Fermi Distribution Explained: Energy Levels in Metals

[apb86]

Hello!

I've just started reading about quantum mechanics, so my question may sound silly.
I'm using the book of physics of Paul Tipler. It's says that inside the metals we have a lot of free electrons, like a electrons cloud (like when we learn in the School). These electrons follow the Fermi-Dirac distribution, in which the electrons occupy the energy levels from the lower to the higher ones.

OK, my doubt is:
If these electrons are "free", I understand that they are not bounded to any nucleus (protons). If they are not bounded, how can we establish energy levels. The levels are related to the particle's potential energy, isn't it? But if we don't have the force field crated by the electrostatic force of the protons (Coulomb Law), how can we establish potential energy levels?

Thanks
Alexandre

 

[nicky.tiangn]

Hi, Alexandre

As what was mentioned, we can use the free electron cloud model to simulate the behavior of "free" electrons in metals.

In my opinion, it tells us that, the "free" model is good enough to describe the electrons in metal, but it does never have to be equivalent to the tale that "free" electrons are FREE as FREE POINT-PARTICLES we studied in CLASSICAL MECHANICS courses.

Nevertheless, if the "free" electrons are totally FREE, how can they always be bounded in the volume (still, with a small area covers it) of metals.

Clearly, the electrons are bounded, and we may successfully solve the Schroedinger Eq. then get the discontinuous energy levels or energy bands of the metal materials.

Yours,
Nicky

 

[Bill_K]

(You'd be better off asking this on the Solid State forum!)

If these electrons are "free", I understand that they are not bounded to any nucleus (protons). If they are not bounded, how can we establish energy levels. The levels are related to the particle's potential energy, isn't it?

apb86, In the free electron theory of metals, the atomic cores are smeared out, and the attractive potential that they exert on the electrons is modeled as a uniform potential well, whose depth is called the work function.

Electrons are fermions, meaning that no two can occupy the same state, so they occupy this well with higher and higher energy states, up to the Fermi level. In addition to the uniform potential energy, the electrons therefore have kinetic energy. In this model the available energy levels are uniformly distributed.

In more sophisticated models the atomic cores are assumed to create a potential which is not uniform but periodic (equally spaced bumps), and in this case the electron states have a band structure.