In our March 17, 2021 blog post we analyzed the Wikipedia formula for bremsstrahlung and the Gaunt factor. There we looked at processes where the electron passes the proton at a distance > the Compton wavelength of the electron.
Bremsstrahlung in a close encounter
Let us in this section analyze closer encounters with relativistic electrons.
Let us calculate the radiated energy when a relativistic electron passes a proton.
The acceleration close to the proton is
a = k e² / (b² m_e γ),
where b is the impact parameter and γ is the Lorentz factor of the electron.
The radiated power is
P = e² a² γ⁴ / (6 π ε₀ c³).
Let us determine numerical values for b = the Compton wavelength of the electron:
b = 2.4 * 10^-12 m.
a = 4.4 * 10^26 m/s² / γ.
P = 1 W * γ²,
For a relativistic electron, the fly-by lasts for
t = b / c = 8 * 10^-21 s
and the radiated energy is
E = 8 * 10^-21 J * γ²,
which is 10^-7 of the rest mass-energy (511 keV) of the electron if γ is close to 1. Bremsstrahlung is very small at that distance. The total radiated energy is 0.05 eV. The acceleration happens in the time t.
The transverse end velocity of the electron is
v = a t = 3 * 10^6 m/s / γ,
and the electron moves a transverse distance
l = 1/2 a t² = 2 * 10^-14 m / γ
during the fly-by.
The radiated energy E scales as
E ~ 1 / b³.
If we set
b = 3 * 10^-14 m
and
γ = 1.5,
then E = 50 keV, and v is only mildly relativistic, and l < b. We see that bremsstrahlung is very large already for b which is 10X the classical radius, or a cross section of 10 barn.
Acceleration of a non-relativistic electron: kinetic energy versus Larmor radiation
Let us assume that an electron is accelerated with a force F for a time t.
The work done by the force is
W = 1/2 F / m * t² * F,
where m is the effective mass of the electron. Part of its field does not have time to react to the acceleration. Therefore, m is smaller than the rest mass of the electron.
The energy radiated according to Larmor is
E = e² (F / m)² / (6 π ε₀ c³) * t.
Obviously, W > E. If we set
W = E,
we get
t = 2 e² / (6 π ε₀ c³) * 1 / m
= 10^-23 s,
if we let m = m_e = 511 keV.
We see that
t c = 3 * 10^-15 m,
or the classical radius of the electron. The Larmor formula does not work properly if acceleration lasts that short a time.
The bremsstrahlung spectrum
The above document (2006) in Figure 3 shows the bremsstrahlung intensity spectrum to be flat up to a cutoff angular velocity ω_cut. The cutoff angular velocity is determined by the interaction time
t = b / v.
The maximum energy that an electron can lose in a collision to a nucleus is the kinetic energy of the electron. The cutoff angular velocity is, of course, restricted by that value.
In the document above, the bremsstrahlung spectrum is calculated by putting cutoffs:
b_min < b < b_max.
If the electron has low energy and comes close to the proton, then Coulomb interaction pulls it significantly closer to the proton during the fly-by. In the calculation we brutally cut off the those cases. The lowest impact parameter in the calculation is denoted b¹_min in the paper.
If the impact parameter b is smaller than the de Broglie wavelength of the electron, then the paper claims that the classical approximation does not work, and brutally cuts off those values of b. The minimum b is in this case denoted b²_min.
In most cases, b²_min is the larger one, and acts as the low-end cutoff.
If we calculate bremsstrahlung for a material where nuclei are at a distance 2 * b_max from each other, that defines the upper cutoff.
In Section 2 the Gaunt factor has an unexplained coefficient sqrt(3) / π in it.
The calculation method is to ignore what happens if the electron comes closer than the de Broglie wavelength to the proton. How can this work?
The "scale problem" once again and the inner field problem
In this spring we wrote about the problem of very close encounters of a proton and an electron, where the impact parameter b is ~ 2.8 * 10^-15 m. Classically, the electric field of the electron gains a very sharp deformation where the maximum photon energy would be around 1 GeV, regardless of the kinetic energy of the electron.
What phenomenon wipes out very sharp features from bremsstrahlung? Uncertainty of the electron position does some of the wiping. If the total energy of the electron is 1 MeV, the uncertainty is ~ 10^-12 m.
The emitted photon at most has the kinetic energy of the electron plus a few electron volts if the electron starts to orbit the proton.
We would like to know the precise bremsstrahlung spectrum, to study how the scale problem is resolved. What is the radiation from very close encounters like? But we have not found anything precise.
The texts that we linked calculate bremsstrahlung by imposing a brutal cut at b = b_min. They do not help us.
Neither were we able to find any good experimental data on Møller or Bhabha scattering, which could reveal the structure of the inner field of the electron. QED calculations seem to be very complicated, and we could not find a comprehensive analysis of all types of bremsstrahlung.
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