### Synchrotron light

The electromagnetic radiation emitted when charged particles are accelerated radially ($\backslash mathbf\{a\}\backslash perp\; \backslash mathbf\{v\}$) is called **synchrotron radiation.** It is produced in synchrotrons using bending magnets, undulators and/or wigglers. It is similar to cyclotron radiation except that synchrotron radiation is generated by the acceleration of ultrarelativistic charged particles through magnetic fields. 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.

## Contents

- 1 History
- 2 Properties of synchrotron radiation
- 3 Emission mechanism
- 4 Synchrotron radiation from accelerators
- 5 Synchrotron radiation in astronomy
- 6 Formulation
- 7 See also
- 8 Notes
- 9 External links
- 10 References

## History

Synchrotron radiation was named after its discovery in 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".^{[1]} Pollock recounts:

- "On April 24, Langmuir and I were running the machine and as usual were trying to push the electron gun and its associated pulse transformer to the limit. Some intermittent sparking had occurred and we asked the technician to observe with a mirror around the protective concrete wall. He immediately signaled to turn off the synchrotron as "he saw an arc in the tube." The vacuum was still excellent, so Langmuir and I came to the end of the wall and observed. At first we thought it might be due to Cherenkov radiation, but it soon became clearer that we were seeing Ivanenko and Pomeranchuk radiation."
^{[2]}

## Properties of synchrotron radiation

- Broad Spectrum (which covers from microwaves to hard X-rays): the users can select the wavelength required for their experiment.
- High Flux: high intensity photon beam allows rapid experiments or use of weakly scattering crystals.
- High Brilliance: highly collimated photon beam generated by a small divergence and small size source (spatial coherence)
- High Stability: submicron source stability
- Polarization: both linear and circular
- 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 rapid motion, 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, $\backslash gamma$. Relativistic length contraction then bumps the frequency observed in the lab by another factor of $\backslash gamma$, 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 focussed 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 7x10^{10}. Each proton may lose 6.7keV per turn due to this phenomenon.^{[3]}

## Synchrotron radiation in astronomy

Synchrotron radiation is also generated by astronomical objects, typically where relativistic electrons spiral (and hence change velocity) through magnetic fields.
Two of its characteristics include non-thermal power-law spectra, and polarization.^{[4]}

### History of detection

It was first detected in a jet emitted by Messier 87 in 1956 by Geoffrey R. Burbidge,^{[5]} 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 ^{[6]} in 1950.

T. K. Breus noted that questions of priority on the history of astrophysical synchrotron radiation is complicated, writing:

- "In particular, the Russian physicist V.L. Ginzburg broke his relationships with I.S. Shklovsky and did not speak with him for 18 years. In the West, Thomas Gold and Sir Fred Hoyle were in dispute with H. Alfven and N. Herlofson, while K.O. Kiepenheuer and G. Hutchinson were ignored by them."
^{[7]}

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 6×c (six times the speed of light) 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 (ten or twenty years) giving the illusion of faster than light travel. There is no violation of special relativity.^{[8]}

### 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,^{[9]} probably due to synchrotron emission by electrons trapped in the strong magnetic field around the pulsar.
Polarization in the Crab^{[10]} at energies from 0.1 to 1.0 MeV illustrates a typical synchrotron radiation.

## Formulation

### Liénard–Wiechert Field

We start with the expressions for the Liénard–Wiechert field :

- $\backslash mathbf\{B\}(\backslash mathbf\{r\},t)=-\backslash frac\{\backslash mu\_0q\}\{4\backslash pi\}\backslash left[\backslash frac\{c\backslash ,\backslash hat\{\backslash mathbf\{n\}\}\backslash times\backslash vec\{\backslash beta\}\}\{\backslash gamma^2R^2(1-\backslash vec\{\backslash beta\}\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\})^3\}+\backslash frac\{\backslash hat\{\backslash mathbf\{n\}\}\backslash times[\backslash ,\backslash dot\{\backslash vec\{\backslash beta\}\}+\backslash hat\{\backslash mathbf\{n\}\}\backslash times(\backslash vec\{\backslash beta\}\backslash times\backslash dot\{\backslash vec\{\backslash beta\}\})]\}\{R\backslash ,(1-\backslash vec\{\backslash beta\}\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\})^3\}\backslash right]\_\{\backslash mathrm\{retarded\}\}\; \backslash qquad\; (1)$
- $\backslash mathbf\{E\}(\backslash mathbf\{r\},t)=\backslash frac\{q\}\{4\backslash pi\backslash varepsilon\_0\}\backslash left[\backslash frac\{\backslash hat\{\backslash mathbf\{n\}\}-\backslash vec\{\backslash beta\}\}\{\backslash gamma^2R^2(1-\backslash vec\{\backslash beta\}\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\})^3\}+\backslash frac\{\backslash hat\{\backslash mathbf\{n\}\}\backslash times[(\backslash hat\{\backslash mathbf\{n\}\}-\backslash vec\{\backslash beta\})\backslash times\backslash dot\{\backslash vec\{\backslash beta\}\}\backslash ,]\}\{c\backslash ,R\backslash ,(1-\backslash vec\{\backslash beta\}\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\})^3\}\backslash right]\_\{\backslash mathrm\{retarded\}\}\; \backslash qquad\; \backslash qquad\; (2)$

