Biomedical Engineering Reference
In-Depth Information
The atoms or molecules in a crystalline solid are arranged in a regular order, and for this
reason we usually start a discussion of the solid state from the properties of regular solids.
Once such patterns were truly understood at the beginning of the twentieth century, the
theory of the solid state made rapid progress.
Liquids are much harder to model and to study experimentally than solids and gases;
elementary textbooks usually state that liquids show neither complete order nor complete
disorder. The basis of this remark concerns a property called the
radial distribution function
g
(
r
). Consider Figure 10.1, which is a snapshot of the particles in a simple atomic liquid.
r
r
+
d
r
Figure 10.1
Radial distribution function
We take a typical atom (the grey one, designated
i
) and draw two spheres of radii
r
and
r
d
r
. We then count the number of atoms whose centres lie between these two spheres,
and repeat the process for a large number
N
of atoms. If the result for atom
i
is
g
i
(
r
)d
r
then
the radial distribution function is defined as
+
N
1
N
g
(
r
) d
r
=
g
i
(
r
) d
r
(10.3)
i
=
1
This process then has to be repeated for many complete shells over the range of values of
r thought to be significant.
In the case of an ideal gas, we would expect to find the number of particles to be propor-
tional to the volume enclosed by the two spheres, which is 4π
r
2
d
r
. This gives
g
(
r
)
=
4π
r
2
,
a simple quadratic curve.
Consider now the simple cubic solid shown in Figure 10.2 whose nearest neighbour
distance is
a
. Each atom is surrounded by 6 nearest neighb
o
urs at a distance
a
,12ata
distance
√
2
a
, 8 next-next nearest neighbours at a distance
√
3
a
, 6 at a further distance 2
a
and so on. We would therefore expect to find a radial distribution function similar to the
one shown in Figure 10.3. The height of each peak is proportional to the number of atoms
distant
r
from any given atom.
