1. 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

2. BREMSSTRAHLUNG RADIATION
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

3. Bremsstrahlung

4. Short Review of Radiation from an Accelerated Charge – e.g. brems
A charged particle accelerating in a vacuum radiates power, as described by the Larmor formula and its relativistic generalizations. Although the term "bremsstrahlung" is usually reserved for charged particles accelerating in matter, not vacuum, the formulas are similar.

5. Review continued: Angular Distribution
where n is a unit vector pointing from the particle towards the observer, and dW is a infinitesimal bit of solid angle.
In the case where velocity is parallel to acceleration (for example, linear motion), this simplifies to

6. Quantum Process 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.

7. For a short constant acceleration
Spectrum produced in the Bremsstrahlung process.
The spectrum is flat up to a cutoff frequency wcut,
and falls off exponentially at higher frequencies.

8. X-ray tube
In an X-ray tube, electrons are accelerated in a vacuum by an electric field and shot into a piece of metal called the "target". X-rays are emitted as the electrons slow down (decelerate) in the metal. The output spectrum consists of a continuous spectrum of X-rays, with additional sharp peaks at certain energies (see graph on right). The continuous spectrum is due to bremsstrahlung, while the sharp peaks are characteristic X-rays associated with the atoms in the target. For this reason, bremsstrahlung in this context is also called continuous X-rays.
pm is picometers

9. Simple Bremsstrahlung Formula for free-free radiation
Makes it easy to estimate the bremsstrahlung produced radiation field for a reasonable temperature plasma.
For much hotter plasma one must take the gaunt factor in to more careful account.
For reasonable temperature there is log rise

10. Bremsstrahlung
Probability of bremsstrahlung production per atom is proportional to the square of Z of the absorber
Energy emission via bremsstrahlung varies inversely with the square of the mass of the incident particle
Protons and alpha particles produce less than one-millionth the amount of bremsstrahlung radiation as electrons of the same energy

11. Bremsstrahlung
Ratio of electron energy loss by bremsstrahlung production to that lost by excitation and ionization = EZ/820
E = kinetic energy of incident electron in MeV
Z = atomic number of the absorber
Bremsstrahlung x-ray production accounts for ~1% of energy loss when 100 keV electrons collide with a tungsten (Z = 74) target in an x-ray tube

12. Thermal Bremsstrahlung
The emission of a single electron, with impact parameter b and velocity v.
We would now like to generalize the results to a population of electrons with a certain velocity and density distribution. An astrophysically useful case is for a population with a uniform temperature T. The total emission by all particles in this population is called thermal bremsstrahlung.
Consider an cloud of ionised gas with a characteristic temperature T. The velocity distribution of the particles in this cloud is given by the Maxwell distribution.
In cgs units the power emitted per cubic cm is
Here, gB is the frequency averaged gaunt factor for the thermal distribution of velocities, and is of order unity.

13. Thermal Bremsstrahlung Power Spectrum
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 .

15. Bethe-Heitler Eqn
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.

16. Thermal Issues 2.898 x106 nm K T = max c  = 
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.
To calculate the thermal source temperature we use Wien’s law:
2.898 x106 nm K
max
T =
Recall that wavelength and frequency  are related via: c 
 =
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 constant
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.

17. Gamma & X-rays Ranges
Armed with these equations, we can now see what thermal temperatures and energies are involved in high energy radiation:
Gamma radiation is generally defined by photons with   < nm, which corresponds E >1.24 MeV and T >108 K. Such high energy photons are created in nuclear reactions and other very high energy processes.
X-rays are those photons in within the wavelength range nm <  < 10 nm, with eV < E < 1.24 MeV and 106 K < T < 108 K These high energy photons are created, for example, in supernovae remnants and the solar corona, as well as in the hot gas between galaxy clusters.

18. Types of Thermal Radiation
You are familiar with atomic excitation and de-excitation as a means of producing photons. Thermal atomic excitation is generally via collisions, which excite atoms, and as they de-excite, they emit photons. The hotter the medium, the higher the kinetic energy of the impacts, and the higher the resulting photon energy.
photon

19. Thermal Bremsstrahlung
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.
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: H+ e-
photon
This type of emission form is called free-free emission, or thermal bremsstrahlung - which is German for “braking radiation”.