where

- $\backslash mathbf\; R(t\text{'})=\backslash mathbf\; r-\backslash mathbf\; r\_0(t\text{'}),$

- $R(t\text{'})=|\backslash mathbf\; R|,$

- $\backslash hat\{\backslash mathbf\; n\}(t\text{'})=\backslash mathbf\; R/R,$

which is the unit vector between the observation point and the position of the charge at the retarded time, and $t\text{'}$ is the retarded time.

In equation (1), and (2), the first terms 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*. And the second terms fall off as the inverse *first* power of the distance from the source, and it is called the *radiation field* or *acceleration field*.
If we ignore the *velocity* field, the radial component of Poynting's Vector resulted from the Liénard–Wiechert field can be calculated to be

- $[\backslash mathbf\{S\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\}]\_\{\backslash mathrm\{retarded\}\}\; =\; \backslash frac\{q^2\}\{16\backslash pi^2\backslash varepsilon\_0\; c\}\backslash left\backslash \{\backslash frac\{1\}\{R^2\}\backslash left|\backslash frac\{\backslash hat\{\backslash mathbf\{n\}\}\backslash times[(\backslash hat\{\backslash mathbf\{n\}\}-\backslash vec\{\backslash beta\})\backslash times\backslash dot\{\backslash vec\{\backslash beta\}\}]\}\{(1-\backslash vec\{\backslash beta\}\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\})^3\}\backslash right|^2\backslash right\backslash \}\_\{\backslash text\{not\; retarded\}\}\; \backslash qquad\; \backslash qquad\; (3)$

Note that

- The spatial relationship between $\backslash vec\{\backslash beta\}$ and $\backslash dot\{\backslash vec\{\backslash beta\}\}$ determines the detailed angular power distribution.
- The relativistic effect of transforming from the rest frame of the particle to the observer's frame manifests itself by the presence of the factors $(1-\backslash vec\{\backslash beta\}\backslash mathbf\{\backslash cdot\}\backslash vec\{\backslash mathbf\{n\}\})$ in the denominator of Eq. (3).
- For ultrarelativistic particles the latter effect dominates the whole angular distribution.

The energy radiated into per solid angle during a finite period of acceleration from $t\text{'}=T\_1$ to $t\text{'}=T\_2$ is

- $\backslash frac\{\backslash mathrm\{d\}P\}\{\backslash mathrm\{d\}\backslash mathit\{\backslash Omega\}\}\; =\; R(t\text{'})^2\backslash ,[\backslash mathbf\{S\}(t\text{'})\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\}(t\text{'})]\backslash ,\backslash frac\{\backslash mathrm\{d\}t\}\{\backslash mathrm\{d\}t\text{'}\}\; =\; R(t\text{'})^2\backslash ,\backslash mathbf\{S\}(t\text{'})\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\}(t\text{'})\backslash ,[1-\backslash vec\{\backslash beta\}(t\text{'})\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\}(t\text{'})]$
- $=\; \backslash frac\{q^2\}\{16\backslash pi^2\backslash varepsilon\_0c\}\backslash ,\backslash frac\{|\backslash hat\{\backslash mathbf\{n\}\}(t\text{'})\backslash times\backslash (t\text{'})-\backslash vec\{\backslash beta\}(t\text{'})]\backslash times\backslash dot\{\backslash vec\{\backslash beta\}\}(t\text{'})\backslash \}|^2\}(t\text{'})]^5\}\; \backslash qquad\; \backslash qquad\; (4)$

