Bremsstrahlung
Encyclopedia
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
by an atomic nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
. The moving particle loses kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
, which is converted into a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
because energy is conserved
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum
Continuous spectrum
The spectrum of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum.If H is a topological vector space and A:H \to H is a linear map, the spectrum of A is the set of complex numbers \lambda such that A - \lambda I : H \to H is not...
, which becomes more intense and shifts toward higher frequencies when the energy of the accelerated particles is increased.
Strictly speaking, bremsstrahlung refers to any radiation due to the acceleration of a charged particle, which includes synchrotron radiation
Synchrotron radiation
The electromagnetic radiation emitted when charged particles are accelerated radially is called synchrotron radiation. It is produced in synchrotrons using bending magnets, undulators and/or wigglers...
; however, it is frequently used in the more narrow sense of radiation from electrons stopping in matter.
Bremsstrahlung emitted from plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
is sometimes referred to as free-free radiation. This refers to the fact that the radiation in this case is created by charged particles that are free both before and after the deflection (acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
) that causes the emission.
Dipole approximation
Suppose that a particle of charge
experiences an acceleration 
which is collinear with its velocity 
(this is the relevant case for linear acceleratorsLinear particle accelerator
A linear particle accelerator is a type of particle accelerator that greatly increases the velocity of charged subatomic particles or ions by subjecting the charged particles to a series of oscillating electric potentials along a linear beamline; this method of particle acceleration was invented...
). Then, the relativistic expression for the angular distribution of the bremsstrahlung (considering only the dominant dipole radiation contribution), is
-
,
- where
and 
is the angle between 
and the point of observation.
Integrating
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
over all angles then gives the total power emitted as
-
,
where
is the Lorentz factorLorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
.
The general expression for the total radiated power is
-
where
signifies a time derivative of 
. Note, this general expression for total radiated power simplifies to the above expression for the specific case of acceleration parallel to velocity (
), by noting that 
and 
. For the case of acceleration perpendicular to the velocity (
) (a case that arises in circular particle accelerators known as synchrotronSynchrotronA synchrotron is a particular type of cyclic particle accelerator in which the magnetic field and the electric field are carefully synchronised with the travelling particle beam. The proton synchrotron was originally conceived by Sir Marcus Oliphant...
s), the total power radiated reduces to-
.
The total power radiated in the two limiting cases is proportional to
(
) or 
(
). Since 
, we see that the total radiated power goes as 
or 
, which accounts for why electrons lose energy to bremsstrahlung radiation much more rapidly than heavier charged particles (e.g., muons, protons, alpha particles). This is the reason a TeV energy electron-positron collider (such as the proposed International Linear ColliderInternational Linear ColliderThe International Linear Collider is a proposed linear particle accelerator. It is planned to have a collision energy of 500 GeV initially, and, if approved after the project has published its Technical Design Report, planned for 2012, could be completed in the late 2010s. A later upgrade to 1000...
) cannot use a circular tunnel (requiring constant acceleration), while a proton-proton collider (such as the Large Hadron ColliderLarge Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....
) can utilize a circular tunnel. The electrons lose energy due to bremsstrahlung at a rate
times higher than protons do.
Thermal bremsstrahlung
In a plasma the free electrons constantly produce bremsstrahlung in collisions with the ions. A complete analysis requires accounting for both binary Coulomb collisions as well as collective (dielectric) behavior. A detailed treatment is given in, some of which is summarized in [?], while a simplified one is given in. In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber
.
Consider a uniform plasma, with thermal electrons (distributed according to the Maxwell–Boltzmann distribution with the temperature
). Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole 
srSteradianThe steradian is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane...
of solid angle, and in both polarizations) of the bremsstrahlung radiated, is calculated to be
where
is the electron plasma frequency, 
is the number density of electrons and ions, 
is the classical radius of electronClassical electron radiusThe classical electron radius, also known as the Lorentz radius or the Thomson scattering length, is based on a classical relativistic model of the electron...
,
is its mass, 
is the Boltzmann constant, and 
is the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. The first bracketed factor is the index of refraction of a light wave in a plasma, and shows that emission is greatly suppressed for
(this is the cutoff condition for a light wave in a plasma; in this case the light wave is evanescentEvanescent waveAn evanescent wave is a nearfield standing wave with an intensity that exhibits exponential decay with distance from the boundary at which the wave was formed. Evanescent waves are a general property of wave-equations, and can in principle occur in any context to which a wave-equation applies...
). This formula thus only applies for
. Note that the second bracketed factor has units of 1/volume and the third factor has units of energy, giving the correct total units of energy/volume. This formula should be summed over ion species in a multi-species plasma.
