Synchrotron radiation

As synchrotron radiation is called the electromagnetic radiation , the tangential , of charged particles to the movement direction is radiated when they are deflected from a straight path. Since the deflection represents an acceleration in the physical sense (change in the velocity vector), it is a special form of bremsstrahlung .

Synchrotron radiation becomes relevant at relativistic speeds of the particles close to the speed of light . The name synchrotron radiation comes from its occurrence in synchrotron particle accelerators, in which particles are accelerated to such speeds on circular orbits. Nowadays synchrotron radiation is widely used to determine the structure of a wide variety of materials. One advantage over many other methods is the high level of brilliance with which this radiation can be generated.

Synchrotron radiation can naturally also arise in space during a wide variety of processes. A natural source of synchrotron radiation in space is e.g. B. Jupiter , which bombarded its moons with this type of radiation.

history

Synchrotron radiation was predicted in 1944 by Dmitri Dmitrijewitsch Ivanenko and Isaak Jakowlewitsch Pomerantschuk . At about the same time, their appearance in 1945 was calculated by Julian Schwinger at MIT , who lectured on it in Los Alamos in the summer of 1945 and on other occasions such as the APS meeting in New York in autumn 1946 and published it with David Saxon. At the time, some physicists questioned the existence of such radiation. It was argued that the circulating current of electrons would generate destructive interfering radiation, which Schwinger was able to refute.

First observed experimentally it was in 1946 at a synchrotron by General Electric in the ring accelerating electrons. At first it was only seen as a disruptive loss of energy from the accelerated particles. It was not until the late 1970s that planning began for special accelerators for generating synchrotron radiation.

properties

The loss of energy through synchrotron radiation becomes relevant at relativistic speeds, i.e. when the speed is no longer small compared to the speed of light. Then it is bundled and emitted in the direction of movement of the particles and is largely linearly polarized in the plane of the direction of movement . Their spectrum is continuous and extends from infrared to deep into the X-ray range.

Angle dependence

Coordinate system with polar angle and azimuthal angle , the particle moves in the plane ${\displaystyle \theta }$ ${\displaystyle \phi }$ ${\displaystyle (x,z)}$

The differential radiated power per solid angle element of a charge moving on a circular path is ${\displaystyle \mathrm {d} P/d\Omega }$

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} \Omega }}={\frac {q^{2}}{16\pi ^{2}\varepsilon _{0}c}}{\frac {{\dot {\vec {\beta }}}^{2}}{(1-\beta \cos \theta )^{3}}}\left[1-{\frac {\sin ^{2}\theta \cos ^{2}\phi }{\gamma ^{2}(1-\beta \cos \theta )^{2}}}\right]}$

The coordinate system is chosen so that the moving load is currently at the origin, the velocity vector points in the direction and the acceleration vector points in the direction. Then the azimuthal angle and the polar angle are. The remaining quantities in the equation are: ${\displaystyle z}$ ${\displaystyle x}$ ${\displaystyle \phi }$ ${\displaystyle \theta }$

• the electrical charge of the particle , ${\displaystyle q}$
• the electric field constant , ${\displaystyle \varepsilon _{0}}$
• the speed of light , ${\displaystyle c}$
• the speed of the particle in units of the speed of light and ${\displaystyle {\vec {\beta }}={\vec {v}}/c}$
• the Lorentz factor ${\displaystyle \gamma =(1-\beta ^{2})^{-1/2}}$

For highly relativistic speeds,, the maximum of the radiated power inclines more and more in the forward direction . This borderline case lets itself through ${\displaystyle \beta \to 1}$ ${\displaystyle \theta \to 0}$

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} \Omega }}\approx {\frac {q^{2}}{2\pi ^{2}\varepsilon _{0}c}}\gamma ^{6}{\frac {{\dot {\vec {\beta }}}^{2}}{(1+\gamma ^{2}\theta ^{2})^{3}}}\left[1-{\frac {4\gamma ^{2}\theta ^{2}\cos ^{2}\phi }{(1+\gamma ^{2}\theta ^{2})^{2}}}\right]}$

describe. After averaging over the azimuthal angle , the root of the mean square opening angle results ${\displaystyle \phi }$

