# Fermi-Dirac statistics valid for electron gas in metals?

• blue2script
In summary, the conversation discusses the use of fermi-dirac statistics in solid state physics and how it relates to the coherence length of electrons in metals. It is explained that FD statistics is based on the exclusion principle and the symmetry of the total wavefunction, which is related to the spin of the electron. The concept of quantum state is also clarified, with the understanding that electrons in a metal have a large number of states available due to their freedom to move. The distinction between quantum numbers and quantum state is discussed and how it applies to electrons in a crystal. The conversation ends with a request for further clarification on the topic.
blue2script
Hello!

In my course of solid states physics we use the fermi-dirac statistics for a free electron gas in metals. The fermi wave length of the electrons is about 1 Angström. Now, the wavelength may be intepreted as something as a coherence range - the electron should forget about the state of the other electrons on a scale of about 1 Angström, right? But then, how can we use FD-statistics if there is no coherence of the electron gas along the whole metal (or crystal)?

Blue2script

FD statistics has nothing to do with the coherence length.
Two electrons can not occupy the same quantum state, i.e. they obey the exclusion principle which is why they obey FD statistics.
Mathematically this is related to the the symmetry of the total wavefunction (more specifically if it changes sign when you swap the particles around) which in turn is related to the spin of the electron; electrons are fermions since they have half-integer spin (1/2)

Yeah, ok, but how do I know if electrons are to be described by the same wave function? I mean, taking two separated crystals would give separate FD statistics. But bringing them together would yield all electrons to be described by one wavefunction thus obeying one FD statistic.

So, the question is: When is the distance between two electrons big enough so that the two can be in the same quantum numbers? Any relation to the coherence length?

Thanks again!

Blue2script

It is not the same quantum number; it is the same quantum state. It is not the same thing.
Two electrons that are spatially separated are not in the same state.

Hmmm... ? But isn't quantum state und the set of all quantum numbers the same? I mean, the state of an electron is determined by its quantum numbers, right? And so the state of two electrons is the same if they have the same quantum number? I am confused...

Sure this only holds if the two electrons are not separated spatially. But so are electrons in a crystal? Could you explain your statement in more detail, f95toli? I have the feeling to be one the wrong path...

Thank you!

Blue2script

blue2script said:
Hmmm... ? But isn't quantum state und the set of all quantum numbers the same?

Not quite. it is true that if the electron is bound to an atom we can use quantum numbers to do the "bookkeeping", two electrons with the same numers would indeed be in the same state which is forbidden. However, electrons in a metal are free to move so you can't assign numbers like n,l,m etc to them. Instead we have to start from the beginning and solve the SE for electrons in a periodic potential (in the simplest case), we then get new numbers that I guess you could call "quantum numbers" that are associated with Bloch states which in turn depend on the wavevector k. This rather simple description still give rise to quite complicated physics, specficially a very large number of energy bands in k-space that can be occuiped by electrons (the bands are very densily packed, which is why we talk about the density of states as a continuum). The main point here is that whereas we can still only have one electron per state there is a huge number of states available and I suppose two electrons could at least in principle be in the "same place" (whatever that means) as long as they had different momentum.
Note that "real space" descriptions become rather complicated, it is much easier to describe what is going on in k-space.

## 1. What is Fermi-Dirac statistics?

Fermi-Dirac statistics is a statistical model used to describe the behavior of particles with half-integer spin, such as electrons. It takes into account the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously.

## 2. How is Fermi-Dirac statistics applicable to electron gas in metals?

In metals, the conduction electrons can be treated as a gas of free particles, and their behavior can be described using Fermi-Dirac statistics. This model helps to understand various physical properties of metals, such as their electrical and thermal conductivity.

## 3. What is the Fermi energy in the context of Fermi-Dirac statistics?

The Fermi energy is the highest energy level occupied by electrons at absolute zero temperature in a system described by Fermi-Dirac statistics. It is also known as the Fermi level and is a crucial parameter in understanding the electronic properties of metals.

## 4. How does temperature affect the behavior of electrons in metals according to Fermi-Dirac statistics?

At absolute zero temperature, all available energy levels below the Fermi energy are occupied by electrons, and no energy levels above the Fermi energy are occupied. As the temperature increases, some electrons gain enough energy to occupy higher energy levels, leading to changes in the electrical and thermal conductivity of the metal.

## 5. Can Fermi-Dirac statistics be applied to other systems besides electron gas in metals?

Yes, Fermi-Dirac statistics can be applied to any system of fermions, such as protons, neutrons, and quarks. It is also used in other fields of physics, such as nuclear physics and astrophysics, to describe the behavior of particles with half-integer spin.

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