- $\backslash frac\{\backslash mathrm\{d\}P\}\{\backslash mathrm\{d\}\backslash mathit\{\backslash Omega\}\}\; =\; R(t\text{'})^2\backslash ,[\backslash mathbf\{S\}(t\text{'})\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\}(t\text{'})]\backslash ,\backslash frac\{\backslash mathrm\{d\}t\}\{\backslash mathrm\{d\}t\text{'}\}\; =\; R(t\text{'})^2\backslash ,\backslash mathbf\{S\}(t\text{'})\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\}(t\text{'})\backslash ,[1-\backslash vec\{\backslash beta\}(t\text{'})\backslash mathbf\{\backslash cdot\}\backslash hat\{\backslash mathbf\{n\}\}(t\text{'})]$

Integrating Eq. (4) over the all solid angles, we get **relativistic generalization of Larmor's formula**

- $P=\backslash frac\{e^2\}\{6\backslash pi\; \backslash varepsilon\; \_0\; c\}\backslash gamma\; ^6$

\left [ \left | \dot{\vec{\beta }} \right |^2 -\left | \vec{\beta}\times \dot{\vec{\beta }}\right |^2 \right ]\qquad (5) However, this also can be derived by relativistic transformation of the 4-acceleration in Larmor's formula.

### Velocity ⊥ acceleration: synchrotron radiation

When the charge is in instantaneous circular motion, its acceleration $\backslash dot\{\backslash vec\{\backslash beta\}\}$ is perpendicular to its velocity $\backslash vec\{\backslash beta\}$. Choosing a coordinate system such that instantaneously $\backslash vec\{\backslash beta\}$ is in the z direction and $\backslash dot\{\backslash vec\{\backslash beta\}\}$ is in the x direction, with the polar and azimuth angles $\backslash theta$ and $\backslash phi$ defining the direction of observation, the general formula Eq. (4) reduces to

- $\backslash frac\{\backslash mathrm\{d\}P\}\{\backslash mathrm\{d\}\backslash mathit\{\backslash Omega\}\}\; =\; \backslash frac\{q^2\}\{16\backslash pi^2\backslash epsilon\_0\; c\}\backslash frac\{|\backslash dot\{\backslash vec\{\backslash beta\; \}\}|^2\}\{(1-\backslash beta\backslash cos\backslash theta)^3\}\backslash left[1-\backslash frac\{\backslash sin^2\backslash theta\backslash cos^2\backslash phi\}\{\backslash gamma^2(1-\backslash beta\backslash cos\backslash theta)^2\}\backslash right].\; \backslash qquad\; (6)$

In the relativistic limit $(\backslash gamma\backslash gg\; 1)$, the angular distribution can be written approximately as

- $\backslash frac\{\backslash mathrm\{d\}P\}\{\backslash mathrm\{d\}\backslash mathit\{\backslash Omega\}\}\; \backslash simeq\; \backslash frac\{2\}\{\backslash pi\}\backslash frac\{e^2\}\{c^3\}\backslash gamma^6\backslash frac\{|\backslash dot\{\backslash mathbf\; v\}|^2\}\{(1+\backslash gamma^2\backslash theta^2)^3\}\backslash left[1-\backslash frac\{4\backslash gamma^2\backslash theta^2\backslash cos^2\backslash phi\}\{(1+\backslash gamma^2\backslash theta^2)^2\}\backslash right].\; \backslash qquad\; \backslash qquad\; (7)$

The factors $(1-\backslash beta\backslash cos\backslash theta)$ 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 (dP/dΩ vs. γθ) shows a sharp peak around θ=0.

Integration over the whole solid angle yields the total power radiated by one electron

- $P=\backslash frac\{e^2\}\{6\backslash pi\backslash epsilon\_0c\}\backslash left\; |\; \backslash dot\{\backslash vec\{\backslash beta\; \}\}\; \backslash right\; |^2\backslash gamma\; ^4=\backslash frac\{e^2c\}\{6\backslash pi\backslash epsilon\_0\}\backslash frac\{\backslash gamma\; ^4\}\{\backslash rho\; ^2\}=\backslash frac\{e^4\}\{6\backslash pi\backslash epsilon\_0m^4c^5\}E^2B^2,\backslash qquad\; (8)$

where E is the electron energy, B is the magnetic field, and ρ is the radius of curvature of the track in the field. Note that the radiated power is proportional to $1/m^4$, $1/\backslash rho^2$, and $B^2$. In some cases the surfaces of vacuum chambers hit by synchrotron radiation have to be cooled because of the high power of the radiation.