${\displaystyle {\sqrt {\overline {\theta ^{2}}}}={\frac {1}{\gamma }}={\frac {mc^{2}}{E}}}$

and decreases with increasing energy of the particle. In the momentary rest system, the radiation takes place according to the characteristics of a Hertzian dipole across the acceleration of the particle. The increasingly sharp bundling along the direction of movement observed in the laboratory system with increasing energy can be understood through a Lorentz transformation . ${\displaystyle E}$

Frequency distribution and polarization

The differential intensity distribution of the radiation of a charged particle follows the Liénard-Wiechert potentials for moving charges as:

${\displaystyle {\frac {\mathrm {d} ^{2}I}{\mathrm {d} \omega \,\mathrm {d} \Omega }}={\frac {q^{2}}{16\pi ^{3}\varepsilon _{0}c}}\omega ^{2}\left|\int _{-\infty }^{\infty }\mathrm {d} t\,{\vec {n}}\times ({\vec {n}}\times {\vec {\beta }})e^{\mathrm {i} \omega (t-{\vec {n}}\cdot {\vec {r}}/c)}\right|^{2}}$

Since in the highly relativistic Limes only small opening angles are of interest and the synchrotron radiation sweeps over the observation point in only short pulses, similar to a headlight, a Taylor expansion leads to and to ${\displaystyle t}$ ${\displaystyle \theta }$

${\displaystyle {\frac {\mathrm {d} ^{2}I}{\mathrm {d} \omega \,\mathrm {d} \Omega }}\approx {\frac {q^{2}}{12\pi ^{3}\varepsilon _{0}c}}\left({\frac {\omega R}{c}}\right)^{2}\left({\frac {1}{\gamma ^{2}}}+\theta ^{2}\right)^{2}\left[K_{2/3}^{2}(\xi )+{\frac {\theta ^{2}}{(1/\gamma ^{2})+\theta ^{2}}}K_{1/3}^{2}(\xi )\right]}$,

where and denote the Airy's integrals . Your argument is . The first term in brackets is the intensity of the polarized radiation in the plane of the orbit, the second term of the radiation polarized perpendicular to it. The result is independent of the azimuthal angle , since it is integrated over a complete orbit and the direction is averaged out relative to the acceleration vector. ${\displaystyle K_{1/3}}$ ${\displaystyle K_{2/3}}$ ${\displaystyle \textstyle \xi ={\frac {\omega R}{3c}}\left(1/\gamma ^{2}+\theta ^{2}\right)^{3/2}}$ ${\displaystyle \phi }$ ${\displaystyle r(t)}$

The integration over all angles gives

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} \omega }}={\sqrt {3}}{\frac {q^{2}}{4\pi \varepsilon _{0}c}}\gamma {\frac {\omega }{\omega _{c}}}\int _{\omega /\omega _{c}}^{\infty }\mathrm {d} x\,K_{5/3}(x)}$

with the critical angular frequency and the integration over all frequencies ${\displaystyle \textstyle \omega _{c}={\frac {3}{2}}\gamma ^{3}{\frac {c}{R}}}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} \Omega }}={\frac {7}{16}}{\frac {1}{4\pi \varepsilon _{0}}}{\frac {q^{2}}{R}}{\frac {1}{(1/\gamma ^{2}+\theta ^{2})^{5/2}}}\left[1+{\frac {5}{7}}{\frac {\theta ^{2}}{(1/\gamma ^{2})+\theta ^{2}}}\right]}$.

Complete integration over both angles and frequencies results

${\displaystyle I\approx {\frac {7}{24\varepsilon _{0}}}{\frac {q^{2}}{R}}\gamma ^{4}\left[1+{\frac {1}{7}}\right]}$;

the intensity of the polarization parallel to the plane of the orbit is thus seven times greater than that of the polarization orthogonal to the plane of the orbit.

