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| {{about|physical phenomenon of synchrotron radiation|details on the production of this radiation and applications in laboratories|Synchrotron light source}}
| | If you've gained a limited pounds in recent years, you are wondering, "How do I [http://safedietplansforwomen.com/how-to-lose-weight-fast lose weight]?" And, you possibly want to do so immediately and easily. It's also a well-known (and normally correct) perception that in the event you want to lose fat, we do so slowly plus carefully in order to be healthy about it. There is a method to lose weight, though, where you can do it immediately and be healthy about it -- and have it be permanent as well.<br><br>Your initially step if you would like to lose weight fast and easy is to make small changes to your everyday routine. If you stick with the following practices and not deviate from the goal of losing fat you'll see a favourable weight loss inside a brief amount of time.<br><br>The best time to do any cardio exercise is initially thing each morning before we eat anything. What occurs is the fact that considering you haven't eaten anything during the night, because you're asleep, your body has to employ calories that are stored inside the body plus burning calories is what we want if you need to lose weight.<br><br>One important tip for losing fat naturally is to exercise everyday. Cardiovascular exercises like aerobics, swimming, jogging plus cycling must be completed everyday as they help in losing weight from all over the body. These ought to be combined with exercises aimed at a particular body piece, which strengthen plus tone up muscles of the specific body region. Weight training, flexibility exercises plus resistance training could be a piece of the exercise regimen. But, these should be undertaken just under the guidance plus supervision of the trained pro. So, when you could afford, hire a trainer that would come and teach you effective fat reduction exercises. Otherwise, get some exercise videos within the marketplace and undertake exercises at home, watching them.<br><br>We can have watched photos of woman body builders. They usually have protruding biceps, extremely defined abdominal muscles, plus sometimes protruding veins. In brief, they can look masculine, plus this impact will turn away other females whom wish To train with weights. The wise news is that it simply is impossible for a woman to bulk up like a guy. To achieve bulky muscles, a woman usually might consume muscle building compounds or chemicals and exercise over once a day. Women could feel free to take benefit of the benefits that come with fat lifting, considering they usually just tone up.<br><br>Every mom which chooses to breastfeeds usually wonders if the baby is getting enough to consume plus when all is going well. Since you cannot keep track of how numerous ounces the baby is taking in, it's frequently a wonder on a modern mom's notice. Some are even told which they require to consume a lot of extra food whilst nursing to keep up with the milk supply. This really is turn scares several female into thinking which losing fat while breastfeeding is not a wise decision. Breastfeeding shouldn't be looked on as a scary event. It is a breathtaking time in a woman's life.<br><br>This is a fun exercise plus anybody could do it. All you want is some wise significant beat music. We can additionally join a class, enjoy learning a hot dance form plus lose fat side by side. Quick step salsa, is a dance form which burns the maximum calories. |
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| The [[electromagnetic radiation]] emitted when charged particles are accelerated radially (<math>\mathbf{a}\perp \mathbf{v}</math>) is called '''synchrotron radiation.''' It is produced in [[synchrotron]]s using bending magnets, [[undulator]]s and/or [[wiggler (synchrotron)|wiggler]]s. It is similar to [[cyclotron radiation]] except that synchrotron radiation is generated by the acceleration of [[ultrarelativistic limit|ultrarelativistic]] charged particles through magnetic fields. Synchrotron radiation may be achieved artificially in synchrotrons or [[storage ring]]s, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic [[polarization (waves)|polarization]] and the frequencies generated can range over the entire [[electromagnetic spectrum]].
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| [[File:Syncrotron.png|thumb|Synchrotron radiation from a bending magnet]] [[File:Undulator.png|thumb|Synchrotron radiation from an undulator]]
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| ==History==
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| [[Image:ge-synchrotron-acclerator.jpg|thumb|300px|General Electric [[synchrotron accelerator]] built in 1946, the origin of the discovery of synchrotron radiation. The arrow indicates the evidence of radiation.]]