The special function
is defined in the exponential integralExponential integralIn mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.-Definitions:For real, nonzero values of x, the exponential integral Ei can be defined as...
article, and the unitless quantity
is
is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, 
when 
(typical in plasmas that are not too cold), where 
eV is the Hartree energyAtomic unitsAtomic units form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article...
, and
is the electron thermal de Broglie wavelength. Otherwise, 
where 
is the classical Coulomb distance of closest approach.
For the usual case
, we find
.
The formula for
is approximate, in that it neglects enhanced emission occurring for 
slightly above 
.
In the limit
, we can approximate E1 as
where
is the Euler-Mascheroni constantEuler-Mascheroni constantThe Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
. The leading, logarithmic term is frequently used, and resembles the Coulomb logarithm that occurs in other collisional plasma calculations. For
the log term is negative, and the approximation is clearly inadequate. Bekefi gives corrected expressions for the logarthmic term that match detailed binary-collision calculations.
The total emission power density, integrated over all frequencies, is
and decreases with 
; it is always positive. For 
, we find
The first bracketed factor has units of 1/volume, while the second has units of power. Note the appearance of the fine structure constant
due to the quantum nature of 
. In practical units, a commonly used version of this formula for 
is
.
This formula is 1.59 times the one given above, with the difference due to details of binary collisions. Such ambiguity is often expressed by introducing Gaunt factorGaunt factorThe Gaunt factor is used as a multiplicative correction to the continuous absorption or emission results when calculated using classical physics techniques. In cases where classical physics provides a close approximation, the Gaunt factor can be set to 1.0...
, e.g. in one finds
where everything is expressed in the CGS units.
Relativistic corrections
For very high temperatures there are relativistic corrections to this formula, that is, additional terms of the order of
http://theses.mit.edu/Dienst/UI/2.0/Page/0018.mit.theses/1995-130/25?npages=306
Bremsstrahlung cooling
If the plasma is optically thinOptical depthOptical depth, or optical thickness, is a measure of transparency. Optical depth is defined by the negative logarithm of the fraction of radiation that is not scattered or absorbed on a path...
, the bremsstrahlung radiation leaves the plasma, carrying part of the internal plasma energy. This effect is known as the bremsstrahlung cooling. It is a type of radiative coolingRadiative coolingRadiative cooling is the process by which a body loses heat by thermal radiation.- Earth's energy budget :In the case of the earth-atmosphere system it refers to the process by which long-wave radiation is emitted to balance the absorption of short-wave energy from the sun.The exact process by...
. The energy carried away by bremsstrahlung is called bremsstrahlung losses and represent, respectively, a type of radiative losses. One generally uses the term bremsstrahlung losses in the context when the plasma cooling is undesired, as e.g. in fusion plasmasNuclear fusionNuclear fusion is the process by which two or more atomic nuclei join together, or "fuse", to form a single heavier nucleus. This is usually accompanied by the release or absorption of large quantities of energy...
.
X-ray tube
In an X-ray tubeX-ray tubeAn X-ray tube is a vacuum tube that produces X-rays. They are used in X-ray machines. X-rays are part of the electromagnetic spectrum, an ionizing radiation with wavelengths shorter than ultraviolet light...
, electrons are accelerated in a vacuum by an electric fieldElectric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
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-raysCharacteristic x-rayA high energy electron interacts with a bound electron in an atom and ejects it. The incident electron is scattered and the target electron gets displaced from its shell. The incident electron energy must exceed the binding energy of the electron to eject it...
associated with the atoms in the target. For this reason, bremsstrahlung in this context is also called continuous X-rays.
The spectrum has a sharp cutoff at low wavelength, which is due to the limited energy of the incoming electrons. For example, if an electron in the tube is accelerated through 60 kV, then it will acquire a kinetic energy of 60 keV, and when it strikes the target it can create X-rays with energy of at most 60 keV, by conservation of energyConservation of energyThe nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
. (This upper limit corresponds to the electron coming to a stop by emitting just one X-ray photonPhotonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
. Usually the electron emits many photons, and each has an energy less than 60 keV.) A photon with energy of at most 60 keV has wavelength of at least 21 pm, so the continuous X-ray spectrum has exactly that cutoff, as seen in the graph. More generally the formula for the low-wavelength cutoff is:
where h is Planck constantPlanck constantThe Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
, c is the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, V is the voltageVoltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
that the electrons are accelerated through, e is the elementary chargeElementary chargeThe elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
, and pm is picometers. This is called the Duane–Hunt law.
Beta decay
Beta particle-emitting substances sometimes exhibit a weak radiation with continuous spectrum that is due to bremsstrahlung.