### Radiation integral

The energy received by an observer (per unit solid angle at the source) is

$\backslash frac\{d^2W\}\{d\backslash Omega\; \}=\backslash int\_\{-\backslash infty\; \}^\{\backslash infty\; \}\backslash frac\{d^2P\}\{d\backslash Omega\; \}dt=c\backslash varepsilon\; \_0\backslash int\_\{-\backslash infty\; \}^\{\backslash infty\; \}\backslash left\; |\; R\backslash vec\{E\}(t)\; \backslash right\; |^2dt$

Using the Fourier Transformation we move to the frequency space $\backslash frac\{d^2W\}\{d\backslash Omega\; \}=2c\backslash varepsilon\; \_0\backslash int\_\{0\; \}^\{\backslash infty\; \}\backslash left\; |\; R\backslash vec\{E\}(\backslash omega)\; \backslash right\; |^2d\backslash omega$

Angular and frequency distribution of the energy received by an observer (consider only the radiation field)

$\backslash frac\{d^3W\}\{d\backslash Omega\; d\backslash omega\; \}=2c\backslash varepsilon\; \_0R^2\backslash left\; |\; \backslash vec\{E\}(\backslash omega)\; \backslash right\; |^2=\backslash frac\{e^2\}\{4\backslash pi\backslash varepsilon\_0\; 4\backslash pi^2\; c\}\backslash left\; |\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash frac\{\backslash hat\{n\}\backslash times\backslash left\; [\; \backslash left\; (\; \backslash hat\{n\}-\backslash vec\{\backslash beta\; \}\; \backslash right\; )\backslash times\backslash dot\{\backslash vec\{\backslash beta\; \}\}\; \backslash right\; ]\}\{\backslash left\; (\; 1-\backslash hat\{n\}\backslash cdot\; \backslash vec\{\backslash beta\; \}\; \backslash right\; )^2\}e^\{i\backslash omega(t-\backslash hat\{n\}\backslash cdot\backslash vec\{r\}(t)/c)\}dt\backslash right\; |^2\; \backslash qquad\; (9)$

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 (even for simple cases as for the radiation emitted by an electron in a bending magnet, they require Airy function or the modified Bessel functions).

### Example 1: bending magnet

#### Integrating

Trajectory of the arc of circumference is

$\backslash vec\{r\}(t)=\backslash left\; (\; \backslash rho\; \backslash sin\backslash frac\{\backslash beta\; c\}\{\backslash rho\}t,\; \backslash rho\backslash left\; (\; 1-\backslash cos\backslash frac\{\backslash beta\; c\}\{\backslash rho\}t\; \backslash right\; ),\; 0\; \backslash right)$

In the limit of small angles we compute

$\backslash hat\{n\}\backslash times\backslash left\; (\; \backslash hat\{n\}\backslash times\backslash vec\{\backslash beta\}\; \backslash right\; )\; =\backslash beta\backslash left\; [\; -\backslash vec\{\backslash varepsilon\; \}\_\backslash parallel\; \backslash sin\backslash left\; (\; \backslash frac\{\backslash beta\; c\; t\}\{\backslash rho\}\; \backslash right\; )+\backslash vec\{\backslash varepsilon\}\_\backslash perp\; \backslash cos\backslash left\; (\; \backslash frac\{\backslash beta\; c\; t\}\{\backslash rho\}\backslash right\; )\backslash sin\backslash theta\; \backslash right\; ]$

$\backslash omega\backslash left\; (\; t-\backslash frac\{\backslash hat\{n\}\backslash cdot\; \backslash vec\{r\}(t)\}\{c\}\; \backslash right\; )\; =\; \backslash omega\backslash left\; [\; t-\backslash frac\{\backslash rho\}\{c\}\backslash sin\backslash left\; (\; \backslash frac\{\backslash beta\; c\; t\}\{\backslash rho\}\; \backslash right\; )\backslash cos\backslash theta\; \backslash right\; ]$

