Mark Dijkstra (2017) in Figure 16 has calculated the cross section with classical methods as 7 * 10^-15 m^2.
We suggested in a blog post that the angular momentum, which the electron in the hydrogen atom can absorb, restricts the absorption rate. Let the atom A emit a 122 nm Lyman alpha photon. The atom A recoils to a random direction.
The speed of the 1s electron in the hydrogen atom is 1 / 137 c. Its momentum is 2 * 10^-24 kg m/s.
The distance of the electron from the proton is the Bohr radius r_0, 5 * 10^-11 m. We conclude that the electron may absorb an angular momentum
J = 2 p r_0 = 2 * 10^-34 kg m^2 / s.
The coefficient 2 comes from the fact that the electron can make a U-turn.
The momentum of a 122 nm photon is
p = c * h f / c^2 = h / λ
= 5 * 10^-27 kg m/s.
The maximum distance at which the absorbed photon may pass is
r = J / p = 4 * 10^-8 m = 40 nm.
The maximum possible cross section is
π r^2 = 5 * 10^-15 m^2.
The figure given by Dijkstra was 7 * 10^-15 m^2. Why is the Dijkstra figure larger than ours? We assumed that the electron would stay at the Bohr radius, but it climbs to the second orbital.
The radius of the second Bohr orbital is 4 r_0 and the electron speed there is 1/2 of the speed of the first orbital. The angular momentum is 2X of the first orbital. We must raise our figure for J by 50%. Then we get a maximum cross section
π (60 nm)^2 = 11 * 10^-15 m^2,
which agrees with Dijkstra.
The Dijkstra cross section is surprisingly large. The silhouette of a hydrogen atom only has an area ~ 10^-20 m^2. The cross section is 700,000 times larger.
If the photon passes the hydrogen atom at a distance < 0.4 λ, it gets absorbed.
Cross section for photon-photon scattering
We can get an upper limit for photon-photon scattering cross section through calculating the maximum angular momentum that a (virtual) pair can possess if it has the energy of the colliding photons.
If the pair has 0.5 MeV of energy, then the maximum possible distance is 2.8 * 10^-15 m, but then the particles stand still.
The dipole electric field energy is concentrated between the particles.
Let us calculate with a particle-force model and ignore the effect of the electric field energy distribution.
If we put the particles at a distance 1.4 * 10^-15 m, they have 1.0 MeV mass-energy, -1.0 MeV potential energy, and 0.5 MeV of kinetic energy. Their speed is 0.85 c.
Their angular momentum J is the same as the photons passing each other at a distance 3.6 * 10^-15 m.
We conclude that the cross section is less than 3 * 10^-29 m^2, or 0.3 barn.
Our blog post on January 22, 2021 contained figures that have been calculated from Feynman diagrams. The diagrams have four propagators and the cross section has to be very small then. Literature says that the Feynman diagram cross section is less than 10 microbarn.
Individual Feynman diagrams diverge but their sum converges. Regularization (removing the divergence in an ad hoc way) has to be used.
There is a huge difference from our estimate. Let us check what are the known bounds for scattering.
M. Bregant et al. (2008) report that the measured cross section for 532 nm photons is less than 2.7 * 10^-60 m^2. The upper bound is still 20 million times higher than the calculated QED value.
Our maximum angular momentum argument removes the infinity in a Feynman integral, because the cross section gets an upper bound. The bound is very big, though.
We need to find a way to curtail the probability that a virtual pair absorbs long wavelength photons. Intuitively, it is extremely improbable that a pair whose separation is very small absorbs a 532 nm photon.
Rayleigh scattering cross section for nitrogen for 532 nm light is only 5 * 10^-31 m^2.
A virtual pair lives an extremely short time. That might be the reason why the probability of photon-photon scattering is very small.
Conclusions
The angular momentum argument makes individual Feynman integrals (or, more precisely, path integrals) to converge in the position space in photon-photon scattering, because there is a maximum angular momentum which can be held by a virtual pair. The colliding photons have to come quite close - otherwise the angular momentum of their relative motion is too large.
For the hydrogen atom Lyman alpha absorption, the angular momentum argument gives quite a precise correct value.
We need to understand better how a virtual pair absorbs the colliding photons. The measured cross section is very small for 532 nm photons.
No comments:
Post a Comment