In this context, bremsstrahlung is a type of "secondary radiation", in that it is produced as a result of stopping (or slowing) the primary radiation (beta particles). In electron and positronPositronThe positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
emission the photon's energy comes from the electron/nucleonNucleonIn physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...
pair, with the spectrum of the bremsstrahlung decreasing continuously with increasing energy of the beta particle. In electron capture the energy comes at the expense of the neutrinoNeutrinoA neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
, and the spectrum is greatest at about one third of the normal neutrino energy, reaching zero at zero energy and at normal neutrino energy.
Inner and outer bremsstrahlung
The "inner" bremsstrahlung arises from the creation of the electron and its loss of energy (due to the strong electric fieldElectric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
in the region of the nucleus undergoing decay) as it leaves the nucleus. This is to be contrasted with the "outer" bremsstrahlung due to the impingement on the nucleus of electrons coming from the outside (i.e., emitted by another nucleus).
Radiation safety
In some cases, e.g. , the bremsstrahlung produced by shielding the beta radiation with the normally used dense materials (e.g. leadLeadLead is a main-group element in the carbon group with the symbol Pb and atomic number 82. Lead is a soft, malleable poor metal. It is also counted as one of the heavy metals. Metallic lead has a bluish-white color after being freshly cut, but it soon tarnishes to a dull grayish color when exposed...
) is itself dangerous; in such cases, shielding must be accomplished with low density materials, e.g. Plexiglass (lucite), plasticPlasticA plastic material is any of a wide range of synthetic or semi-synthetic organic solids used in the manufacture of industrial products. Plastics are typically polymers of high molecular mass, and may contain other substances to improve performance and/or reduce production costs...
, woodWoodWood is a hard, fibrous tissue found in many trees. It has been used for hundreds of thousands of years for both fuel and as a construction material. It is an organic material, a natural composite of cellulose fibers embedded in a matrix of lignin which resists compression...
, or waterWaterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
; because the rate of deceleration of the electron is slower, the radiation given off has a longer wavelengthWavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
and is therefore less penetrating.
In astrophysics
The dominant luminous component in a cluster of galaxies is the 107 to 108 kelvin intracluster medium. The emission from the intracluster medium is characterized by thermal bremsstrahlung. This radiation is in the energy range of X-rays and can be easily observed with space-based telescopes such as Chandra X-ray ObservatoryChandra X-ray ObservatoryThe Chandra X-ray Observatory is a satellite launched on STS-93 by NASA on July 23, 1999. It was named in honor of Indian-American physicist Subrahmanyan Chandrasekhar who is known for determining the maximum mass for white dwarfs. "Chandra" also means "moon" or "luminous" in Sanskrit.Chandra...
, XMM-NewtonXMM-NewtonThe XMM-Newton is an orbiting X-ray observatory launched by ESA in December 1999 on a Ariane 5 rocket...
, ROSATROSATROSAT was a German Aerospace Center-led satellite X-ray telescope, with instruments built by Germany, the UK and the US...
, ASCAAdvanced Satellite for Cosmology and AstrophysicsASCA is the fourth cosmic X-ray astronomy mission by Japan's , and the second for which the United States is providing part of the scientific payload. The satellite was successfully launched on February 20, 1993. The first eight months of the ASCA mission were devoted to performance verification...
, EXOSATEXOSATThe European X-ray Observatory Satellite , originally named HELOS, was operational from May 1983 until April 1986 and in that time made 1780 observations in the X-ray band of most classes of astronomical object including active galactic nuclei, stellar coronae, cataclysmic variables, white dwarfs,...
, Suzaku, RHESSIReuven Ramaty High Energy Solar Spectroscopic ImagerReuven Ramaty High Energy Solar Spectroscopic Imager is the sixth mission in the line of NASA Small Explorer missions...
and future missions like IXO http://constellation.gsfc.nasa.gov/ and Astro-H http://www.astro.isas.ac.jp/future/NeXT.
Bremsstrahlung is also the dominant emission mechanism for HII Regions at radio wavelengths.
See also
- Cyclotron radiationCyclotron radiationCyclotron radiation is electromagnetic radiation emitted by moving charged particles deflected by a magnetic field. The Lorentz force on the particles acts perpendicular to both the magnetic field lines and the particles' motion through them, creating an acceleration of charged particles that...
- Free-electron laser
- Nuclear fusion: bremsstrahlung losses
- Radiation lengthRadiation lengthIn physics, the radiation length is a characteristic of a material, related to the energy loss of high energy, electromagnetic-interacting particles with it.-Definition:High-energy electrons predominantly lose energy in matter...
characterising energy loss by bremsstrahlung by high energy electrons in matter - Synchrotron lightSynchrotron lightA synchrotron light source is a source of electromagnetic radiation produced by a synchrotron, which is artificially produced for scientific and technical purposes by specialized particle accelerators, typically accelerating electrons...
- X-rays: History
Further reading
External links
-