Biomedical Engineering Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
2
4
6
8
10
Reduced temperature
Figure 7.11
Populations as a function of temperature
7.7 Lindemann's Theory of Melting
We are going to catch our breath here, and draw together several strands already discussed.
Figure 7.12 shows a two-dimensional cut through a simple solid. Think of a crystal where
all the particles are the same, such as argon or possibly a metal. Metals actually get us into
deep water (so to speak) because they have a sea of free electrons, which is why many
conduct electricity, but do not worry about this problem for the minute. The important
thing is that the particles do not appear to interact with each other and they are all the
same. (Actually, if the particles did not interact they would not form a crystal, they would
always exist as a perfect gas. That is a matter for discussion later in the text.) The bottom
right-hand particle is constrained to sit in its equilibrium position by the other particles,
and we will allow it to make simple harmonic excursions from its equilibrium position
depending on the temperature. The higher the temperature, the higher are the excursions
from the equilibrium position. So when does this solid melt?
The treatment I am going to give is a very simple one that relies on the
equipartition
of energy principle
. You will probably know that a molecule comprising
N
atoms has 3
N
degrees of freedom, and that we usually think if these as 3 translational, 3 rotational (or 2
if the molecule is linear) and 3
N
5 if the molecule is linear) degrees
of freedom. According to the laws of classical physics, each of these degrees of freedom
has energy of
1
/
2
k
B
T
.
−
6 vibrational (3
N
−
