# Free-fall atomic model

The free-fall atomic model was introduced in 1965 by Michał Gryziński (1930-2004), a Polish nuclear and plasma physicist, as a result of adding electron's magnetic moment to Bohr's atomic picture. Gryziński proposed the model as a consequence of his classical scattering theory. In this classical approximation, electrons don't circulate as in the Bohr model, but enter in almost radial free-fall towards the nucleus. Due to their magnetic moment, the Lorentz force bends the electron trajectories to return them to their initial radius. This model has never been part of mainstream physics.

## Model

For simplicity, most of these considerations neglect small changes of orientation of the spin (physics) axis of electron, assuming that it is firmly oriented in space - it's called rigid top approximation. Magnetic moment of nucleus is thousands of times smaller than electron's, so such hyperfine corrections can be neglected in basic models.

Finally the basic considered Lagrangian for dynamics of single electron in these models is:

$\mathbf{L} = \frac{1}{2}m\mathbf{v}^2+\frac{Ze^2}{r}+\frac{Ze}{c}\left[ \mathbf{v}\cdot\left( \frac{\mu\times \mathbf{r}}{r^3}\right)\right]$

The last term describes interaction between magnetic field of traveling electron's magnetic moment and electric field of nucleus (spin-orbit interaction).