Hello,(adsbygoogle = window.adsbygoogle || []).push({});

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 isEisberg 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.)

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Basic band theory question, free electron model

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - Basic band theory | Date |
---|---|

A Basic questions on crystal lattice: k-vectors and Brillouin | Jun 23, 2016 |

He-Ne Laser -- Basic Questions | Jan 21, 2016 |

Help needed to understand dispersion curve of a 1D lattice with diatomic basic | Oct 21, 2014 |

Some basic problems with energy band | May 13, 2014 |

Doubts about basics of superconductivity. | Apr 16, 2013 |

**Physics Forums - The Fusion of Science and Community**