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| 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".<ref>Elder, F. R.; Gurewitsch, A. M.; Langmuir, R. V.; Pollock, H. C., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1947PhRv...71..829E&db_key=PHY&data_type=HTML&format=&high=42ca922c9c07689 Radiation from Electrons in a Synchrotron]" (1947) ''Physical Review'', vol. 71, Issue 11, pp. 829-830</ref> Pollock recounts:
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| :"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 [[Dmitri Ivanenko|Ivanenko]] and [[Isaak Pomeranchuk|Pomeranchuk]] radiation."<ref>Iwanenko D., Pomeranchuk I., "[http://prola.aps.org/abstract/PR/v65/i11-12/p343_1 On the maximal energy attainable in betatron]", (1944) ''Physical Review'' vol.65, p. 343</ref>
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| ==Properties of synchrotron radiation==
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| # Broad Spectrum (which covers from microwaves to hard X-rays): the users can select the wavelength required for their experiment.
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| # High Flux: high intensity photon beam allows rapid experiments or use of weakly scattering crystals.
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| # High Brilliance: highly collimated photon beam generated by a small divergence and small size source (spatial coherence)
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| # High Stability: submicron source stability
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| # Polarization: both linear and circular
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| # Pulsed Time Structure: pulsed length down to tens of picoseconds allows the resolution of process on the same time scale.
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| ==Emission mechanism==
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| 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]], <math>\gamma</math>.
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| Relativistic [[Lorentz contraction|length contraction]] then bumps the frequency observed in the lab by another factor of <math>\gamma</math>, 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 [[Larmor formula#Relativistic Generalisation|relativistic Larmor formula]] while the force on the emitting electron is given by the [[Abraham-Lorentz-Dirac force]].
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| 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.
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| 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.
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| ==Synchrotron radiation from accelerators==
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| {{main|Synchrotron light source}}
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| Synchrotron radiation may occur in accelerators either as a nuisance, causing undesired energy loss in [[particle physics]] contexts, or as a [[synchrotron light source|deliberately produced radiation source]] for numerous laboratory applications.
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| Electrons are accelerated to high speeds in several stages to achieve a final energy that is typically in the GeV range. In the [[Large Hadron Collider|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<sup>10</sup>. Each proton may lose 6.7keV per turn due to this phenomenon.<ref name="CERN Synchrotron">[http://cdsweb.cern.ch/record/858155/files/lhc-project-report-819.pdf] Synchrotron Radiation Damping in the LHC 2005 Joachim Tuckmantel.</ref>
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| ==Synchrotron radiation in astronomy==
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| [[Image:M87 jet.jpg|thumb|225px|'''[[Messier 87]]'s Energetic Jet''', [[Hubble Space Telescope|HST]] image.
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| The blue light from the jet emerging from the bright [[active galactic nucleus|AGN]] core, towards the lower right, is due to synchrotron radiation.]]
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| Synchrotron radiation is also generated by astronomical objects, typically where relativistic electrons spiral (and hence change velocity) through magnetic fields.
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| Two of its characteristics include non-thermal [[power-law]] spectra, and polarization.<ref>Vladimir A. Bordovitsyn, "[http://books.google.com/books?vid=ISBN9810231563&id=rG9ZWoCtwagC&pg=PA385&lpg=PA385&dq=%22Synchrotron++radiation%22+astronomy Synchrotron Radiation in Astrophysics]" (1999) ''[http://www.worldscibooks.com/physics/3492.html Synchrotron Radiation Theory and Its Development]'', ISBN 981-02-3156-3</ref>
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| ===History of detection===
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| It was first detected in a jet emitted by [[Messier 87]] in 1956 by [[Geoffrey Burbidge|Geoffrey R. Burbidge]],<ref>Burbidge, G. R. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1956ApJ...124..416B&db_key=AST&data_type=HTML&format=&high=42ca922c9c17516 On Synchrotron Radiation from Messier 87.] ''Astrophysical Journal'', vol. 124, p. 416"</ref> who saw it as confirmation of a prediction by [[Iosif Samuilovich Shklovsky|Iosif S. Shklovsky]] in 1953, but it had been predicted earlier by [[Hannes Alfvén]] and Nicolai Herlofson <ref>Alfvén, H.; Herlofson, N. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1950PhRv...78..616A&db_key=PHY&data_type=HTML&format=&high=42ca922c9c25284 Cosmic Radiation and Radio Stars]" ''Physical Review'' (1950), vol. 78, Issue 5, pp. 616–616</ref> in 1950.
