Free-fall atomic model: Difference between revisions

Revision as of 17:14, 20 March 2021

Revision as of 17:15, 20 March 2021

Line 1:

== Free-fall atomic model ==

In the [[Bohr model]] electrons are imagined as traveling in circular orbits, which explains the quantized energy levels but leads to several other disagreements with experimental results. For example, in the observed [[electron capture]] process the nucleus captures an electron from an orbital, what needs this electron to get to a distance of the range of [[nuclear forces]] (femtometers), which is many orders of magnitude smaller than in Bohr model. Another fundamental disagreement for the circulating electron is the created magnetic field not observed for hydrogen. In contrast, the angular momentum of the electron in quantum ground state of hydrogen is zero.

Gryzinski presents many other arguments, especially for agreement with various scattering scenarios, to focus on nearly zero angular momentum trajectories: with electrons traveling through nearly radial trajectories. Attracted by the Coulomb field they free-fall to the nucleus, then increase the distance up to some turning point and so on.

The free-fall atomic model focuses on Kepler-like orbits for very low angular momentum. They are not exactly ellipses due to adding the [[magnetic dipole moment]] of the [[electron]] ([[electron magnetic moment]]) into considerations, which results in a [[Lorentz force]] proportional to {\displaystyle <math>v/r^{3}}{\displaystyle v</r^{3}}math> and perpendicular to the velocity and spin of the electron. This [[spin-orbit interaction]] is nearly negligible unless the electron passes very close to the nucleus (small {\displaystyle <math>r\approx 10^{-13} m}{\displaystyle r\approx 10^{-13}m}</math>, large {\displaystyle v}<math>v</math>). This force bends the trajectory of the electron, preventing any collision with the nucleus.

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

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

{\displaystyle<math> \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]}{\displaystyle \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]}</math>

The last term describes the interaction between the magnetic field of the traveling electron's magnetic moment and the electric field of the nucleus ([[spin-orbit interaction]]).<ref name="free">{{cite journal |doi=10.1063/1.430847 |access-date=20 March 2021}}</ref>

==Reflist==