Synchrotron radiation


Synchrotron radiation is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.
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In astrophysics, synchrotron emission occur, for instance, due to ultra-relativistic motion of a source around a black hole. When the source performs a circular geodesic around the black hole, the synchrotron radiation occur for orbits close to the photosphere where the motion is in ultra-relativistic regime.

History

Syncradiation was named after it was discovered in Schenectady, New York from a General Electric synchrotron accelerator built in 1946 and announced in May 1947 by Frank Elder, Anatole Gurewitsch, Robert Langmuir and Herb Pollock in a letter entitled "Radiation from Electrons in a Synchrotron". Pollock recounts:

Properties of synchrotron radiation

  1. Broad Spectrum : the users can select the wavelength required for their experiment;
  2. High Flux: high intensity photon beam allows rapid experiments or use of weakly scattering crystals;
  3. High Brilliance: highly collimated photon beam generated by a small divergence and small size source ;
  4. High Stability: submicron source stability;
  5. Polarization: both linear and circular;
  6. Pulsed Time Structure: pulsed length down to tens of picoseconds allows the resolution of process on the same time scale.

    Emission mechanism

When high-energy particles are in acceleration, including electrons forced to travel in a curved path by a magnetic field, synchrotron radiation is produced. This is similar to a radio antenna, but with the difference that, in theory, the relativistic speed will change the observed frequency due to the Doppler effect by the Lorentz factor,. Relativistic length contraction then bumps the frequency observed by another factor of, thus multiplying the GHz frequency of the resonant cavity that accelerates the electrons into the X-ray range. The radiated power is given by the relativistic Larmor formula while the force on the emitting electron is given by the Abraham–Lorentz–Dirac force.
The radiation pattern can be distorted from an isotropic dipole pattern into an extremely forward-pointing cone of radiation. Synchrotron radiation is the brightest artificial source of X-rays.
The planar acceleration geometry appears to make the radiation linearly polarized when observed in the orbital plane, and circularly polarized when observed at a small angle to that plane. Amplitude and frequency are however focused to the polar ecliptic.

Synchrotron radiation from accelerators

Synchrotron radiation may occur in accelerators either as a nuisance, causing undesired energy loss in particle physics contexts, or as a deliberately produced radiation source for numerous laboratory applications.
Electrons are accelerated to high speeds in several stages to achieve a final energy that is typically in the GeV range. In the LHC proton bunches also produce the radiation at increasing amplitude and frequency as they accelerate with respect to the vacuum field, propagating photoelectrons, which in turn propagate secondary electrons from the pipe walls with increasing frequency and density up to 7×1010. Each proton may lose 6.7 keV per turn due to this phenomenon.

Synchrotron radiation in astronomy

Synchrotron radiation is also generated by astronomical objects, typically where relativistic electrons spiral through magnetic fields.
Two of its characteristics include non-thermal power-law spectra, and polarization. It is considered to be one of the most powerful tools in the study of extra-solar magnetic fields wherever relativistic charged particles are present. Most known cosmic radio sources emit synchrotron radiation. It is often used to estimate the strength of large cosmic magnetic fields as well as analyze the contents of the interstellar and intergalactic media.

History of detection

It was first detected in a jet emitted by Messier 87 in 1956 by Geoffrey R. Burbidge, who saw it as confirmation of a prediction by Iosif S. Shklovsky in 1953, but it had been predicted earlier by Hannes Alfvén and Nicolai Herlofson in 1950.
Solar flares accelerate particles that emit in this way, as suggested by R. Giovanelli in 1948 and described critically by J.H. Piddington in 1952.
T. K. Breus noted that questions of priority on the history of astrophysical synchrotron radiation are complicated, writing:
.' The bluish glow from the central region of the nebula is due to synchrotron radiation.
Supermassive black holes have been suggested for producing synchrotron radiation, by ejection of jets produced by gravitationally accelerating ions through the super contorted 'tubular' polar areas of magnetic fields. Such jets, the nearest being in Messier 87, have been confirmed by the Hubble telescope as apparently superluminal, travelling at from our planetary frame. This phenomenon is caused because the jets are travelling very near the speed of light and'' at a very small angle towards the observer. Because at every point of their path the high-velocity jets are emitting light, the light they emit does not approach the observer much more quickly than the jet itself. Light emitted over hundreds of years of travel thus arrives at the observer over a much smaller time period giving the illusion of faster than light travel. There is no violation of special relativity.

Pulsar wind nebulae

A class of astronomical sources where synchrotron emission is important is the pulsar wind nebulae, a.k.a. plerions, of which the Crab nebula and its associated pulsar are archetypal.
Pulsed emission gamma-ray radiation from the Crab has recently been observed up to ≥25 GeV, probably due to synchrotron emission by electrons trapped in the strong magnetic field around the pulsar.
Polarization in the Crab at energies from 0.1 to 1.0 MeV illustrates a typical synchrotron radiation.

Interstellar and Intergalactic Media

Much of what is known about the magnetic environment of the interstellar medium and intergalactic medium is derived from observations of synchrotron radiation. Cosmic ray electrons moving through the medium interact with relativistic plasma and emit synchrotron radiation which is detected on Earth. The properties of the radiation allow astronomers to make inferences about the magnetic field strength and orientation in these regions, however accurate calculations of field strength cannot be made without knowing the relativistic electron density.

