Vai alla Home Page About me Courseware Federica Living Library Federica Federica Podstudio Virtual Campus 3D Le Miniguide all'orientamento Gli eBook di Federica La Corte in Rete
 
I corsi di Scienze Matematiche Fisiche e Naturali
 
Il Corso Le lezioni del Corso La Cattedra
 
Materiali di approfondimento Risorse Web Il Podcast di questa lezione

Maurizio Paolillo » 3.Absorption and scattering processes – Part II


Contents

Interaction of radiation with matter (cont’ed):

  • Thomson and Compton scattering.
  • Inverse Compton scattering.
  • Pair production.

Thomson Scattering

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.

Figure 1: Schematic representation of the electric field produced by an accelerated charge.

Figure 1: Schematic representation of the electric field produced by an accelerated charge.

Figure 2: Larmor formula for the energy radiated by an accelerated charge.

Figure 2: Larmor formula for the energy radiated by an accelerated charge.


Thomson Scattering (cont’ed)

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):

\alpha = \frac 83 \pi r{^2}_e

where r_e=\frac{e^2}{m_e c^2} is the classical electron radius

r_e=2.82 \times 10^{-15}

Schematic representation of Thomson scattering.

Schematic representation of Thomson scattering.

Thomson scattering cross-section.

Thomson scattering cross-section.


Thomson scattering (cont’ed)

If N = number of particles per m3, then the area blocked by a square meter of material is 6.65 \times 10^{-29} N/m

If the absorbing material extends for a distance R along the line of sight then the optical depth will be given by:

\tau = 6.65 \times 10^{-29}N/m

and the transmitted flux will be

F=F_0 \ exp  (- \tau)

Area blocked to the passage of radiation due to Thomson scattering.

Area blocked to the passage of radiation due to Thomson scattering.


Thomson scattering (cont’ed)

Main properties of Thomson scattering:

  1. It is symmetric with respect to the scattering angle, i.e. as much radiation is scattered forwards as backwards.
  2. Electrons present the same Thomson cross-section to 100% polarized light as they do to unpolarized light.
  3. The scattered radiation is polarized however: 100% in the plane orthogonal to the direction of travel of the incoming photon. (0% in the direction of travel of the incoming photon).
  4. Thomson scattering is one of the most important processes for impeding the escape of photons through a medium. For example, it is Thomson scattering which builds up radiation pressure in very luminous sources and defines the Eddington limit.

Compton 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 is the final energy (i.e. after being scattered).

Schematic representation of Compton scattering on a free electron. This scheme applies when the total energy of the electron is much smaller than the one of the photon.

Schematic representation of Compton scattering on a free electron. This scheme applies when the total energy of the electron is much smaller than the one of the photon.

Frequency change due to Compton scattering.

Frequency change due to Compton scattering.


Compton scattering (cont’ed)

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.

Average frequency change due to Compton scattering.

Average frequency change due to Compton scattering.

Realtive frequency change of the scattered photon.

Realtive frequency change of the scattered photon.


Compton scattering (cont’ed)

Main properties of Compton scattering:

  1. It is asymmetric with respect to the scattering angle, i.e. the likelyhood of a specific scattering angle and thus a specific energy transfer depends on the angle itself (see Figure).
  2. Forward scattering has the maximum likelihood to occur and corresponds to minimum energy loss, while background scatter yields the maximum energy transfer to the electron.
The experimental scattering cross section at different incident energies. At the low energy range of about l Kev, the cross section shows minima at  = 90o and 270o, and maxima at  = 0o and 180o. Asymmetric patterns appear at higher energy with the maximum shifted to the forward direction.

The experimental scattering cross section at different incident energies. At the low energy range of about l Kev, the cross section shows minima at = 90o and 270o, and maxima at = 0o and 180o. Asymmetric patterns appear at higher energy with the maximum shifted to the forward direction.


Compton scattering (cont’ed)

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’.

The Klein-Nishina cross-section dependence on energy. At low energies we retrieve the Thomson value, while at high energies there is an inverse dependence on energy.

The Klein-Nishina cross-section dependence on energy. At low energies we retrieve the Thomson value, while at high energies there is an inverse dependence on energy.


Electron-positron pair production

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.

Energy of the γ-ray required to produce an e+/e- pair.

Energy of the γ-ray required to produce an e+/e- pair.


Electron-positron pair production (cont’ed)

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.


And this is…

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:
E_{\gamma  min} = \frac {(m_0c^2)^2}{E_p}

A head-on collision yelds the minimum energy required to produce a e-/e+ pair ~0.511 MeV.

A head-on collision yelds the minimum energy required to produce a e-/e+ pair ~0.511 MeV.


Photon-nucleus pair production (cont’ed)

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).

  • Contenuti protetti da Creative Commons
  • Feed RSS
  • Condividi su FriendFeed
  • Condividi su Facebook
  • Segnala su Twitter
  • Condividi su LinkedIn
Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion