I Basic band theory question, free electron model

[pierce15]

Hello,

I am trying to figure out the width of bands in a 1-dimensional lattice. Here is a short derivation from the book I am reading: if we approximate the free electrons as being in a square well then the energy levels are $\frac{\pi^2 \hbar^2 n^2}{2mL^2}$. If there are $N$ ions separated by a distance $a$ in the metal, then $N = L/a$. The number of energy levels in the band is $N$, so $$E_\text{max} \approx \frac{\pi^2 \hbar^2 N^2}{2mL^2} = \frac{ \pi^2 \hbar^2}{2 ma^2},$$
since the lowest energy is nearly 0. Thus, the width of the band depends not on $N$, but on $a$.

I'm a bit confused about what the calculation actually says. My understanding was that there are many bands, and that the energy width of each successive band increases. Is this the calculation for the width of the lowest energy band? Or is this just a nonsense derivation?

Thank you for the help. In case you are curious, the source is Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, which I have found to be a good source for nearly everything except results requiring math. (In fact, nearly every calculation requiring more than a page is relegated to a ~100 page long appendix.)

 

[drvrm]

If there are NNN ions separated by a distance aaa in the metal, then N=L/aN=L/aN = L/a. The number of energy levels in the band is NNN, so

Emax≈π2ℏ2N22mL2=π2ℏ22ma2,Emax≈π2ℏ2N22mL2=π2ℏ22ma2,​

E_\text{max} \approx \frac{\pi^2 \hbar^2 N^2}{2mL^2} = \frac{ \pi^2 \hbar^2}{2 ma^2},
since the lowest energy is nearly 0. Thus, the width of the band depends not on NNN, but on aaa.

in one dimensional lattic the quantized momentum k=( 2.Pi/L) n , therefor this n=0, 1,2,...+N/2 and -1,-2,...-N/2
so the width will be the energy difference between two consecutive states..
check in your expression n and N are identical or not.

 

[nasu]

This is for free electron (Fermi) gas in a box. There is no band structure in this model. You need to introduce interaction with lattice for that.
In this model (free electrons) you have energy levels (not bands), given by that formula, with various values of n.
What you have there is the highest energy level occupied by an electron, at zero Kelvin. It is called the Fermi energy.
If you want, you can consider all these levels as forming a band.
The value of this energy depends on the electron density (for 1D is just N/L) but you can write this density in terms of the lattice constant, as you did here.

 

[pierce15]

Ah, I see. Thanks.