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| T. K. Breus noted that questions of priority on the history of astrophysical synchrotron radiation is complicated, writing:
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| :"In particular, the Russian physicist [[Vitaly Ginzburg|V.L. Ginzburg]] broke his relationships with [[Iosif Samuilovich Shklovsky|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 [[Hannes Alfvén|H. Alfven]] and N. Herlofson, while K.O. Kiepenheuer and G. Hutchinson were ignored by them."<ref>Breus, T. K., "[http://adsabs.harvard.edu/abs/2001IAIss..26...88B Istoriya prioritetov sinkhrotronnoj kontseptsii v astronomii %t] (Historical problems of the priority questions of the synchrotron concept in astrophysics)" (2001) in ''Istoriko-Astronomicheskie Issledovaniya'', Vyp. 26, p. 88 - 97, 262 (2001)</ref>
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| [[Image:Crab Nebula.jpg|225px|thumb|'''[[Crab Nebula]].''' The bluish glow from the central region of the nebula is due to synchrotron radiation.]]
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| [[Supermassive black hole]]s 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]].<ref>{{cite web|last=Chase|first=Scott I.|title=Apparent Superluminal Velocity of Galaxies|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/Superluminal/superluminal.html|accessdate=22 August 2012}}</ref>
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| ===Pulsar wind nebulae===
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| A class of [[astronomical source]]s where synchrotron emission is important is the [[pulsar wind nebula]]e, a.k.a. [[plerion]]s, of which the [[Crab nebula]] and its associated [[pulsar]] are archetypal.
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| Pulsed emission gamma-ray radiation from the Crab has recently been observed up to ≥25 GeV,<ref>"Observation of Pulsed {gamma}-Rays Above 25 GeV from the Crab Pulsar with MAGIC", ''Science'' 21 November 2008:
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| Vol. 322. no. 5905, pp. 1221–1224"</ref> probably due to synchrotron emission by electrons trapped in the strong magnetic field around the pulsar.
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| Polarization in the Crab<ref>
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| ''Dean et al.'',"Polarized Gamma-Ray Emission from the Crab", ''Science'' 29 August 2008: Vol. 321. no. 5893, pp. 1183–1185</ref> at energies from 0.1 to 1.0 MeV illustrates a typical synchrotron radiation.
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| ==Formulation==
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| ===Liénard–Wiechert Field===
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| We start with the expressions for the [[Liénard–Wiechert potential#Corresponding values of electric and magnetic fields|Liénard–Wiechert field]] :
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| ::<math>\mathbf{B}(\mathbf{r},t)=-\frac{\mu_0q}{4\pi}\left[\frac{c\,\hat{\mathbf{n}}\times\vec{\beta}}{\gamma^2R^2(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}})^3}+\frac{\hat{\mathbf{n}}\times[\,\dot{\vec{\beta}}+\hat{\mathbf{n}}\times(\vec{\beta}\times\dot{\vec{\beta}})]}{R\,(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}})^3}\right]_{\mathrm{retarded}} \qquad (1)</math>
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| ::<math>\mathbf{E}(\mathbf{r},t)=\frac{q}{4\pi\varepsilon_0}\left[\frac{\hat{\mathbf{n}}-\vec{\beta}}{\gamma^2R^2(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}})^3}+\frac{\hat{\mathbf{n}}\times[(\hat{\mathbf{n}}-\vec{\beta})\times\dot{\vec{\beta}}\,]}{c\,R\,(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}})^3}\right]_{\mathrm{retarded}} \qquad \qquad (2)</math>
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| where
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| ::<math>\mathbf R(t')=\mathbf r-\mathbf r_0(t'),</math>
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| ::<math>R(t')=|\mathbf R|,</math>
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| ::<math>\hat{\mathbf n}(t')=\mathbf R/R,</math>
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| which is the unit vector between the observation point and the position of the charge at the retarded time, and <math>t'</math> is the retarded time. | |
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| 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''.
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| If we ignore the ''velocity'' field, the radial component of Poynting's Vector resulted from the Liénard–Wiechert field can be calculated to be
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| ::<math>[\mathbf{S\cdot}\hat{\mathbf{n}}]_{\mathrm{retarded}} = \frac{q^2}{16\pi^2\varepsilon_0 c}\left\{\frac{1}{R^2}\left|\frac{\hat{\mathbf{n}}\times[(\hat{\mathbf{n}}-\vec{\beta})\times\dot{\vec{\beta}}]}{(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}})^3}\right|^2\right\}_{\text{not retarded}} \qquad \qquad (3) </math>
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| Note that
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| *The spatial relationship between <math>\vec{\beta}</math> and <math>\dot{\vec{\beta}}</math> determines the detailed angular power distribution.
