Let us then analyze pair production in the Bethe and Heitler 1934 paper:
Pair production from a virtual photon
This time the encounter of the electron with the nucleus does not produce a real photon like in bremsstrahlung. It produces a virtual (off-shell) photon which excites another electron in a negative energy state to become a real electron. The "hole" which is left behind the electron is the positron.
It looks like we can conjure up a negative energy electron at will - there is no shortage of them. A suitable virtual photon will always produce a pair.
--------- e-
/
/ \
/ --------- e+
/
/ virtual photon
e- --------------------------------------
|
| virtual
| photon
Z+ --------------------------------------
Above is the Feynman diagram of the process.
How would we interpret this classically?
The virtual photon has to contain a lot of energy compared to its momentum because it has to create the rest masses of the electron and the positron. The electric field of such a photon oscillates mainly in time and the oscillation does not move at the speed of light.
The obvious classical candidate for such a photon is the stretching of the electric field lines of the electron as it makes a sharp turn close to the proton. The far field of the electron lags behind and the electric field lines stretch and bend. Creating a pair breaks the stretched field lines and reduces the energy of the electric field.
Pair creation would mean that the "rubber plate" of the electron electric field is torn apart. The edges of the torn rubber are the new electron and the positron.
An alternative classical interpretation is that the rapidly changing electric field of the electron tears apart a zero-energy pair which initially is at a distance 1.4 * 10⁻¹⁵ m or half the electron classical radius from each other.
Let us assume that the incoming electron has the energy ~ 2 MeV.
If the new created electron has the energy ~ 511 keV, then its wavelength is ~ 2.4 * 10⁻¹² m. The new electron is born in a volume and time which is ~ 3 * 10⁻¹⁵ m. It has problems emerging from such a small volume of spacetime. We expect the cross section to be inversely proportional to h, just as in the case of bremsstrahlung of a large photon.
Does the positron have problems emerging from the small volume? Let us check the Bethe-Heitler formula.
The formulae (21) and (22) in their paper contain the familiar factor 1 / 137. Thus, the cross section is inversely proportional to h.
What is the intuitive reason why the positron has no problem emerging from a small spacetime volume? Maybe it really is a "hole" and not a particle at all?
Pair production "between" the colliding particles
Bethe and Heitler do not cover this process in their paper.
e- ------------------------------------
| virtual photon
|
|----------------------- e-
|
| virtual electron
|
|----------------------- e+
|
| virtual photon
Z+ -----------------------------------
We assume that the positron is available at will. If we reverse time, the nucleus scatters the positron into a virtual (off-shell) electron. We then restore time to the normal order. The electron scatters the virtual electron into a real (on-shell) electron.
Another interpretation is that the nucleus excites an electron in the negative energy state. The incoming electron further excites the newly created electron, so that it becomes on-shell. The hole which is left behind is the positron.
What is the classical analogue for this peculiar process?
We have an obvious candidate. When the incoming electron comes very close to the nucleus, the common electric field of the particles becomes very strong between them. Then it may reduce energy to create a new pair. The electric field lines break.
Alternatively, the colliding particles kick the components of a pre-existing zero-energy pair.
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