Biomedical Engineering Reference
In-Depth Information
Figure 7.12
Part of a three-dimensional crystal
Now back to Figure 7.12. For the minute, assume that the particles only vibrate simple
harmonically along the horizontal axis, which I will call the
x
axis. The energy of each
particle is
1
2
mv
x
+
1
2
k
s
(
x
x
e
)
2
ε
tot
=
ε
kinetic
+
ε
potential
=
−
(7.27)
According to classical physics, the average translational energy of any atom or molecule
is always
1
/
2
k
B
T
per degree of freedom. But, because the particle is vibrating simply har-
monically, the average value of the potential energy is also equal to
1
/
2
k
B
T
per degree of
freedom. In other words
2
k
s
(
x
x
e
)
2
=
1
1
2
k
B
T
−
(7.28)
Taking the square root of both sides gives
(
x
x
e
)
2
=
k
B
T
k
s
−
(7.29)
The expression on the left-hand side is the root mean square of
x
and so
k
B
T
k
s
x
rms
=
(7.30)
Most authors agree that a solid will melt when
x
rms
achieves a substantial percentage of
the natural spacing
d
between the particles in a crystal. Many take 10%, and so
k
s
d
2
100
k
B
T
melt
=
(7.31)
Table 7.8 gives data for five metals, and the agreement between Lindemann's model and
experiment is very reasonable for such a simple model.
