Interaction of radiation with matter (cont’ed):
The scattering of a photon by an electron where the photon energy is much less than the rest mass of the electron.
The classical treatement assumes that the electric charge oscillates under the effect of an incoming sinusoidal electromagnetic wave.
In this case the Thomson cross section can be derived starting from the Larmor formula, describing the total energy radiated by an accellerated charge (Figure 2)
This radiation is polarized with the electric field oriented as the acceleration vector (Figure 1) and its spectrum depends on the time-dependent particle acceleration.
The full derivation of the Thomson cross-section presented by an electron to an incoming photon in the case of Thomson scattering is given in Longair (2nd Edition, Volume 1, p92-96).
The total Thomson cross-section, σT, is given by the following expression (re is the classical electron radius):
where is the classical electron radius
If N = number of particles per m3, then the area blocked by a square meter of material is
If the absorbing material extends for a distance R along the line of sight then the optical depth will be given by:
and the transmitted flux will be
Main properties of Thomson scattering:
If photons have a much higher energy we have Compton scattering, where photon lose some of their energy which is transferred to the electron.
The Compton treatement must be used when hν~mec2 or ve~c in the electron rest-frame.
If hν << γ mec2 instead, we are back in the Thomson regime.
In direct Compton scattering, wavelength increases and frequency decreases i.e. photon energy decreases.
The frequency change of the photon in the Compton scattering process is given by the relationship shown, where me is the mass of an electron, c is the speed of light and h is Planck’s constant. ν0 is the initial frequency of the photon, ν is the final frequency and hν is the final energy (i.e. after being scattered).
This effect of cooling the radiation (ie because it loses energy) and transferring the energy to the electron is sometimes called the recoil effect.
For very high energy electrons, the approximation of electrons at rest does not hold anymore and the proper quantum relativistic cross-section for scattering must be used.
Also, if the electron is moving ultrarelativistically, the quantum relativistic cross-section must be used and this is given by the Klein-Nishina formula. The details of the calculations can be found on Longair, Volume 1.
Compton scattering has the effect of broadening spectral lines and washing out edges.
Main properties of Compton scattering:
Main properties of Compton scattering (cont’ed):
The precise calculation of the Compton cross-section, the Klein-Nishina cross-section, is complex but its qualitative dependence on the photon energy is reproduced in the Figure: it reduces to the Thomson value at low energies while it shows an inverse dependence on the frequency above the ‘knee’.
An electron-positron pair can be created when a gamma-ray collides with another photon. In the process, the gamma-ray is absorbed.
Using Energy and Momentum conservations, and relativistic invariants it is possible to derive the minimum energy of the gamma-ray required for this process to occur, as shown here on the right and schematized in next slide.
Two photons, one of which must be a γ-ray with E > 2mec2, collide and create an electron-positron (e-/e+) pair. This is therefore a form of γ-ray absorption.
The minimum energy is obtained for a head-on collision (theta=180 deg) and zero momentum of the e+/e- pair.
Its value is found by simply making the denominator of the expression shown in the previous slide as large as possible, i.e. when cos(θ)= -1, or when θ=180 degrees.
And the minimum γ-ray energy is given by:
Note that the minimum energy is also inversely proportional to the energy of the photon with which it interacts.
While in the laboratory, it is more usual to consider photon-nucleus production, this process is usually ignored in astrophysics.
In fact while photons and nuclei have a similar cross-section (~10-29 m2), and the γ-ray does not differentiate much between another photon or a nucleus, the space density of background (3K) photons in space is much larger (~109 m-3) than nuclei (~106 m-3).
13. X-ray binaries