Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 8.1, pp. 189-190   | 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 single fibres, 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. An example of a machine lattice (the ESRF) is shown in Fig.[link].


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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 angular brightness (also well collimated), high spectral brightness (also a small source size and well collimated), tunable, polarized, defined time structure (fine time resolution) and exactly calculable spectra. The precise definition of the spectral brightness is[\eqalignno{\hbox{Spectral brightness} &= \hbox{photons per s per mm}^2 \hbox{ per mrad}^{2}&\cr&\quad \hbox{ per (0.1\% bandwidth)}.&(}]Care needs to be exercised to check precisely the definition in use (Mills et al., 2005[link]). The mrad2 term refers to the radiation solid angle delivered from the source, and the mm2 term to the source cross-sectional area. Mills et al. (2005[link]) concluded that the units given in equation ([link], which do not follow the SI code for units, are so ensconced in the field `that a drive to change this would only lead to more confusion rather than more clarity in the descriptions of synchrotron-radiation sources'. At the ESRF the term brilliance is firmly ensconced in house and with its large user community, and so is the label for the y axes used in Fig[link].


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SR spectra. (a) Spectral brightness (also referred to as brilliance) of different SR source types (undulator, multipole wiggler and bending magnet) as exemplified by such types of sources at the ESRF. For the undulator, the tuning range (i.e. as the magnet gap is changed) is indicated. (b) Brilliance produced by the in-vacuum undulator of cell 27 of the ESRF dedicated to high-pressure studies. The plain curve corresponds to the condition in use as of September 2010. Further increase in brilliance (dotted curve) is expected in the years to come by increasing the ring current, increasing the length of the undulator and further decreasing the vertical emittance. Kindly provided by Dr Pascal Elleaume, ESRF, Grenoble, France.

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 critical wavelength and half is below this value.

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. Schiltz & Bricogne (2009[link]) advocated definitions to use in the analysis of polarization-dependent phenomena that are instrument-independent and completely general. They have implemented these methods in the macromolecular phasing program SHARP for exploiting the polarization anisotropy of anomalous scattering in protein crystals.


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.
Mills, D. M., Helliwell, J. R., Kvick, Å., Ohta, T., Robinson, I. A. & Authier, A. (2005). Report of the Working Group on Synchrotron Radiation Nomenclature – brightness, spectral brightness or brilliance? J. Synchrotron Rad. 12, 385.
Schiltz, M. & Bricogne, G. (2009). Instrument-independent specification of the diffraction geometry and polarization state of the incident X-ray beam. J. Appl. Cryst. 42, 101–108.
Schwinger, J. (1949). On the classical radiation of accelerated electrons. Phys. Rev. 75, 1912–1925.

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