Tables for
Volume F
Crystallography of biological molecules
Edited by M. G. Rossmann and E. Arnold

International Tables for Crystallography (2006). Vol. F, ch. 8.1, p. 155   | 1 | 2 |

Section 8.1.2. The physics of SR

J. R. Helliwella*

aDepartment of Chemistry, University of Manchester, M13 9PL, England
Correspondence e-mail:

8.1.2. The physics of SR

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The physics of the SR source spectral emission was predicted by Iwanenko & Pomeranchuk (1944)[link] and Blewett (1946)[link], and was fully described by Schwinger (1949)[link]. It is `universal' to all machines of this type, i.e., wherever charged particles such as electrons (or positrons) travel in a curved orbit under the influence of a magnetic field, and are therefore subject to centripetal acceleration. At a speed very near the speed of light, the relativistic particle emission is concentrated into a tight, forward radiation cone angle. There is a continuum of Doppler-shifted frequencies from the orbital frequency up to a cutoff. The radiation is also essentially plane-polarized in the orbit plane. However, in high-energy physics machines, the beam used in target or colliding-beam experiments would be somewhat unstable; thus, while pioneering experiments ensued through the 1970s, a considerable appetite was stimulated for machines dedicated to SR with stable source position, for fine focusing onto small samples such as crystals, and with a long beam lifetime for more challenging data collection. Crystallography has been both an instigator and major beneficiary of these developments through the 1970s and 1980s onwards. The evolution of new machines and the massive increase in source brilliance, year after year, are shown in Fig.[link]; the most recent additions are SPring-8 (8 GeV) and MAX2 (1.5 GeV), thus illustrating the need for a range of machine energies today (Fig.[link]. A general view of an SR source as exemplified by the SRS at Daresbury is shown in Fig.[link]. An example of a machine lattice (the ESRF) is shown in Fig.[link].


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(a) Evolution of X-ray source brilliance [(\hbox{photons s}^{-1} \;\hbox{mrad}^{-2} \;\hbox{mm}^{-2}\hbox{ per }0.1\hbox{\%} \ \delta\lambda/\lambda)] in the hundred years since Rontgen's discovery of X-rays in 1895. Adapted from Coppens (1992)[link]. (b) The evolution of storage-ring synchrotron-radiation sources over the decades, as illustrated by their increasing number and range of machine energies up to the present (Suller, 1998[link]).


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Overall layout of the Daresbury SRS facility, including LINAC, booster synchrotron, main storage ring and experimental beamlines (as in 1985 for clarity). Reproduced with the permission of CLRC Daresbury Laboratory.


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The ring tunnel and part of the machine lattice at the ESRF, Grenoble, France.

The properties of synchrotron radiation can be described in terms of the well defined quantities of high flux (a large number of photons), high brightness (also well collimated), high brilliance (also a small source size and well collimated), tunable, polarized, defined time structure (fine time resolution) and exactly calculable spectra. The more precise definitions of these quantities are [\displaylines{\hbox{Flux} \quad\! \ \ \ \ \;= \hbox{ photons per s per 0.1\% } {\delta \lambda /\lambda}, \hfill (\cr \noalign{\vskip5pt}\hbox{Brightness} = \hbox{ photons per s per 0.1\% } {\delta \lambda / \lambda} \hbox{ per mrad}^{2}, \hfill (\cr \noalign{\vskip5pt}\hbox{Brilliance } = \hbox{ photons per s per 0.1\% } {\delta \lambda / \lambda} \hbox{ per mrad}^{2} \hbox{ per mm}^{2}.\hfill\cr\hfill \ (}] Care needs to be exercised to check precisely the definition in use. The mrad2 term refers to the radiation solid angle delivered from the source, and the mm2 term to the source cross-sectional area.

Another useful term is the machine emittance, [epsilon]. This is an invariant for a given machine lattice and electron/positron machine energy. It is the product of the divergence angle, σ′, and the source size, σ: [\varepsilon = \sigma \sigma'. \eqno(] The horizontal and vertical emittances need to be considered separately.

The total radiated power, Q (kW), is expressed in terms of the machine energy, E (GeV), the radius of curvature of the orbiting electron/positron beam, ρ (m), and the circulating current, I (A), as [Q = 88.47 E^{4}I/\rho. \eqno(] The opening half-angle of the synchrotron radiation is [{1/\gamma}] and is determined by the electron rest energy, [mc^{2}], and the machine energy, E: [\gamma^{-1} = mc^{2}/E. \eqno(] The basic spectral distribution is characterized by the universal curve of synchrotron radiation, which is the number of photons per s per A per GeV per horizontal opening in mrad per 1% [{\delta\lambda /\lambda}] integrated over the vertical opening angle, plotted versus [{\lambda /\lambda_{c}}]. Here the critical wavelength, [\lambda_{c}\;(\hbox{\AA})], is given by [\lambda_{c} = 5.59\rho/E^{3}, \eqno(] again with ρ in m and E in GeV. Examples of SR spectral curves are shown in Fig.[link]. The peak photon flux occurs close to [\lambda_{c}], the useful flux extends to about [\lambda_{c}/10], and exactly half of the total power radiated is above the characteristic wavelength and half is below this value.


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SR spectra. (a) Brilliances of different SR source types (undulator, multipole wiggler and bending magnet) as exemplified by such sources at the ESRF. For the undulator, the tuning range (i.e. as the magnet gap is changed) is indicated. (b) Undulator-emitted spectra at the ESRF, shown as photon fluxes through a 1 × 0.5 mm aperture at 30 m, for three different gaps, i.e. widening the gap shifts the emitted fundamental and associated harmonics in each case to higher photon energies. Kindly provided by Dr Pascal Elleaume, ESRF, Grenoble, France.

In the plane of the orbit, the beam is essentially 100% plane polarized. This is what one would expect if the electron orbit was visualized edge-on. Away from the plane of the orbit there is a significant (several per cent) perpendicular component of polarization.


Blewett, J. P. (1946). Radiation losses in the induction electron accelerator. Phys. Rev. 69, 87–95.
Iwanenko, D. & Pomeranchuk, I. (1944). On the maximal energy attainable in a betatron. Phys. Rev. 65, 343.
Schwinger, J. (1949). On the classical radiation of accelerated electrons. Phys. Rev. 75, 1912–1925.

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