann
statistics if the exponential factor in the FermiDirac distribution function is significantly larger than one.
Using
the relationship
EE
F
=nK
b
T
for various values of
n
where
n
is an integer, calculate the value of the FermiDirac
distribution and the Boltzmann distribution.
State at which value of
n
,
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
T
k
E
E
B
F
exp
can be considered to be
significantly larger than one.
Hint
: Create a table with the calculated error between the full
F(E)
expression and
the Boltzmann distribution for different values of
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '10
 Takamura
 Atom, Electron, Statistical Mechanics, Fundamental physics concepts, Boltzmann Distribution, Paul Dirac

Click to edit the document details