Substituting into the radiation integral and introducing $\backslash xi\; =\backslash frac\{\backslash rho\; \backslash omega\}\{3c\backslash gamma^3\}\backslash left\; (\; 1+\backslash gamma^2\; \backslash theta^2\; \backslash right\; )^\{3/2\}$

$\backslash frac\{d^3\; W\}\{d\backslash Omega\; d\backslash omega\}=\backslash frac\{e^2\}\{16\backslash pi^3\backslash varepsilon\_0\; c\}\; \backslash left\; (\; \backslash frac\{2\backslash omega\backslash rho\}\{3c\backslash gamma^2\}\; \backslash right\; )^2\; \backslash left\; (\; 1+\backslash gamma^2\; \backslash theta^2\; \backslash right\; )^2\; \backslash left\; [\; K\_\{2/3\}^2(\backslash xi\; )\; +\; \backslash frac\{\backslash gamma^2\; \backslash theta^2\}\{1+\backslash gamma^2\; \backslash theta^2\}K\_\{1/3\}^2(\backslash xi)\backslash right\; ]\backslash qquad\; (10)$

, where the function $K$ is a modified Bessel function of the second kind.

#### Frequency distribution of radiated energy

From Eq.(10), we observe that the radiation intensity is negligible for $\backslash xi\backslash gg\; 1$.
*Critical frequency* is defined as the frequency when $\backslash xi=\backslash frac\{1\}\{2\}$ and $\backslash theta=0$. So,

- $\backslash omega\_\backslash text\{c\}=\backslash frac\{3\}\{2\}\backslash frac\{c\}\{\backslash rho\}\backslash gamma^3$

, and *critical angle is defined as*

- $\backslash theta\_c=\backslash frac\{1\}\{\backslash gamma\}\backslash left\; (\; \backslash frac\{\backslash omega\_\backslash text\{c\}\}\{\backslash omega\}\; \backslash right\; )^\{1/3\}$

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.

- $\backslash frac\{dW\}\{d\backslash omega\}=\backslash oint\; \backslash frac\{d^3\; W\}\{d\backslash omega\; d\backslash Omega\; \}d\backslash Omega$

=\frac{\sqrt{3}e^2}{4\pi\varepsilon_0 c}\gamma\frac{\omega}{\omega_\text{c}}\int_{\omega/\omega_\text{c}}^{\infty}K_{5/3}(x)dx

If we define

- $S(y)\backslash equiv\; \backslash frac\{9\backslash sqrt\{3\}\}\{8\backslash pi\}y\backslash int\_\{y\}^\{\backslash infty\}K\_\{5/3\}(x)dx$ $\backslash int\_\{0\}^\{\backslash infty\}S(y)dy=1$

, where $y=\backslash frac\{\backslash omega\}\{\backslash omega\_\backslash text\{c\}\}$. Then,

- $\backslash frac\{dW\}\{d\backslash omega\}=\backslash frac\{2e^2\; \backslash gamma\}\{9\; \backslash varepsilon\_0c\}S(y)\backslash qquad\; (11)$

Note that $\backslash frac\{dW\}\{d\backslash omega\}\backslash sim\; \backslash frac\{e^2\}\{4\backslash pi\; \backslash varepsilon\_0c\}\backslash left\; (\; \backslash frac\{\backslash omega\; \backslash rho\}\{c\}\; \backslash right\; )^\{1/3\}$, if $\backslash omega\backslash ll\; \backslash omega\_\backslash text\{c\}$, and $\backslash frac\{dW\}\{d\backslash omega\}\backslash approx\; \backslash sqrt\{\backslash frac\{3\backslash pi\}\{2\}\}\backslash frac\{e^2\}\{4\backslash pi\backslash varepsilon\_0\; c\}\backslash gamma\backslash left\; (\; \backslash frac\{\backslash omega\}\{\backslash omega\_\backslash text\{c\}\}\; \backslash right\; )^2\; e^\{-\backslash omega/\backslash omega\_\backslash text\{c\}\}$, if $\backslash omega\backslash gg\; \backslash omega\_\backslash text\{c\}$

Formula for spectral distribution of synchrotron radiation, given above, can be expressed in terms of rapidly coverged integral with no special functions involved ^{[11]} (see also modified Bessel functions ) by means of the relation:

- $$

\int_{\xi}^\infty K_{5/3} (x) dx = \frac{1}{ \sqrt{3}} \, \int_0^\infty \, \frac{9+36x^2+16x^4}{(3+4x^2) \sqrt{1+x^2/3}} \exp \left[- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right] \ dx