Formulation

Liénard–Wiechert Field

We start with the expressions for the Liénard–Wiechert field of a point charge of mass and charge :
where, and , which is the unit vector between the observation point and the position of the charge at the retarded time, and is the retarded time.
In equation, and, the first terms for B and E resulting from the particle fall off as the inverse square of the distance from the particle, and this first term is called the generalized Coulomb field or velocity field. These terms represents the particle static field effect, which is a function of the component of its motion that has zero or constant velocity, as seen by a distant observer at r. By contrast, the second terms fall off as the inverse first power of the distance from the source, and these second terms are called the acceleration field or radiation field because they represent components of field due to the charge's acceleration, and they represent E and B which are emitted as electromagnetic radiation from the particle to an observer at r.
If we ignore the velocity field in order to find the power of emitted EM radiation only, the radial component of Poynting's vector resulting from the Liénard–Wiechert fields can be calculated to be
Note that
The energy radiated into per solid angle during a finite period of acceleration from to is
Integrating Eq. over the all solid angles, we get the relativistic generalization of Larmor's formula
However, this also can be derived by relativistic transformation of the 4-acceleration in Larmor's formula.

Velocity perpendicular to acceleration (v ⟂ a): synchrotron radiation

When the charge is in instantaneous circular motion, its acceleration is perpendicular to its velocity. Choosing a coordinate system such that instantaneously is in the direction and is in the direction, with the polar and azimuth angles and defining the direction of observation, the general formula Eq. reduces to
In the relativistic limit, the angular distribution can be written approximately as
The factors in the denominators tip the angular distribution forward into a narrow
cone like the beam of a headlight pointing ahead of the particle. A plot of the angular distribution shows a sharp peak around.
If we neglect any electric force on the particle, the total power radiated from Eq. is
where is the particle's total energy, is the magnetic field, and is the radius of curvature of the track in the field. Note that the radiated power is proportional to,, and. In some cases the surfaces of vacuum chambers hit by synchrotron radiation have to be cooled because of the high power of the radiation.
Using
where is the angle between the velocity and the magnetic field and is the radius of the circular acceleration, the power emitted is:
Thus the power emitted scales as energy to the fourth, and decreases with the square of the radius and the fourth power of particle mass. This radiation is what limits the energy of an electron-positron circular collider. Generally, proton-proton colliders are instead limited by the maximum magnetic field; this is why, for example, the LHC has a center-of-mass energy 70 times higher than the LEP even though the proton mass is 2000 times the electron mass.

Radiation integral

The energy received by an observer is
Using the Fourier transformation we move to the frequency space
Angular and frequency distribution of the energy received by an observer
Therefore, if we know the particle's motion, cross products term, and phase factor, we could calculate the radiation integral. However, calculations are generally quite lengthy.

Example 1: bending magnet

Integrating

Trajectory of the arc of circumference is
In the limit of small angles we compute
Substituting into the radiation integral and introducing
where the function is a modified Bessel function of the second kind.

Frequency distribution of radiated energy

From Eq., we observe that the radiation intensity is negligible for. Critical frequency is defined as the frequency when and. So,
and critical angle is defined as the angle for which and is approximately
For frequencies much larger than the critical frequency and angles much larger than the critical angle, the synchrotron radiation emission is negligible.
Integrating on all angles, we get the frequency distribution of the energy radiated.
If we define
where. Then
Note that, if, and, if
The formula for spectral distribution of synchrotron radiation, given above, can be expressed in terms of a rapidly converging integral with no special functions involved by means of the relation:

Synchrotron radiation emission as a function of the beam energy

First, define the critical photon energy as
Then, the relationship between radiated power and photon energy is shown in the graph on the right side. The higher the critical energy, the more photons with high energies are generated. Note that, there is no dependence on the energy at longer wavelength.

Polarization of synchrotron radiation

In Eq., the first term is the radiation power with polarization in the orbit plane, and the second term is the polarization orthogonal to the orbit plane.
In the orbit plane, the polarization is purely horizontal. Integrating on all frequencies, we get the angular distribution of the energy radiated
Integrating on all the angles, we find that seven times as much energy is radiated with parallel polarization as with perpendicular polarization. The radiation from a relativistically moving charge is very strongly, but not completely, polarized in the plane of motion.

Example 2: undulator

Solution of equation of motion and undulator equation

An undulator consists of a periodic array of magnets, so that they provide a sinusoidal magnetic field.
Solution of equation of motion is
where
and
and the parameter is called the undulator parameter.
Condition for the constructive interference of radiation emitted at different poles is
Expanding and neglecting the terms in the resulting equation, one obtains
For, one finally gets
This equation is called the undulator equation.

Radiation from the undulator

Radiation integral is
Using the periodicity of the trajectory, we can split the radiation integral into a sum over terms, where is the total number of bending magnets of the undulator.
where 　
, and
,　,　and　

The radiation integral in an undulator can be written as
where is the frequency difference to the n-th harmonic.
The sum of generates a series of sharp peaks in the frequency spectrum harmonics of fundamental wavelength
and depends on the angles of observations and
On the axis, the radiation integral becomes
and
where
Note that only odd harmonics are radiated on-axis, and as increases higher harmonic becomes stronger.