Free electron energy levels

In summary, the extended zone scheme in solid state physics displays energy levels for the free electron approximation in band calculations as a collection of many parabolas centered on the reciprocal lattice sites. However, this is not an accurate representation of the free electron solution, which would only have a single parabola with a vertex at k=0. This is due to the assumption of a finite periodicity in the potential, while the truly free electron model has an infinitesimal periodicity. This allows for comparison with real band structures and serves as a starting point for understanding reciprocal space.
  • #1
nista
11
0
In many textbooks and sites of solid state physics energy levels for the free electron approximation in band calculations are displayed. The result is a collection of many parabolas, each of these parabolas being centered on a site of the reciprocal lattice.
This is not anyway the picture that should emerge from the free electron solution where we should find a single parabola with vertex at k=0.
Can anyone help me explain this discrepancy?
Thanks a lot
 
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  • #2
I'm not quite sure what you're asking, but it sounds like you're comparing the extended zone scheme to the free electron theory? The extended zone scheme (or any zone scheme) assumes that there is a fixed periodicity in your real space lattice, ie. the potential V(x+a) = V(x) with a > 0. This gives a periodicity in reciprocal space, in 1D the periodicity is [tex]g = 2\pi/a[/tex]. This gives us a periodicity in the energy bands, [tex]\varepsilon(k) = \varepsilon(k+g)[/tex]. But the truly free electron model doesn't have just a finite periodicity, it has an infinitesimal periodicity, so V(x+a) = V(x) for any arbitrarily small a. If you put this into reciprocal space by taking the limit as a -> 0, you would find that [tex]g \rightarrow \infty[/tex], destroying the periodicity in reciprocal space.

So the thing to remember is that with the free electron model applied to a lattice, we assume the potential has some fixed finite periodicity, but we neglect the actual effect of that potential on the electron energy bands. This is done as a starting point for learning about reciprocal space, and it also provides something to compare real band structures to in order to see how free-electron like they are. For instance, compare the band structures on page 6 of http://www.mse.ncsu.edu/WideBandgaps/classes/MSE%20704/Handouts/BZs&Bands.pdf" [Broken].
 
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  • #3
Ok that was exactly what I was asking.. I think this point is quite subtle.. thanks a lot for the explanation
 

1. What are free electron energy levels?

Free electron energy levels refer to the different energy levels that an electron can occupy within a material or substance, without being bound to a specific atom or molecule. These energy levels are associated with the movement of electrons within a material, and can be influenced by external factors such as temperature and electric fields.

2. How do free electron energy levels affect the properties of a material?

The energy levels of free electrons play a crucial role in determining the electrical and thermal conductivity of a material. Higher energy levels allow for easier movement of electrons, resulting in a higher conductivity. Additionally, the spacing and distribution of energy levels can impact the optical and magnetic properties of a material.

3. How are free electron energy levels determined?

The energy levels of free electrons can be calculated using quantum mechanical models, such as the Schrödinger equation. These models take into account the interactions between electrons and the surrounding atoms or molecules, and can provide valuable insights into the energy levels and properties of a material.

4. Can free electron energy levels be manipulated?

Yes, free electron energy levels can be manipulated through various methods such as applying an external electric or magnetic field, or altering the temperature of the material. These changes can result in a shift in the energy levels and subsequently, affect the properties of the material.

5. How do free electron energy levels differ from energy levels in bound electrons?

Bound electrons have specific, discrete energy levels that are determined by the structure and composition of the atom they are bound to. Free electrons, on the other hand, have a spectrum of energy levels that can vary depending on their environment. Additionally, bound electrons are not able to move freely within a material, while free electrons have more mobility and can contribute to the conductivity of a material.

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