#### Synchrotron radiation emission as a function of the beam energy

First, define the critical photon energy as
$\backslash varepsilon\_c=\backslash hbar\; \backslash omega\_\backslash text\{c\}=\backslash frac\{3\}\{2\}\backslash frac\{\backslash hbar\; c\}\{\backslash rho\}\backslash gamma^3$

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.(10), the first term $K\_\{2/3\}^2(\backslash xi)$ is the radiation power with polarization in the orbit plane, and the second term $\backslash frac\{\backslash gamma^2\; \backslash theta^2\}\{1+\backslash gamma^2\; \backslash theta^2\}K\_\{1/3\}^2(\backslash xi)$ is the polarization orthogonal to the orbit plane. In the orbit plane $\backslash theta=0$, the polarization is purely horizontal. Integrating on all frequencies, we get the angular distribution of the energy radiated

- $$

\frac{d^2 W}{d\Omega }=\int_{0}^{\infty}\frac{d^3W}{d\omega d\Omega }d\omega =\frac{7e^2 \gamma^5}{64\pi\varepsilon_0\rho}\frac{1}{(1+\gamma^2\theta^2)^{5/2}}\left [1+\frac{5}{7}\frac{\gamma^2\theta^2}{1+\gamma^2\theta^2} \right ] \qquad (12)

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.

- $\backslash vec\{B\}=\backslash left\; (\; 0,\; B\_0\; \backslash sin(k\_\backslash text\{u\}\; z),0\; \backslash right\; )$

Solution of equation of motion is

- $\backslash vec\{r\}(t)=\backslash frac\{\backslash lambda\_\backslash text\{u\}\; K\}\{2\backslash pi\backslash gamma\}\backslash sin\; \backslash omega\_\backslash text\{u\}t\backslash cdot\; \backslash hat\{x\}$

+\left ( \bar{\beta_z}ct+\frac{\lambda_\text{u}K^2}{16\pi\gamma^2}\cos(2\omega_\text{u}t) \right )\cdot \hat{z}

where, $K=\backslash frac\{eB\_0\backslash lambda\_\backslash text\{u\}\}\{2\backslash pi\; mc\}$ , and $\backslash bar\{\backslash beta\_z\}=1-\backslash frac\{1\}\{2\backslash gamma^2\}\backslash left\; (\; 1+\backslash frac\{K^2\}\{2\}\; \backslash right\; )$

, and the parameter $K$ is called the *undulator parameter*.

Condition for the constructive interference of radiation emitted at different poles is

- $d=\backslash frac\{\backslash lambda\_\backslash text\{u\}\}\{\backslash bar\{\backslash beta\}\}-\backslash lambda\_\backslash text\{u\}\backslash cos\backslash theta=n\backslash lambda$

Therefore,

- $\backslash lambda\_n\; =\; \backslash frac\{\backslash lambda\_\backslash text\{u\}\}\{2\backslash gamma^2n\}\backslash left\; (\; 1+\backslash frac\{K^2\}\{2\}+\backslash gamma^2\backslash theta^2\; \backslash right\; )\backslash qquad\; (13)$

This equation is called the *undulator equation*.

#### Radiation from the undulator

Radiation integral is

- $$

\frac{d^3W}{d\Omega d\omega}=\frac{e^2}{4\pi\varepsilon_0 4\pi^2 c}\left|\int_{-\infty}^{\infty}\frac{\hat{n}\times\left[\left(\hat{n}-\vec{\beta}\right)\times\dot{\vec{\beta}}\right]}{\left(1-\hat{n}\cdot\vec{\beta}\right)^2}e^{i\omega(t-\hat{n}\cdot\vec{r}(t)/c)}dt\right|^2

Using the periodicity of the trajectory, we can split the radiation integral into a sum over $N\_u$ terms.