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| *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 <math>(1-\vec{\beta}\mathbf{\cdot}\vec{\mathbf{n}})</math> in the denominator of Eq. (3).
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| *For ultrarelativistic particles the latter effect dominates the whole angular distribution.
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| The energy radiated into per solid angle during a finite period of acceleration from <math>t'=T_1</math> to <math>t'=T_2</math> is
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| ::<math>\frac{\mathrm{d}P}{\mathrm{d}\mathit{\Omega}} = R(t')^2\,[\mathbf{S}(t')\mathbf{\cdot}\hat{\mathbf{n}}(t')]\,\frac{\mathrm{d}t}{\mathrm{d}t'} = R(t')^2\,\mathbf{S}(t')\mathbf{\cdot}\hat{\mathbf{n}}(t')\,[1-\vec{\beta}(t')\mathbf{\cdot}\hat{\mathbf{n}}(t')]</math>
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| :::::<math> = \frac{q^2}{16\pi^2\varepsilon_0c}\,\frac{|\hat{\mathbf{n}}(t')\times\{[\hat{\mathbf{n}}(t')-\vec{\beta}(t')]\times\dot{\vec{\beta}}(t')\}|^2}{[1-\vec{\beta}(t')\mathbf{\cdot}\vec{\mathbf{n}}(t')]^5} \qquad \qquad (4)</math>
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| Integrating Eq. (4) over the all solid angles, we get '''relativistic generalization of [[Larmor formula|Larmor's formula]]'''
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| ::<math>P=\frac{e^2}{6\pi \varepsilon _0 c}\gamma ^6
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| \left [ \left | \dot{\vec{\beta }} \right |^2
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| -\left | \vec{\beta}\times \dot{\vec{\beta }}\right |^2 \right ]\qquad (5)
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| </math>
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| However, this also can be derived by relativistic transformation of the 4-acceleration in Larmor's formula.
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| ===Velocity ⊥ acceleration: synchrotron radiation===
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| [[File:Syncrotron radiation energy flux.png|thumb|500px|When the electron velocity approaches the speed of light, the emission pattern is sharply collimated forward.]] When the charge is in instantaneous circular motion, its acceleration <math>\dot{\vec{\beta}}</math> is perpendicular to its velocity <math>\vec{\beta}</math>. Choosing a coordinate system such that instantaneously <math>\vec{\beta}</math> is in the z direction and <math>\dot{\vec{\beta}}</math> is in the x direction, with the [[spherical coordinate system|polar and azimuth]] angles <math>\theta</math> and <math>\phi</math> defining the direction of observation, the general formula Eq. (4) reduces to
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| :<math>\frac{\mathrm{d}P}{\mathrm{d}\mathit{\Omega}} = \frac{q^2}{16\pi^2\epsilon_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]. \qquad (6)</math>
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| In the relativistic limit <math>(\gamma\gg 1)</math>, the angular distribution can be written approximately as | |
| :<math>\frac{\mathrm{d}P}{\mathrm{d}\mathit{\Omega}} \simeq \frac{2}{\pi}\frac{e^2}{c^3}\gamma^6\frac{|\dot{\mathbf v}|^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]. \qquad \qquad (7)</math>
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| The factors <math>(1-\beta\cos\theta)</math> 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.
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| Integration over the whole solid angle yields the total power radiated by one electron
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| :<math>P=\frac{e^2}{6\pi\epsilon_0c}\left | \dot{\vec{\beta }} \right |^2\gamma ^4=\frac{e^2c}{6\pi\epsilon_0}\frac{\gamma ^4}{\rho ^2}=\frac{e^4}{6\pi\epsilon_0m^4c^5}E^2B^2,\qquad (8)</math>
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| <!-- -->
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| 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 <math>1/m^4</math>, <math>1/\rho^2</math>, and <math>B^2</math>. In some cases the surfaces of vacuum chambers hit by synchrotron radiation have to be cooled because of the high power of the radiation.