Approximations for large and small frequencies

The representation of the frequency distribution as an integral over a Bessel function is exact, but unwieldy in practical use. Therefore, the following approximations exist for small frequencies

${\displaystyle \left.{\frac {\mathrm {d} I}{\mathrm {d} \omega }}\right|_{\omega \ll \omega _{c}}\approx \left({\frac {\Gamma (2/3)}{\pi }}\right)^{2}\left({\frac {3}{2}}\right)^{1/3}{\frac {q^{2}}{2\varepsilon _{0}c}}\gamma \left({\frac {\omega }{\omega _{c}}}\right)^{1/3}}$

with the gamma function taking the value and for large frequencies ${\displaystyle \Gamma }$ ${\displaystyle \Gamma (2/3)\approx 1{,}354}$

${\displaystyle \left.{\frac {\mathrm {d} I}{\mathrm {d} \omega }}\right|_{\omega \gg \omega _{c}}\approx \left({\frac {3}{8\pi }}\right)^{1/2}{\frac {q^{2}}{2\varepsilon _{0}c}}\gamma \left({\frac {\omega }{\omega _{c}}}\right)^{1/2}e^{-\omega /\omega _{c}}}$

Total performance

The total power results from an integration of the power over the complete solid angle to the Larmor formula

${\displaystyle P={\frac {q^{2}}{6\pi \varepsilon _{0}c}}\gamma ^{6}\left(({\dot {\vec {\beta }}})^{2}-({\vec {\beta }}\times {\dot {\vec {\beta }}})^{2}\right)}$.

With a circular path, the acceleration is always orthogonal to the current direction of movement. It is therefore possible to calculate with amounts instead of vectors. Furthermore is the amount of centripetal acceleration . It follows ${\displaystyle {\dot {\beta }}=c\beta ^{2}/R}$

${\displaystyle P={\frac {c}{6\pi \varepsilon _{0}}}{\frac {q^{2}}{R^{2}}}\beta ^{4}\gamma ^{4}}$.

The total energy loss of a particle per revolution is therefore

${\displaystyle \Delta E=P\Delta t=P{\frac {2\pi R}{\beta c}}={\frac {1}{3\varepsilon _{0}}}{\frac {q^{2}}{R}}\beta ^{3}\gamma ^{4}}$

For relativistic velocities, that is , this becomes: ${\displaystyle \beta \approx 1}$

${\displaystyle \Delta E\approx {\frac {1}{3\varepsilon _{0}}}{\frac {q^{2}}{R}}\gamma ^{4}={\frac {1}{3\varepsilon _{0}}}{\frac {q^{2}}{R}}\left({\frac {E}{mc^{2}}}\right)^{4}}$

Due to the law of conservation of energy, this agrees with the total energy of the emitted radiation. At speeds close to the speed of light, the energy loss of the particle due to radiation increases very strongly with the kinetic energy.

Generation and application

Synchrotron radiation generated by a deflecting magnet

Synchrotron radiation generated by alternating magnetic fields in an undulator
1: magnets,
2: electron beam,
3: radiation lobe

Synchrotron radiation is used for application purposes by deflection of electrons with kinetic energies of the order of 1 giga electron volts generated (GeV). This is done by using electron storage rings and free electron lasers with specially designed magnetic structures ( undulators ).

There are around 30 laboratories around the world for the generation of synchrotron radiation . In Germany these include BESSY in Berlin , DESY in Hamburg , DELTA at the Technical University of Dortmund and ANKA in Karlsruhe .

A number of aspects are advantageous for applications of synchrotron radiation in science and technology:

• A suitable frequency window can be selected from the wide range for different purposes.
• It can be generated with high intensity compared to other radiation sources besides lasers ,
• in addition, with very high brilliance .
• It can be generated in a pulsed manner ; Frequency and pulse duration can be set (within narrow limits).
• Due to its good predictability, it is suitable as a radiation standard for the calibration of radiation sources or detectors.

