The fine structure constant: an update

In our previous blog post we claimed that in a classical model an electron might lose most of its kinetic energy if it passes the proton at a distance

       < 0.9 * 10^-15 m.

The idea was that all of the inertia of the electron is located in its static electric field farther than

       r₀ / 2 = 1.4 * 10^-15 m

from the pointlike electron. There

       r₀ = 2.8 * 10^-15 m

is the classical radius of the electron.

A more detailed analysis reveals that it is enough that a half of the inertia of the electron "lags behind" in the movement. Imagine that a half of the mass of the electron is attached to it with an elastic rubber band. The rest of the mass is rigidly fixed to the pointlike electron.

              ● half of the electron mass

               |

               |  rubber band

               |

              ● e- electron

              v  ------->

Let the initial velocity vector of the system be v.

If the pointlike electron suddenly bounces back in the field of a proton, so that its velocity vector becomes -v, then all of the original kinetic energy of the system will go to stretching the rubber band. That is, the kinetic energy is totally converted to vibration or electromagnetic waves.

Where is the inertia of the electron located? We again meet the mystery of the "inner field" of the electron. We do not know the inertia distribution close to the classical radius of the electron. The Larmor formula suggests that in the far field of the electron, the inertia is in the mass-energy of the static electric field.

In our previous blog post, the cross section, which we calculated to be 25 millibarn, could be off by a factor 100, depending on the distribution of the inertia of the electron. That is bad news for our claim that the fine structure constant is determined by the geometry of the electron electric field.