- $$

\frac{d^3W}{d\Omega d\omega }=\frac{e^2\omega^2}{4\pi\varepsilon_0 4\pi^2 c}\left | \int_{-\lambda_u/2\bar{\beta}c}^{\lambda_u/2\bar{\beta}c}\hat{n}\times\left ( \hat{n}\times\vec{\beta} \right )e^{i\omega(t-\hat{n}\cdot\vec{r}(t)/c)}dt\right|^2 \left|1+e^{i\delta}+e^{2i\delta}+\cdots +e^{i(N_u-1)\delta} \right |^2 \qquad (14)

, where $\backslash bar\{\backslash beta\}=\backslash beta\backslash left\; (\; 1-\backslash frac\{K^2\}\{4\backslash gamma^2\}\; \backslash right\; )$

, and $\backslash delta=\backslash frac\{2\backslash pi\; \backslash omega\}\{\backslash omega\_\backslash text\{res\}\; (\backslash theta)\}$, $\backslash omega\_\backslash text\{res\}(\backslash theta)=\backslash frac\{2\backslash pi\; c\}\{\backslash lambda\_\backslash text\{res\}(\backslash theta)\}$, and $\backslash lambda\_\backslash text\{res\}(\backslash theta)=\backslash frac\{\backslash lambda\_u\}\{2\backslash gamma^2\}\backslash left\; (\; 1+\backslash frac\{K^2\}\{2\}+\backslash gamma^2\backslash theta^2\; \backslash right\; )$

The radiation integral in an undulator can be written as

- $$

\frac{d^3 W}{d\Omega d\omega}=\frac{e^2\gamma^2N^2}{4\pi\varepsilon_0 c} L\left ( N\frac{\Delta \omega_n}{\omega_\text{res}(\theta)} \right ) F_n (K, \theta, \phi) \qquad (15)

The sum of $\backslash delta$ generates a series of sharp peaks in the frequency spectrum harmonics of fundamental wavelength

- $L\backslash left\; (\; N\backslash frac\{\backslash Delta\; \backslash omega\_k\}\{\backslash omega\_\backslash text\{res\}(\backslash theta)\}\; \backslash right\; )$

=\frac{\sin^2\left ( N\pi\Delta \omega_k / \omega_\text{res}(\theta) \right )}{N^2 \sin^2\left ( \pi\Delta \omega_k/\omega_\text{res}(\theta) \right )}

, and $F\_n$ depends on the angles of observations and $K$

- $F\_n(K,\backslash theta,\backslash phi)\backslash propto$

\left | \int_{-\lambda_u/2\bar{\beta} c}^{\lambda_u/2\bar{\beta} c}\hat{n}\times\left ( \hat{n}\times\vec{\beta} \right )e^{i\omega(t-\hat{n}\cdot\vec{r}(t)/c)}dt\right|^2

On the axis($\backslash theta=0$, $\backslash phi=0$), the radiation integral becomes

$\backslash frac\{d^3\; W\}\{d\backslash Omega\; d\backslash omega\}=\backslash frac\{e^2\backslash gamma^2N^2\}\{4\backslash pi\backslash varepsilon\_0\; c\}L\backslash left\; (\; N\backslash frac\{\backslash Delta\; \backslash omega\_n\}\{\backslash omega\_\backslash text\{res\}(0)\}\; \backslash right\; )F\_n(K,0,0)$

and,

$F\_n(K,0,0)=\backslash frac\{n^2K^2\}\{1+K^2/2\}\; \backslash left[J\_\{\backslash frac\{n+1\}\{2\}\}(Z)-J\_\{\backslash frac\{n-1\}\{2\}\}(Z)\; \backslash right\; ]^2$

, where $Z=\backslash frac\{nK^2\}\{4(1+K^2/2)\}$

Note that only odd harmonics are radiated on-axis, and as K increases higher harmonic becomes stronger.

## See also

- Bremsstrahlung
- Cyclotron Radiation
- Free-electron laser
- Radiation reaction
- Relativistic beaming
- Sokolov–Ternov effect
- Synchrotron for this type of particle accelerator
- Synchrotron light source for laboratory generation and applications of synchrotron radiation

## Notes

## External links

- Cosmic Magnetobremsstrahlung (synchrotron Radiation), by Ginzburg, V. L., Syrovatskii, S. I., ARAA, 1965
- Developments in the Theory of Synchrotron Radiation and its Reabsorption, by Ginzburg, V. L., Syrovatskii, S. I., ARAA, 1969
- Lightsources.org
- X-Ray Data Booklet

## References

- Brau, Charles A. Modern Problems in Classical Electrodynamics. Oxford University Press, 2004. ISBN 0-19-514665-4.
- Jackson, John David. Classical Electrodynamics. John Wiley & Sons, 1999. ISBN 0-471-30932-X