20. Energy 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 thermal process.
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 A

21. Bremsstrahlung Radiation
Movie of the bremsstrahlung process

22. BREMSSTRAHLUNG ON HEAVY NUCLEUS
Movie of bremsstrahulung on a heavy nucleus with bound electrons.
This is more relevant to detectors and man-generated x-rays than astro

23. DIFFERENT DEGREES OF DECCELERATION
X-RAYS
HEAT

24. BREMS RADIATION IS: POLYENERGETIC !
90% OF X-RAYS ARE PRODUCED THROUGH BREMS
INTERACTIONS WHEN KVP APPLIED

25. BREMS EMISSION-CONTINUOUS

26. Characteristic X-rays
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.

27. Characteristic Radiation
KE OF PROJECTILE ELECTRON > BINDING ENERGYORBITAL ELECTRON

28. CHARACTERISTIC CASCADE

29. OF DIFFERENT SHELL ELECTRONS
TUNGSTEN-74
BINDING ENERGIES
OF DIFFERENT SHELL ELECTRONS
K-70 KEV
L-12 KEV
M-2.8 KEV

30. CHARACTERISTIC X-RAYSL K
70-12 = 58 keV
70-3 = 67 keV M K M
12-3 = 9 keV L

31. Bremsstrahlung Bremsstrahlung “Free-free Emission” “Braking” Radiation
Radiation due to acceleration of charged particle by the Coulomb
field of another charge.
Relevant for
(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

32. Method of Attack:
(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-
 thermal bremsstrahlung
or non-thermal bremsstrahlung
(3) Relativistic bremsstrahlung (Virtual Quanta)

33. A qualitative picture

34. Emission from Single-Speed Electronsv
Electron moves past ion, assumed to be stationary.
b= “impact parameter” b R Ze
ion
- Suppose the deviation of the e- path is negligible
 small-angle scattering
The dipole moment is a function of time during
the encounter.
- Recall that for dipole radiation
where
is the Fourier Transform of

35. After some straight-forward algebra, (R&L pp
After some straight-forward algebra, (R&L pp. 156 – 157), one can derive
in terms of impact parameter, b.

36. Now, suppose you have a bunch of electrons, all with the same
speed, v, which interact with a bunch of ions.

37. Let ni = ion density (# ions/vol.)
ne = electron density (# electrons / vol)
The # of electrons incident on one ion is
d/t
# e-s /Vol
around one ion, in terms of b

38. So total emission/time/Vol/freq is
Again, evaluating the integral is discussed in detail in
R&L p
We quote the result 

39. 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)

40. Naturally, in most situations, you never have electrons with just
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

41. where
In cgs units, we can write the emission coefficient
ergs /s /cm3 /Hz
Free-free emission coefficient

42. Integrate over frequency:
where
In cgs:
Ergs sec-1 cm-3

43. The Gaunt factors - Analytical approximations exist to evaluate them
- Tables exist you can look up
- For most situations,
so just take

44. Numerical Values of gaunt factor

45. Handy table, from Tucker: Radiation Processes in Astrophysics

46. Important Characteristics of Thermal Bremsstrahlung Emissivity
(1) Usually optically thin. Then
(2) is ~ constant with hν at low frequencies
(3) falls of exponentially at

47. VLA (radio region) ``image'' of Perseus A,
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.

48. Examples: Important in hot plasmas where the gas is mostly ionized, so
that bound-free emission can be neglected.
T (oK)
Obs. of
Solar flare
107 (~ 1keV)
radio flat
X-ray  exponential
H II region
105
Orion
104
radio-flat
Sco X-1
108
optical-flat
X-ray  flat/exp.
Coma Cluster ICM

49. Bremsstrahlung (free-free) absorption
Brems emission
photon e- e-
ion
photon
Inverse Bremss.
free-free abs. e-
collateral
Recall the emission coefficient, jν, is related to the absorption
coefficient αν for a thermal gas:
is isotropic, so
and thus
in cgs:

50. Important Characteristics of
(1)
(e.g. X-rays)
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)

51. (2)
e.g. Radio: Rayleigh Jeans holds
Absorption can be important, even for low ne
in the radio regime.

52. From Bradt’s book: BB spectrum is optically thick limit of
Thermal Bremss.

53. HII Regions, showing free-free absorption in their radio spectra:

55. Spherical source of X-rays, radius R
distance L=10 kpc
flux F= 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

56. (b) Assume the cloud is in hydrostatic equilibrium around a
central mass, M.
Find M, and the density of the cloud, ρ
Vol. emission coeff.
1/r2
Vol.

57. - Since T=109 K, the gas is completely ionized
- Assume it is pure hydrogen, so ni = ne, then
ρ=mass density, g/cm3
Z=1 since pure hydrogen
(1)

58. - Hydrostatic equilibrium  another constraint upon ρ, R
Virial Theorem:
For T=109 K
(2)
- Eqn (1) & (2) 
Substituting L=10 kpc, F=10-8 erg cm-2 s-1