using fermi-dirac distribution function?

Question

Fermi-Dirac distribution function is a continuous distribution function.So how it can predict discrete quantum states in the band structure of a material.As far as a semiconductor or insulator is concerned as per Pouli's exclusion principle there is a band gap where there is no allowed energy state for electrones,but the distribution function predicts probability>0 at that regeon also.Why?

Answer 1

What's usually called the Fermi-Dirac distribution gives the expected occupation number (between 0 and 1) for each single particle state or equivalently the probability that a particular single particle state is occupied. It's applicable to a system of non-interacting fermionic particles in thermal equilibrium.

Even though it depends (continuously) on the energy of the state and the temperature, that doesn't make it a continuous probability distribution. In fact it could hardly be more discrete since it's over only two possibilities: either the state is occupied or it's not.

The Fermi-Dirac distribution is unusable by itself until the actual single particle states are enumerated for the particular system being analyzed. In particular the number of actual states n(E) in each interval of (single particle) energies (i.e. between E and E+dE) is essential.

For an ideal gas this density of states over energies n(E) is more or less continuous. For the electron states in a solid there can be gaps in the energy spectrum where there are no single particle states - so the FD distribution with those energy values substituted into it is irrelevant. For example to calculate the total energy (heat capacity) you would sum over all E:

n(E) * FD probability * E

If E is within a band gap then n(E)=0 so those terms contribute nothing to the sum.

Answer 2

Fermi Dirac Probability Distribution Function

Answer 3

The discrepancy shows that what can be measured is acontinuous function .However It is impossible to measure a discreet quanta ,because a quanta is a lump. And there is always somthing inside the lump that can only be calculated but not measured.
So Discreet Quanta in reality is something that the whole world of physics has not got that right yet..