Bremsstrahlung

1. Review – accelerated rad.

2. Principles of brems.

3. Refer to Rybicki & Lightman Chapter 5 for discussion of thermal brems.

Professor George F. Smoot

Extreme Universe Lab, SINP

Moscow State University

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

Bremsstrahlung

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.

Review continued: Angular Distribution

• where n is a unit vector pointing from the particle towards the observer, and d is a infinitesimal bit of solid angle.

• In the case where velocity is parallel to acceleration (for example, linear motion), this simplifies to

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.

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.

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

Simple Bremsstrahlung Formulafor 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

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

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

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.

Thermal Bremsstrahlung Power Spectrum

The bremsstrahlung power spectrum rapidly decreases for large photon energy, and is also suppressed near theElectron plasma frequency p. This plot is for the quantum case .

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 In forward direction while electrons with a small energy emit photons isotropically.

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 Kmax T =

The photon energy and momentum are related to frequency via:

E = h and Momentum = h / c where h is Planck’s constant

Recall that wavelength and frequency are related via:

c

= where c is the speed of light

Click here to be reminded about the physical constants and units used.

This means that when a photon loses energy and momentum, its frequency decreases.

Thermal Issues

• Gamma radiation is generally defined by photons with < 0.001 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 0.001 nm < < 10 nm, with

124 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.

Armed with these equations, we can now see what thermal temperatures and energies are involved in high energy radiation:

Gamma & X-rays Ranges

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

H+

e-

photon

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:

This type of emission form is called free-free emission, or thermal bremsstrahlung - which is German for “braking 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.

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

Energy of Thermal Bremsstrahlung

Bremsstrahlung Radiation

Movie of the bremsstrahlung process

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

DIFFERENT DEGREES OF DECCELERATION

X-RAYS

HEAT

BREMS RADIATION IS:

POLYENERGETIC !

90% OF X-RAYS ARE PRODUCED THROUGH BREMS INTERACTIONS WHEN 80-100 KVP APPLIED

BREMS EMISSION-CONTINUOUS

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.

Characteristic X-rays

Characteristic Radiation

KE OF PROJECTILE ELECTRON > BINDING ENERGYORBITAL ELECTRON

CHARACTERISTIC CASCADE

TUNGSTEN-74

K-70 KEVL-12 KEV

M-2.8 KEV

BINDING ENERGIESOF DIFFERENT SHELL ELECTRONS

CHARACTERISTIC X-RAYS

L K70-12 = 58 keV

M K 70-3 = 67 keV

ML 12-3 = 9 keV

Bremsstrahlung “Free-free Emission”“Braking” Radiation

Radiation due to acceleration of charged particle by the Coulombfield 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

Bremsstrahlung

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)

A qualitative picture

Emission from Single-Speed Electrons

ve-

R

Zeion

b

Electron moves past ion, assumed to be stationary.

b= “impact parameter”

- 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 d

After some straight-forward algebra, (R&L pp. 156 – 157), one can derive

dbdW )(

in terms of impact parameter, b.

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

Let ni = ion density (# ions/vol.) ne = electron density (# electrons / vol)

The # of electrons incident on one ion is

# e-s /Vold/t

around one ion, in terms of b

So total emission/time/Vol/freq is

Again, evaluating the integral is discussed in detail inR&L p. 157-158.

We quote the result

Energy per volume per frequency per time due to bremsstrahlungfor electrons, all with same velocity v.

Gaunt factors are quantum mechanical corrections function of e- energy, frequency

Gaunt factors are tabulated (more later)

Naturally, in most situations, you never have electrons with justone velocity v.

Maxwell-Boltzmann Distribution Thermal Bremsstrahlung

Average the single speed expression for dW/dwdtdVover the Maxwell-Boltzmann distribution with temperature T:

The result, with

where

In cgs units, we can write the emission coefficient

Free-free emission coefficientergs /s /cm3 /Hz

Integrate over frequency:

where

In cgs:

Ergs sec-1 cm-3

The Gaunt factors

- Analytical approximations exist to evaluate them- Tables exist you can look up

- For most situations,

so just take

Numerical Values of gaunt factor

Handy table, from Tucker: Radiation Processes in Astrophysics

Important Characteristics of Thermal Bremsstrahlung Emissivity

(1) Usually optically thin. Then

(2) is ~ constant with hν at low frequenciesff

ff

(3) falls of exponentially at ff

ff

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.

Examples: Important in hot plasmas where the gas is mostly ionized, so that bound-free emission can be neglected.

Solar flare 107 (~ 1keV) radio flatX-ray exponential

H II region 105 radio flat

Orion 104 radio-flat

Sco X-1 108 optical-flatX-ray flat/exp.

Coma Cluster ICM 108 X-ray flat/exp.

T (oK) Obs. of ff

Bremsstrahlung (free-free) absorption

Recall the emission coefficient, jν, is related to the absorptioncoefficient αν for a thermal gas:

ff is isotropic, so and thus

in cgs:

Brems emission

Inverse Bremss. free-free abs.

e-

ion

e-

e-

photon

photon

collateral

Important Characteristics of ff

(1) (e.g. X-rays)

Because of term, is very small unless ne is very large.

ff

in X-rays, thermal bremsstrahlung emission can betreated as optically thin

(except in stellar interiors)

(2) e.g. Radio: Rayleigh Jeans holds

Absorption can be important, even for low ne

in the radio regime.

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

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

R&L Problem 5.2Spherical source of X-rays, radius R distance L=10 kpc flux F= 10 -8 erg cm-2 s-1

(a) What is T? Assume optically thin, thermal bremsstrahlung.

Turn-over in the spectrum at log hν (keV) ~ 2

(b) Assume the cloud is in hydrostatic equilibrium around a central mass, M.

Find M, and the density of the cloud, ρ

Vol.1/r2

Vol. emission coeff.

- 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)

- 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