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| ===Radiation integral===
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| The energy received by an observer (per unit solid angle at the source) is
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| <math>\frac{d^2W}{d\Omega }=\int_{-\infty }^{\infty }\frac{d^2P}{d\Omega }dt=c\varepsilon _0\int_{-\infty }^{\infty }\left | R\vec{E}(t) \right |^2dt</math>
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| Using the Fourier Transformation we move to the frequency space
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| <math>\frac{d^2W}{d\Omega }=2c\varepsilon _0\int_{0 }^{\infty }\left | R\vec{E}(\omega) \right |^2d\omega</math>
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| Angular and frequency distribution of the energy received by an observer (consider only the radiation field)
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| <math>\frac{d^3W}{d\Omega d\omega }=2c\varepsilon _0R^2\left | \vec{E}(\omega) \right |^2=\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 \qquad (9)</math>
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| 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 function]]s).
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| ===Example 1: bending magnet===
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| ====Integrating====
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| [[File:Circumference trajectory.png|thumb|300px|Trajectory of the arc of circumference]]
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| Trajectory of the arc of circumference is
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| <math>\vec{r}(t)=\left ( \rho \sin\frac{\beta c}{\rho}t, \rho\left ( 1-\cos\frac{\beta c}{\rho}t \right ), 0 \right)</math>
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| In the limit of small angles we compute
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| <math>\hat{n}\times\left ( \hat{n}\times\vec{\beta} \right )
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| =\beta\left [ -\vec{\varepsilon }_\parallel \sin\left ( \frac{\beta c t}{\rho} \right )+\vec{\varepsilon}_\perp \cos\left ( \frac{\beta c t}{\rho}\right )\sin\theta
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| \right ]</math>
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| <math>\omega\left ( t-\frac{\hat{n}\cdot \vec{r}(t)}{c} \right )
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| =
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| \omega\left [ t-\frac{\rho}{c}\sin\left ( \frac{\beta c t}{\rho} \right )\cos\theta \right ]</math>
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| Substituting into the radiation integral and introducing <math>\xi =\frac{\rho \omega}{3c\gamma^3}\left ( 1+\gamma^2 \theta^2 \right )^{3/2}</math>
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| <math>\frac{d^3 W}{d\Omega d\omega}=\frac{e^2}{16\pi^3\varepsilon_0 c}
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| \left ( \frac{2\omega\rho}{3c\gamma^2} \right )^2
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| \left ( 1+\gamma^2 \theta^2 \right )^2
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| \left [ K_{2/3}^2(\xi ) + \frac{\gamma^2 \theta^2}{1+\gamma^2 \theta^2}K_{1/3}^2(\xi)\right ]\qquad (10)
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| </math>
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| , where the function <math>K</math> is a modified [[Bessel function]] of the second kind.
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| ====Frequency distribution of radiated energy====
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| [[File:Critical frequency and critical angle.jpg|thumb|350px|Angular distribution of radiated energy]]
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| From Eq.(10), we observe that the radiation intensity is negligible for <math>\xi\gg 1</math>.
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| ''Critical frequency'' is defined as the frequency when <math>\xi=\frac{1}{2}</math> and <math>\theta=0</math>. So, | |
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| ::<math>\omega_\text{c}=\frac{3}{2}\frac{c}{\rho}\gamma^3</math>
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| , and ''critical angle is defined as
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| ::<math>\theta_c=\frac{1}{\gamma}\left ( \frac{\omega_\text{c}}{\omega} \right )^{1/3}</math>
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| For frequencies much larger than the critical frequency and angles much larger than the critical angle, the synchrotron radiation emission is negligible.
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| Integrating on all angles, we get the frequency distribution of the energy radiated.
