MSE 2620, Spring ‘09
Homework #2
Instructions:
Due
Tuesday, February 26
th
, 2009.
Turn in one solution per group and be sure to include
the group letter, the names of the group members, and if any group members did not participate
enough to deserve full credit along with the reasons why.
Question 1: Conductivity and The Mean Free Path
a)
Sodium is a monovalent metal with a density of 0.97g/cm
3
and a BCC crystal structure.
Estimate the mean separation (nm) between an electron and one metal ion, assuming that
electrons spend the most time between a metal ion and its second nearest neighbor.
Let x be the atoms in the unit cell.
nm
a
a
x
M
dN
volume
atoms
d
Density
mol
atoms
m
kg
mol
kg
at
A
cm
g
429
.
0
)
10
022
.
6
)(
10
97
.
0
(
)
10
99
.
22
(
2
97
.
0
3
/
1
23
3
3
3
3
3
=
×
×
×
=
=
=
=
=

The average separation s between atom and ions is roughly a/2 or 0.214 nm.
If you went to
great extremes, you could show that the Na nearest neighbor separation is really
2
3
a
, which
will give you a separation of 0.866s, or 0.186 nm.
As this is an estimation
, the above
approximation is fine. I’ll use the simple approximation.
s ≈ a/2 = 0.214 nm
b)
Use the potential energy between this electron and metal ion pair to find the mean speed
of an electron (m/s) (HINT: Virial Theorem).
eV
J
nm
e
s
e
PE
o
o
71
.
6
10
07
.
1
)
214
.
0
(
4
4
18
2
2

=
×

=

=

=

πε
πε
,
the potential energy of the ion
electron pair.
The kinetic energy is related to the potential energy via the Virial Theorem.
s
m
e
e
m
J
v
v
m
J
PE
KE
6
2
/
1
19
2
19
10
09
.
1
)
10
37
.
5
(
2
2
1
10
37
.
5
2
1
×
=
×
=
=
×
=

=


c)
Given that the mean free time between scattering events from thermal vibrations of the
lattice is 4.3E14s, find the drift mobility (m
2
/Vs), the conductivity (1/Ωm) and the mean free
path of an electron (m).
Vs
m
e
e
m
s
e
m
e
2
3
14
10
55
.
7
)
10
3
.
4
(


×
=
×
=
=
τ
μ
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m
Vs
m
mol
kg
A
m
kg
N
e
en
Ω


×
=
×
×
×
=
=
1
7
3
3
3
10
07
.
3
)
10
55
.
7
(
)
10
99
.
22
(
)
10
97
.
0
(
2
3
μ
σ
, as sodium is monovalent, the
concentration of free electrons n equals n
a
, the concentration of atoms.
m
s
v
l
s
m
8
14
6
10
67
.
4
)
10
3
.
4
)(
10
09
.
1
(


×
=
×
×
=
=
τ
d)
When the distance between impurities is approximately equal to the mean free path
between collisions, we will see a significant decrease in the resistivity. Estimate the impurity
content (unitless) needed to substantially increase the resistivity of this metal. (Hint: if
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 Spring '09
 MALLIARAS
 Atom, Electron, Mean free path

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