Fermi Dirac (FD) and Maxwell Boltzmann (MB)

In summary, the two curves for the Fermi level (Ef) at 0.25 eV show a significant difference in the beginning. This difference is largely dependent on the temperature, with quantum mechanical effects such as the Pauli Exclusion Principle being more important at lower temperatures. At higher temperatures, the curves are essentially the same. The temperature being considered in this case is 300 K, which means that quantum mechanics must be taken into consideration.
  • #1
Corneo
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I have a homework problem that asks me to interpret the two curves for when the Fermi level (Ef) is 0.25 eV. I ploted the two graphs and both of them look nothing alike when E < Ef. But both plots predict a probability of essentially zero when E > Ef. I was wondering why is there such a large difference in the beginning? Is it because the MB predicts that almost all the electrons will in the lowest state, while FD takes into account of the Pauli Exclusion principle? It seems that both plots are essentially the same when E is the conduction band energy level.
 
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  • #2
The difference depends largely on the temperature, and you haven't mentioned what temperature you're considering. If the temperature is much larger than the Fermi temperature, EF/kB (in your case about 3000 K), then quantum mechanical effects, like the exclusion principle, aren't important, and the curves are essentially the same. When the temperature is lower than the Fermi temperature, then the quantum effects are important because most of an electron's energy will be due to the exclusion principle. That is, it has the energy it does because all lower energy levels are occupied, not because of the temperature, which is the only classical source of energy.
 
  • #3
I forgot the mention that the temperature I am considering is 300 K. So I guess Quantum Mechanics must be taken into consideration.
 

What is the difference between Fermi Dirac (FD) and Maxwell Boltzmann (MB) distributions?

Fermi Dirac and Maxwell Boltzmann distributions are both used to describe the distribution of particles in a system. However, FD is used for particles with half-integer spin (such as electrons) while MB is used for particles with integer spin (such as atoms). Additionally, FD accounts for the exclusion principle, which states that no two identical particles can occupy the same quantum state, while MB does not.

How are the FD and MB distributions related?

The FD distribution can be approximated by the MB distribution at high temperatures and low particle densities. This is because at these conditions, the exclusion principle has a negligible effect on the distribution of particles.

What is the significance of the Fermi energy in the FD distribution?

The Fermi energy is the highest energy level that is occupied by electrons at absolute zero temperature in a system described by the FD distribution. It is a measure of the energy level at which the distribution of electrons transitions from mostly empty to mostly filled.

What are some real-world applications of the FD and MB distributions?

The FD distribution is commonly used in the study of semiconductors and in the field of condensed matter physics. The MB distribution is used to describe the distribution of atoms and molecules in gases and liquids, and is also used in the study of thermodynamics and statistical mechanics.

How do the FD and MB distributions affect the behavior of particles in a system?

The FD distribution leads to a higher probability of finding particles at lower energy levels, resulting in a more even distribution of particles. The MB distribution, on the other hand, leads to a higher probability of finding particles at higher energy levels, resulting in a more skewed distribution of particles towards higher energies.

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