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| ::<math>\frac{dW}{d\omega}=\oint \frac{d^3 W}{d\omega d\Omega }d\Omega
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| =\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</math>
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| [[File:Frequency distributions of radiated energy.png|thumb|400px|Frequency distribution of radiated energy]]
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| If we define
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| ::<math>S(y)\equiv \frac{9\sqrt{3}}{8\pi}y\int_{y}^{\infty}K_{5/3}(x)dx</math> <math>\int_{0}^{\infty}S(y)dy=1 </math>
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| , where <math> y=\frac{\omega}{\omega_\text{c}} </math>. Then,
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| ::<math>\frac{dW}{d\omega}=\frac{2e^2 \gamma}{9 \varepsilon_0c}S(y)\qquad (11)</math>
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| Note that <math>\frac{dW}{d\omega}\sim \frac{e^2}{4\pi \varepsilon_0c}\left ( \frac{\omega \rho}{c} \right )^{1/3}</math>, if <math>\omega\ll \omega_\text{c}</math>, and <math>\frac{dW}{d\omega}\approx \sqrt{\frac{3\pi}{2}}\frac{e^2}{4\pi\varepsilon_0 c}\gamma\left ( \frac{\omega}{\omega_\text{c}} \right )^2 e^{-\omega/\omega_\text{c}}</math>, if <math>\omega\gg \omega_\text{c}</math>
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| 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 <ref>M.Kh.Khokonov. Cascade Processes of Energy Loss by Emission of Hard Photons // JETP, V.99, No.4, pp. 690-707 \ (2004).</ref> (see also [[Bessel function#Modified Bessel functions|modified Bessel function]]s ) by means of the relation:
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| :<math>
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| \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}}
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| \exp \left[- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right] \ dx </math>
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| ====Synchrotron radiation emission as a function of the beam energy====
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| [[File:Syncrotron radiation emission as a function of the beam energy.png|thumb|420px|Relationship between power radiated and the photon energy]]
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| First, define the critical photon energy as
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| <math>\varepsilon_c=\hbar \omega_\text{c}=\frac{3}{2}\frac{\hbar c}{\rho}\gamma^3</math>
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| 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.
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| ====Polarization of synchrotron radiation====
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| In Eq.(10), the first term <math>K_{2/3}^2(\xi) </math> is the radiation power with polarization in the orbit plane, and the second term <math>\frac{\gamma^2 \theta^2}{1+\gamma^2 \theta^2}K_{1/3}^2(\xi)</math> is the polarization orthogonal to the orbit plane.
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| In the orbit plane <math>\theta=0</math>, the polarization is purely horizontal.
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| Integrating on all frequencies, we get the angular distribution of the energy radiated
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| ::<math>
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| \frac{d^2 W}{d\Omega }=\int_{0}^{\infty}\frac{d^3W}{d\omega d\Omega }d\omega
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| =\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) </math>
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| 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.
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| ===Example 2: undulator===
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| ====Solution of equation of motion and undulator equation====
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| An [[undulator]] consists of a periodic array of magnets, so that they provide a sinusoidal magnetic field.
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| ::<math> \vec{B}=\left ( 0, B_0 \sin(k_\text{u} z),0 \right ) </math>
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| [[File:Undulator wih axis.png|thumb|300px|undulator]]
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| Solution of equation of motion is
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| ::<math>\vec{r}(t)=\frac{\lambda_\text{u} K}{2\pi\gamma}\sin \omega_\text{u}t\cdot \hat{x}
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| +\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} </math>
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| where,
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| <math> K=\frac{eB_0\lambda_\text{u}}{2\pi mc} </math>
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| , and
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| <math>\bar{\beta_z}=1-\frac{1}{2\gamma^2}\left ( 1+\frac{K^2}{2} \right )</math>
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| , and the parameter <math> K </math> is called the ''undulator parameter''.
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| [[File:Undulator constructive interference.png|thumb|300px|Constructive interference of the beam in the undulator]]
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| Condition for the constructive interference of radiation emitted at different poles is
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| ::<math>d=\frac{\lambda_\text{u}}{\bar{\beta}}-\lambda_\text{u}\cos\theta=n\lambda</math>
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| Therefore,
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| ::<math>\lambda_n = \frac{\lambda_\text{u}}{2\gamma^2n}\left ( 1+\frac{K^2}{2}+\gamma^2\theta^2 \right )\qquad (13)</math>
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| This equation is called the ''undulator equation''. | |
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| ====Radiation from the undulator====
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| Radiation integral is
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| ::<math>
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| \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
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| </math>
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| Using the periodicity of the trajectory, we can split the radiation integral into a sum over <math>N_u</math> terms.
