As explained in more detail on the main page (under
Continuum Radiation), the usual interpretation of synchrotron radiation due to the acceleration of charges is logically inconsistent because it would make the emission observer dependent. An alternative mechanism is therefore required. In the following, the possibility is examined that this radiation could actually be due to recombination of air molecules which have been ionized by the synchrotron beam.
In order to make a reasonable estimate, the following values for a typical synchrotron have been taken from the book Synchrotron Radiation (Ed.: C.Kunz, Springer Verlag):
air pressure = 10
-9 - 10
-10 Torr (i.e. one can assume a density of 10
7 cm
-3 )
electron current j = 100 mA (which corresponds to an electron flux F= 6.2
.10
17 /cm
2/sec).
beam cross section = 1 cm
2
electron energy E = 1GeV
synchrotron radius R = 10 m.
The total energy loss of a synchrotron is quoted as:
(1) I[Watt] =88.5 .E4[GeV].j[mA]/R[m] .
For the above values this results in I = 1 kW = 10
10 erg/sec (rounded).
As the complete circle of the synchrotron contains 2
.π
.1000 =6.3
.10
3 cm
3, the volume energy loss rate is therefore
(2) L = 1.6.106 erg/cm3/sec .
Assuming that this energy loss is due to collisional ionization, the required air density N would be determined by the equation
(3) N.F.σion(ΔE).ΔE = 1.6.106 erg/cm3/sec .
The collisional ionization cross section σ
ion(ΔE) has strictly speaking to be treated as a differential cross section (as given by the Rutherford formula), but if one assumes schematically that the ionization energy ΔE has a fixed value, one can use the well known relationship for Coulomb scattering
(4) b.tan(θ/2) = e2/(2E) ,
where b is the impact parameter, θ the scattering angle, e the elementary charge and E the initial electron energy. With the additional relationship for the collision of equal masses (Landau and Lifshitz, Mechanics; see also
https://www.plasmaphysics.org.uk/#enloss).
(5) ΔE/E = sin2(θ/2) ,
and noting that for small scattering angles tan(θ/2)=sin(θ/2), one obtains
(6) b2 = e4/(4E.ΔE) .
The cross section can therefore schematically be written as
(7) σion(ΔE) = πb2 =π.e4/(4E.ΔE) .
If one assumes an initial electron energy of E= 1 GeV and an ionization energy of ΔE=20eV , this would result in a cross section of about 10
-24cm
2.
However, the above electron energy rests on the assumption of a relativistic mass increase as the electron approaches the speed of light, which is a flawed concept (see
Relativity). If one assumes instead that the electromagnetic forces are merely reduced if the relative velocity approaches c (see
A Newtonian Relativistic Electrodynamics), the maximum possible energy E of the electrons in the synchrotron is in fact only 250 keV. Additionally, one has to take into account that each molecule contains about 30 electrons. With this the cross section becomes
(8) σion(ΔE=20eV) = 5.10-20cm2 .
(it is assumed here that the electron-electron scattering cross section is not substantially reduced for relative velocities approaching c; this would be justified if the relativistic γ-factor does in fact depend on the ratio (velocity relative to the center of mass)/(speed of light) which for particles of equal mass could at best be 0.5 (if the particles have relative velocity c).
From Eq.(3), one finds then that N≈10
18cm
-3. This corresponds to a much higher pressure than actually measured (in fact it is almost 10% of the atmospheric pressure), but one has to realize that the vacuum pump would only be able to reduce the pressure of the neutral air, but not the ionized air which is trapped by the strong magnetic field and hence can build up to very high densities. Collisions with this plasma could then cause the observed energy loss for the electron beam (which leads to a corresponding amount of further (higher degree) ionization). As in a state of equilibrium ionization and recombination rates have to be equal, this will be associated with a corresponding amount of recombination radiation (i.e. synchrotron radiation) The observed pronounced angle dependence of this radiation is likely to be caused by the strong magnetic field of the synchrotron and/or the strongly directional velocities of the recombining charges.
It would require further theoretical and experimental study to explain how the strong increase of the energy loss for an increasing (assumed) electron energy (Eq.(1)) can be explained, but it is likely that this is actually caused by the increase of the magnetic field strength which should very much reduce diffusive ionization losses and hence cause an increase in the plasma density (which in turn would increase the collisional energy loss).