Band structure and dispersion relations

[lokofer]

-Let's suppose we have 2 gases ..one is a "Fermi" gas under an Harmonic potential and the other is a "Bose" gas under another Harmonic potential... in both cases (as an approximation) the particles (bosons and electrons are Non-interacting) then we could write the partition functions.

$$\prod _{k=1}^{\infty}(1+be^{\beta \omega (k)})^{-1}$$

Where "b" is a constant equal to 1 (electrons) or -1 (bosons)..my question is HOw could we calculate the "band structure" for the Fermion gas... ¿are the band structure and dispersion relations the "same" concept but one is valid for Bosons and other for Fermions?.. I've read "Ashcroft: Solid State..." where you can find lot's of method to calculate band structure..but what's the best?.., Is there a differential equation or other type of equation satisfied for the $$\omega (k)$$ in the "Fermionic" case?..thanks.

 

[Epicurus]

calculate n(e) i.e the density of states. You keep referring to a dispersion relation, and this euphimism of dispersion relation for w(k). you already know what w(k) is if you know the partition function and hence the energy spectrum. i think you are getting confused with terminogy here as you have presented the same question before and it does not make sense.

 

[charng]

The band structure and dispersion relations are related but not exactly the same concept. The band structure refers to the energy levels available for particles in a solid, while the dispersion relation describes how these energy levels change with respect to the momentum of the particles. In the case of a Fermi gas under an Harmonic potential, the band structure will be determined by the energy levels of the harmonic oscillator, while the dispersion relation will describe how these levels change as the momentum of the particles increases.

To calculate the band structure for a Fermi gas, one can use various methods such as the tight-binding approximation, the nearly-free electron model, or the density functional theory. Each method has its own advantages and limitations, and the best approach will depend on the specific system being studied.

In terms of the differential equation satisfied for the \omega (k) in the Fermionic case, it will depend on the method used to calculate the band structure. For example, in the tight-binding approximation, the energy levels are determined by solving the Schrodinger equation, while in the density functional theory, the energy levels are determined by solving the Kohn-Sham equations. These equations will have different forms and solutions, but they all ultimately describe the band structure of the Fermi gas.

Overall, the band structure and dispersion relations are important concepts in understanding the behavior of particles in a solid, and their calculation requires the use of different methods depending on the system being studied.