Fermi-Dirac Distribution

In summary, the problem is asking for the temperature at which the probability of an energy state at 7.00 eV being populated is 25% for copper with a Fermi energy of 6.95 eV. Using the Fermi-Dirac Distribution formula, f(E) = 1/(1+e^((E-EF)/kT)), and setting f(E) to 0.25 and E-EF to 0.05 eV, the calculated temperature was found to be 3.2979e21 K. However, this answer seems too high. Another user pointed out that metals have very high Fermi temperatures and suggested looking at it in terms of bosons. However, this would result
  • #1
viviane363
17
0

Homework Statement



Pleas can you help me figure out what I do wrong?
At what temperature is the probability that an energy state at 7.00 eV will be populated equal to 25 percent for copper (EF = 6.95 eV)?

Homework Equations


The formula for the fermi-Dirac Distribution is f(E) = 1/(1+e^((E-EF)/kT))


The Attempt at a Solution


Looking at the problem I figured that f(E) = 25%=0.25 and E-EF=7.00 - 6.95 = 0.05eV
solving for T and found that T=3.2979e21 K, but it doesn't seem to be the right answer, why?
 
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  • #2
Hmmm... That seems a bit much. What Boltzmann constant are you using? And double check your work, because I get a different answer than you.
 
  • #3
That could well be the right answer (not sure of the exact answer, but you will get a big number). Metals have very high fermi temperatures - you can look at it in the following way. Fermi energy levels cannot be multiply-occupied. If that metal was made of bosons, it would have a temperature of 10^21 K because of where the highest energy electrons are.
 
  • #4
That is way too large of a temperature. Hah, that is hotter than 1 second after the big bang. Also, the temperature of the Fermi energy is not even close to that. A Fermi energy of 6.95 eV has a Fermi temperature of 80,654 K.

viviane363 must have made a mistake somewhere in the calculation. Because I used the exact same formula and I got a completely different answer. But I think she forgot about this thread.
 

1. What is the Fermi-Dirac Distribution?

The Fermi-Dirac Distribution, also known as the Fermi-Dirac statistics, is a statistical distribution used to describe the behavior of fermions, which are particles with half-integer spin. It is specifically used to describe the distribution of fermions in a system at thermal equilibrium.

2. How is the Fermi-Dirac Distribution different from the Maxwell-Boltzmann Distribution?

The Fermi-Dirac Distribution takes into account the quantum nature of fermions, while the Maxwell-Boltzmann Distribution is based on classical statistics. This means that the Fermi-Dirac Distribution accounts for the exclusion principle, which states that no two fermions can occupy the same quantum state, while the Maxwell-Boltzmann Distribution does not.

3. What is the significance of the Fermi level in the Fermi-Dirac Distribution?

The Fermi level, also known as the Fermi energy, is a key parameter in the Fermi-Dirac Distribution. It represents the highest occupied energy state at absolute zero temperature and serves as a reference point for the distribution of fermions in a system. It is also used to determine the electrical and thermal conductivity of a material.

4. How does temperature affect the Fermi-Dirac Distribution?

As temperature increases, the Fermi-Dirac Distribution shifts to higher energy levels, as more fermions gain enough energy to occupy higher energy states. At absolute zero temperature, the distribution is a step function with all fermions occupying the lowest energy state. As temperature increases, the step becomes smoother, and at high temperatures, it approaches the Maxwell-Boltzmann Distribution.

5. What are some real-life applications of the Fermi-Dirac Distribution?

The Fermi-Dirac Distribution is used in a wide range of fields, including condensed matter physics, semiconductor physics, and astrophysics. It is used to explain the behavior of electrons in materials, such as metals and semiconductors, and to understand the properties of degenerate matter in high-density astrophysical objects, such as white dwarfs and neutron stars.

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