# Free electron Gas shortcomings

I am studying surface states and the Rashba effect. A common model I keep coming across is to implement the free electron model.

In this model we get the spin orbit interaction Hamiltonian by imaging the electron in the rest frame with the nucleus moving around it. With this the electric field the nucleus produces in it's rest frame is lorentz transformed into a magnetic field which interacts with the electrons magnetic dipole moment, and we get the spin orbit interaction energy.

However, this term is much lower than what is observed because "the lorentz transform neglects the contribution from the atomic cores to the spin-orbit interaction felt by the electrons in a solid."

Apparently this is resolved in the a tight binding model approach.

I don't understand the:

"the lorentz transform neglects the contribution from the atomic cores to the spin-orbit interaction felt by the electrons in a solid."

part.

Does this mean we have two SOI terms? One from the moving charge of the nucleus that produces a magnetic field in the electrons rest frame, AND one from ... what exactly - the atomic cores?? What does this mean exactly? thank you.

The spin orbit coupling can be derived from the nonrelativistic limit of the dirac equation and is given by $$H_{\text{s-p}} = \frac{\varepsilon_0}{2m_e^2c^2}\mathbf{\hat{s}}\cdot\left(\mathbf{E}\times\mathbf{\hat{p}} \right)$$ $\mathbf{E}$ is the total electric field acting on an electron, which consist of a microscopic electric field $\mathbf{E}_\text{mic}$ from the nuclei and other electrons in the solid and an macroscopic electric field $\mathbf{E}_\text{mac}$. The macroscopic electric field can for example stem from heterojunctions.