Derivation of Effective Mass from a Parabola

[Morberticus]

There is some fundamental about effective mass that I am misunderstanding about effective masses.

I understand the relation

$E\left(k\right) = E_0 + \frac{1}{2m^*}k^2$

But I'm not sure when it's appropriate to fit this to a parabola.

I would have thought the fitting is only done when you have an independent-particle model, and know the energies associated with each k "orbital".

The data I have is a parabola of energies built by systematically adding electrons and calculating the corresponding Hartree-Fock energy.

For example, the minimum of my data is the HF energy of n electrons. The next point is the HF energy of n+1 electrons. The next point is the energy of n+2 electrons. So these points, strictly speaking, aren't energies of k values. I was told I can still fit this data to a parabola and derive an effective mass. The data fits a parabola shape very well, but I am assuming I can't simply use the above equation, as my energies are N-electron energies and no "k number" energies. Or can I?

 

[BigJDubs]

Can someone explain this to me? Thank you. Yes, you can still fit your data to a parabola and derive an effective mass. The equation you provided relates the dispersion (energy vs. momentum) relation of a single-particle system, which is usually used to describe the motion of electrons in a crystal lattice. This equation can be used to model the behavior of electrons in a many-body system, such as the one you are describing, by fitting the energy values for different electron numbers to a parabolic form. By doing this, you will obtain an effective mass, which is a measure of how the energy of the system changes with electron number.