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| ::<math>
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| \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
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| \left|1+e^{i\delta}+e^{2i\delta}+\cdots +e^{i(N_u-1)\delta} \right |^2 \qquad (14)</math>
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| , where <math>\bar{\beta}=\beta\left ( 1-\frac{K^2}{4\gamma^2} \right )</math> | |
| [[File:Peak frequency by undulator.png|thumb|300px|Peak frequencies become sharp as the number N increases]]
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| , and | |
| <math> \delta=\frac{2\pi \omega}{\omega_\text{res} (\theta)}</math>, <math>\omega_\text{res}(\theta)=\frac{2\pi c}{\lambda_\text{res}(\theta)}</math>, and <math>\lambda_\text{res}(\theta)=\frac{\lambda_u}{2\gamma^2}\left ( 1+\frac{K^2}{2}+\gamma^2\theta^2 \right )</math>
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| [[File:Undulator radiation on axis.png|thumb|300px|Only odd harmonics are radiated on-axis]] [[File:Undulator radiation off axis.png|thumb|300px|Off-axis radiation contains many harmonics]]
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| The radiation integral in an undulator can be written as
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| ::<math>
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| \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)</math>
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| The sum of <math>\delta</math> generates a series of sharp peaks in the frequency spectrum harmonics of fundamental wavelength
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| ::<math>L\left ( N\frac{\Delta \omega_k}{\omega_\text{res}(\theta)} \right )
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| =\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 )}</math>
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| , and <math>F_n</math> depends on the angles of observations and <math>K</math>
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| ::<math>F_n(K,\theta,\phi)\propto
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| \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</math>
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| On the axis(<math>\theta=0</math>, <math>\phi=0</math>), the radiation integral becomes
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| <math>\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}(0)} \right )F_n(K,0,0)</math>
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| and,
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| <math>F_n(K,0,0)=\frac{n^2K^2}{1+K^2/2}
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| \left[J_{\frac{n+1}{2}}(Z)-J_{\frac{n-1}{2}}(Z) \right ]^2
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| </math>
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| , where <math> Z=\frac{nK^2}{4(1+K^2/2)} </math>
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| Note that only odd harmonics are radiated on-axis, and as K increases higher harmonic becomes stronger.
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| ==See also==
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| *[[Bremsstrahlung]]
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| *[[Cyclotron Radiation]]
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| *[[Free-electron laser]]
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| *[[Radiation reaction]]
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| *[[Relativistic beaming]]
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| *[[Sokolov–Ternov effect]]
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| *[[Synchrotron]] for this type of particle accelerator
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| *[[Synchrotron light source]] for laboratory generation and applications of synchrotron radiation
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| ==Notes==
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| <div style="font-size: 95%;">
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| <!--See [[Wikipedia:Footnotes]] for an explanation of how to generate footnotes using the <ref(erences/)> tags-->
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| <references/>
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| </div>
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| ==External links==
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| *[http://cdsads.u-strasbg.fr/cgi-bin/nph-bib_query?bibcode=1965ARA%26A...3..297G Cosmic Magnetobremsstrahlung (synchrotron Radiation)], by Ginzburg, V. L., Syrovatskii, S. I., ARAA, 1965
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| *[http://cdsads.u-strasbg.fr/cgi-bin/nph-bib_query?bibcode=1969ARA%26A...7..375G Developments in the Theory of Synchrotron Radiation and its Reabsorption], by Ginzburg, V. L., Syrovatskii, S. I., ARAA, 1969
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| *[http://www.lightsources.org Lightsources.org]
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| *[http://xdb.lbl.gov X-Ray Data Booklet]
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| ==References==
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| *Brau, Charles A. Modern Problems in Classical Electrodynamics. Oxford University Press, 2004. ISBN 0-19-514665-4.
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| *Jackson, John David. Classical Electrodynamics. John Wiley & Sons, 1999. ISBN 0-471-30932-X
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| *{{cite web|last=[[Ishfaq Ahmad{{!}}Ishfaq Ahmad]], D.Sc.|title=Measurements of the Relative Oscillator Strengths using the Synchrotron Radiation|url=http://pps-pak.org/proceedings/Eleventh-Proc-2009.pdf|work=Proceedings of the National Syposium on Frontier of Physics, National Centre for Theoretical Physics|publisher=Pakistan Physical Society|accessdate=16 January 2012}}
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| {{DEFAULTSORT:Synchrotron Radiation}}
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| [[Category:Particle physics]]
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| [[Category:Synchrotron radiation]]
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| [[Category:Synchrotron-related techniques]]
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| [[Category:Electromagnetic radiation]]
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| [[Category:Experimental particle physics]]
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