Distribution of electrons below the Fermi energy

In summary, the conversation discusses the difficulty of finding the solution to a problem involving the Fermi function and the expected percentage of electrons lying below the Fermi energy at a given temperature. The attempt at a solution yielded a value of 30%, but the book's answer was a much smaller fraction. The conversation ends with a question about the energy boundaries used in integrating the Fermi function.
  • #1
taishar
3
0
I feel dumb that I can't figure this out. I'm sure its something simple that I'm just not seeing, but its really frustrating.

Homework Statement


How many electrons (in percent of the total number of electrons per mole) lie KbT (ev) below the Fermi energy? Take Ef=5eV and T=300K

Homework Equations


Not quite sure, since the Fermi function did not work.


The Attempt at a Solution


I tried using the Fermi function and end up with values around 30%. The answer (from the back of the book) is [tex]\Delta[/tex]N/Ntot=.566%

Any ideas ?

Thanks!
 
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  • #2
Since Ef is much larger than kT, you wouldn't expect 30% of the electrons to lie between Ef-kT and Ef, right? You'd expect a much smaller fraction, like the answer from the book.

When you integrated the Fermi function to get [tex]\Delta[/tex]N and Ntot, what were the energy boundaries of your integrals?
 

1. How does the distribution of electrons below the Fermi energy affect the properties of a material?

The distribution of electrons below the Fermi energy, also known as the Fermi-Dirac distribution, determines the electronic properties of a material. This distribution describes the probability of finding electrons at different energy levels and is essential in understanding phenomena such as electrical conductivity, thermal conductivity, and magnetic properties of a material.

2. What is the significance of the Fermi energy in the distribution of electrons?

The Fermi energy is the energy level at which the probability of finding an electron is 0.5. It marks the boundary between the filled and unfilled energy states in a material, and any electron with energy below the Fermi energy is considered to be in the filled states. The value of the Fermi energy plays a crucial role in determining the electronic and thermal properties of a material.

3. How does temperature affect the distribution of electrons below the Fermi energy?

At absolute zero temperature, all electrons occupy the lowest energy levels, and the distribution follows a step function. As temperature increases, electrons gain thermal energy and move to higher energy levels, leading to a smoother distribution. The shape of the distribution function also changes with temperature, becoming more symmetrical as temperature increases.

4. Can the distribution of electrons below the Fermi energy be altered?

Yes, the distribution of electrons can be altered by changing the temperature or applying an external electric or magnetic field. For example, at low temperatures, a magnetic field can cause the distribution to become more anisotropic, meaning the distribution becomes more elongated in one direction. This effect is known as the de Haas-van Alphen effect.

5. How does the distribution of electrons below the Fermi energy relate to the Pauli exclusion principle?

The Pauli exclusion principle states that two electrons cannot occupy the same quantum state. As a result, the distribution of electrons below the Fermi energy follows a specific pattern, with one electron occupying each energy level, and no electron can have the same set of quantum numbers. This principle is crucial in understanding the electronic properties of materials and plays a significant role in determining the behavior of electrons in solids.

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