Question

we have the following statistics. maxwell-boltzmann, fermi-dirac and bose-einstein.

when then sonsidering different systems which above statistics would fit the best, give a short answer of why.

a) He4 under normal room conditions of temperature and pressure

b) electrons in copper under normal room conditions

c) He4 at lambda point

d) electrons and holes in a semiconductor Ge in room temperature, band gap 1V

Answer 1

The Maxwell Distribution function arises in low-density gaseous system. This is when the intermolecular forces are negligible as compared to the kinetic energy of the molecules. The particles can be distinguished. For instance, you can easily tell the difference when particle A is in state 1 and particle B is in state 2 as compared to particle A in state 2 and particle b in state 1. Here each particle can be distinguished from the other by some physical property. Examples where particles can be distinguished is sound waves in solid or electromagnetic waves where each particle can be distinguished by wavelength or some other physical property. A gas containing only one type of gas molecule evidently doesn't belong to this case.

When we consider quantum mechanical effects, there are two different statistics to consider:

1. Bose-Einstein Statistics

Here the particles occupy the same state and cannot be distinguished from each other. They do no follow Pauli's Exclusion Principle.Examples include H,He4 or any particle with spin integer value. At low temperatures these are generally distributed over lower energy states.

2. Fermi-Dirac Statistics

Here each state consists only of one particle and cannot be distinguished from each other. This is because they follow Pauli's Exclusion Principle. Examples include electrons and He3. The most frequent application of these kinds of statistics is to "free" electrons in a solid.

An important point to note is that at low density or high pressure particles that generally follow the above two Quantum statistics will now follow Maxwell-Boltzmann distribution since quantum effects are no longer considered.

With this we can now solve the question,

1. He4 at normal room temperature and Pressure

This will follow Bose Einstein Statistics since He4is a boson as explained in the above description of Bose Einstein Statistics.

2.Electrons in Copper at normal room conditions

These follow Fermi Dirac Statistics. Electrons are fermions since they follow the Pauli's Exclusion Principle and occupy only one state. These follow the Fermi Dirac Quantum Statistics and as explained above these are free electrons in a solid which is copper.

3.He4 at lambda point

Lambda point is the temperature at which we get superfluid helium and is about 2.7 K. Superfluid is a state of matter when it behaves like a fluid with zero visco city. He4 follow Bose Einstein Statistics and when there at brought to a temperature as low as 2.7 K they occupy the lower states as mentioned in the description and continue to follow Bose Einstein Statistics.

4. Electrons and holes in a semiconductor Ge in room temperature band gap 1 eV

Electron and holes are fermions since they follow Pauli's Exclusion principle. One electron is indistinguishable from the other hence it follows Fermi-Dirac Statistics as explained above.