Introduction to Nanotechnology

4.4. RARE GAS AND MOLECULAR CLUSTERS 95
This is known as Lennard-Jones potential, and it is used to calculate the structure of
inert gas clusters. The force between the atoms arising from this potential is a
minimum for the equilibrium distance Rmin = (2B/C)''6, which is attractive for
larger separations and repulsive for smaller separations of the atoms. More generally,
it is weaker than the forces that bind metal and semiconducting atoms into clusters.
4.4.2. Superfluid Clusters
Clusters of 4He and 3He atoms formed by supersonic free-jet expansion of helium
gas have been studied by mass spectrometry, and magic numbers are found at cluster
sizes of N=7,10,14,23,30 for 4He, and N=7,10,14,21,30 for 3He. One of the
more unusual properties displayed by clusters is the observation of superfluidity in
He clusters having 64 and 128 atoms. Superfluidity is the result of the difference in
the behavior of atomic particles having half-integer spin, called fermions and
particles having integer spin called bosons. The difference between them lies in
the rules that determine how they occupy the energy levels of a system. Fermions
such as electrons are only allowed to have two particles in each energy level with
their spins oppositely aligned. Bosons on the other hand do not have this restriction.
This means as the temperature is lowered and more and more of the lower levels
become occupied bosons can all occupy the lowest level, whereas fermions will be
distributed in pairs at the lowest sequence of levels. Figure 4.22 illustrates the
difference. The case where all the bosons are in the lowest level is referred to as
Bose-Einstein condensation. When this occurs the wavelength of each boson is the
same as every other, and all of the waves are in phase.
-+----
+
+--
-t- u-
++
++
-t-*
++
HIGH T T=O
FERMIONS
-- -.--
--
-e---
*-
- e---
U^-""'-"
HIGH T T=O
BOSONS
Figure 4.22. Illustration of how fermions and bosons distribute over the energy levels of a
system at high and low temperature.