Bremsstrahlung. Review – accelerated rad. Principles of brems. Refer to Rybicki & Lightman Chapter 5 for discussion of thermal brems. Professor George F. Smoot Extreme Universe Lab, SINP Moscow State University. BREMSSTRAHLUNG RADIATION.
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Review – accelerated rad.
Principles of brems.
Refer to Rybicki & Lightman Chapter 5 for discussion of thermal brems.
Professor George F. Smoot
Extreme Universe Lab, SINP
Moscow State University
Bemsstrahlung is a German word directly describing the process: "Strahlung" means "radiation", and "Bremse" means "brake.
An incoming free electron can get close to the nucleus of an atom (or other charged particle), the strong electric field of the nucleus will attract the electron, thus changing direction and speed of the electron – accelerating it.
An energetic electron loses energy by emitting an X-ray photon and less energetic will emit lower energy photons. The energy of these photons will depend on the degree of interaction between nucleus and electron, i.e. the passing distance as well as the initial energy of the electron. Several subsequent interactions between one and the same electron and different nuclei are possible.
X-rays originating from this process are called bremsstrahlung.
Lower energy photons from this process are often called free-free emission. Free-free because the electron is free before and free after.
Compare to bound-bound and free-bound interactions
Where Z is atomic number
An analysis of the doubly differential cross section above shows that
electrons whose kinetic energy is larger than the rest energy (511 keV)
emit photons in forward direction while electrons with a small energy emit photons isotropically.
Spectrum produced in the Bremsstrahlung process.
The spectrum is flat up to a cutoff frequency wcut,
and falls off exponentially at higher frequencies.
pm is picometers
The bremsstrahlung power spectrum rapidly decreases
for large photon energy, and is also suppressed near the
Electron plasma frequency wp.
This plot is for the quantum case .
An analysis of the doubly differential cross section above shows that electrons
whose kinetic energy is larger than the rest energy (511 keV) emit photons I
n forward direction while electrons with a small energy emit photons isotropically.
2.898 x106 nm K
Recall that wavelength and frequency are related via:
where c is the speed of light
The photon energy and momentum are related to frequency via:
E = h and Momentum = h / c
where h is Planck’s constantThermal Issues
So what sort of temperatures and energies are involved in high energy thermal radiation? First let’s review a few equations related to electromagnetic radiation.
This means that when a photon loses energy and momentum, its frequency n decreases.
Click here to be reminded about the physical constants and units used.
Armed with these equations, we can now see what thermal temperatures and energies are involved in high energy radiation:
As well as atomic excitation, another thermal process is bremsstrahlung radiation, which occurs when free electrons interact with ions in, for example, the hot atmospheres of stars.
This type of emission form is called free-free emission, or thermal bremsstrahlung - which is German for “braking radiation”.
If a negative electron approaches a positive ion, they will be attracted to each other and the strong electric force will alter the trajectory of the electron (i.e. accelerating it), which leads to electromagnetic radiation being emitted:
An example of high energy thermal bremsstrahlung is the X-ray emission from giant elliptical galaxies and hot intercluster gas.
X-ray image of hot intercluster gas in Hydra AEnergy of Thermal Bremsstrahlung
The frequency range of the radiation depends on how much the electron’s trajectory is bent by the interaction with the positive ion. This depends on several things, including the relative velocities of the two bodies, which in turn depends on the temperature of the gas, which is why free-free emission is a thermalprocess.
Movie of the bremsstrahlung process
Movie of bremsstrahulung on a heavy nucleus with bound electrons.
This is more relevant to detectors and man-generated x-rays than astro
90% OF X-RAYS ARE PRODUCED THROUGH BREMS
INTERACTIONS WHEN 80-100 KVP APPLIED
The high energy electron can also cause an electron close to the nucleus in a metal atom to be knocked out from its place. This vacancy is filled by an electron further out from the nucleus. The well defined difference in binding energy, characteristic of the material, is emitted as a monoenergetic photon. When detected this X-ray photon gives rise to a characteristic X-ray line in the energy spectrum.
KE OF PROJECTILE ELECTRON > BINDING ENERGYORBITAL ELECTRON
OF DIFFERENT SHELL ELECTRONS
70-12 = 58 keV
70-3 = 67 keV
12-3 = 9 keV
Radiation due to acceleration of charged particle by the Coulomb
field of another charge.
(i) Collisions between unlike particles: changing dipole emission
e-e-, p-p interactions have no net dipole moment
(ii) e- - ions dominate: acc(e-) > acc(ions) because m(e-) << m(ions)
recall P~m-2 ion-ion brems is negligible
(1) emission from single e-
pick rest frame of ion
calculate dipole radiation
correct for quantum effects (Gaunt factor)
(2) Emission from collection of e-
or non-thermal bremsstrahlung
(3) Relativistic bremsstrahlung (Virtual Quanta)
Electron moves past ion, assumed to be stationary.
b= “impact parameter”
- Suppose the deviation of the e- path is negligible
The dipole moment is a function of time during
- Recall that for dipole radiation
is the Fourier Transform of
After some straight-forward algebra, (R&L pp. 156 – 157), one can derive
in terms of impact parameter, b.
speed, v, which interact with a bunch of ions.
ne = electron density (# electrons / vol)
The # of electrons incident on one ion is
# e-s /Vol
around one ion, in terms of b
Again, evaluating the integral is discussed in detail in
R&L p. 157-158.
We quote the result
Energy per volume per frequency per time due to bremsstrahlung
for electrons, all with same velocity v.
Gaunt factors are quantum mechanical corrections
function of e- energy, frequency
Gaunt factors are tabulated (more later)
one velocity v.
Maxwell-Boltzmann Distribution Thermal Bremsstrahlung
Average the single speed expression for dW/dwdtdV
over the Maxwell-Boltzmann distribution with temperature T:
The result, with
In cgs units, we can write the emission coefficient
ergs /s /cm3 /Hz
Free-free emission coefficient
Ergs sec-1 cm-3
- Analytical approximations exist to evaluate them
- Tables exist you can look up
- For most situations,
so just take
Handy table, from Tucker: Radiation Processes in Astrophysics
(1) Usually optically thin. Then
(2) is ~ constant with hν at low frequencies
(3) falls of exponentially at
overlayed on the X-ray image from the Chandra telescope.
The central source is clearly seen, as well as radio lobes
which loosely coincide with the two circum-nuclear
``bubbles'' in the X-ray image.
Important in hot plasmas where the gas is mostly ionized, so
that bound-free emission can be neglected.
Recall the emission coefficient, jν, is related to the absorption
coefficient αν for a thermal gas:
is isotropic, so
Because of term,
is very small unless ne is very large.
in X-rays, thermal bremsstrahlung emission can be
treated as optically thin
(except in stellar interiors)
e.g. Radio: Rayleigh Jeans holds
Absorption can be important, even for low ne
in the radio regime.
distance L=10 kpc
flux F= 10 -8 erg cm-2 s-1
R&L Problem 5.2
(a) What is T? Assume optically thin, thermal bremsstrahlung.
Turn-over in the spectrum at log hν (keV) ~ 2
central mass, M.
Find M, and the density of the cloud, ρ
Vol. emission coeff.
- Assume it is pure hydrogen, so ni = ne, then
ρ=mass density, g/cm3
Z=1 since pure hydrogen
For T=109 K
- Eqn (1) & (2)
Substituting L=10 kpc, F=10-8 erg cm-2 s-1