Polarization of synchrotron radiation

The radiation, which is linearly polarized in the direction of the plane of the ring , is well suited, for example, to characterizing magnetic materials by means of micromagnetic investigation. The linear polarization can be converted into circular polarization by means of mechanical phase shifting of the magnetization regions in an undulator ; this enables higher contrasts when examining the magnetization regions of magnetic materials. The irradiation of racemic organic compounds with circularly polarized synchrotron radiation makes it possible, for example, to achieve an enantiomeric excess in chiral amino acids .

Differences in brilliance

A distinction is made between sources of the first , second , third and fourth generation. These differ essentially in the brilliance of the radiation emitted.

• In the first generation, particle accelerators from particle physics were used “parasitically”, that is, only the radiation that was inevitably generated during particle physics operation was used.
• In the second generation, synchrotron radiation sources are built solely to generate radiation. The accelerated particles are stored in storage rings for several hours , thus achieving constant working conditions. The radiation is generated not only in the dipole magnets that are already present , but also in special magnetic structures, the wigglers .
• The third generation consists of synchrotrons with undulators in the storage ring. With undulators a more brilliant radiation can be generated than with wigglers.
• Free electron lasers (FEL) represent the fourth generation. The first systems are FELICITA at the DELTA at TU Dortmund University and the FLASH at DESY in Hamburg.

Fields of application

The synchrotron radiation can be used for

• Surface physics
• Materials science
• X-ray diffraction
• Photochemistry
• X-ray depth lithography
• Metrology
• mineralogy
• Molecular biology
• Biophysics and
• medicine

Natural sources of synchrotron radiation

In astronomy , synchrotron radiation occurs when a hot plasma moves in a magnetic field . Examples of cosmic “synchrotron sources” are pulsars , radio galaxies and quasars .

literature

• Albert Hofmann: The Physics of Synchrotron Radiation. Cambridge University Press, 2004, ISBN 0-521-30826-7 .
• Helmut Wiedemann: Synchrotron Radiation. Springer, 2003, ISBN 3-540-43392-9 .

Web links

Commons : Synchrotron  - collection of images, videos and audio files

• TU Dortmund University: Properties of synchrotron radiation pdf 11.9 MB
• List of large accelerators worldwide, including many synchrotron radiation laboratories
• Website of the "united" light sources
• Radiation from accelerated particles (very good overview; PDF, 553 KiB)

Individual evidence

1. ↑ Ivanenko, Pomeranchuk: On the maximal energy attainable in betatron. In: Physical Review. 65 (1944), p. 343.
2. ^ Saxon, Schwinger: Electron orbits in the Synchrotron. In: Phys. Rev. Volume 69, 1946, p. 702. Just a brief note. At that time the complete calculations only circulated as a manuscript and Schwinger did not publish a larger article until 1949: On the classical radiation of accelerated electrons. In: Phys. Rev. Volume 75, 1949, p. 1912.
3. ↑ Jagdish Mehra, Kimball A. Milton: Climbing the mountain. Oxford University Press 2000, p. 138ff (Schwingers biography).
4. ^ Frank Elder, Anatole Gurewitsch, Robert Langmuir, Herb Pollock: Radiation from Electrons in a Synchrotron. In: Physical Review. Volume 71, 1947, pp. 829-830.
5. ^ History of Synchrotron X-rays, ESRF .
6. ↑ ^{a b c d e f g} John David Jackson: Classical electrodynamics . 3. Edition. de Gruyter, Berlin, New York 2002, pp. 771-788 .
7. ^ Bogdan Povh, Klaus Rith, Christoph Scholz, Frank Zetsche: Particles and cores. An introduction to the physical concepts. 2009, ISBN 978-3-540-68075-8 , p. 365.