International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 
International Tables for Crystallography (2018). Vol. H, ch. 2.2, pp. 5154
Section 2.2.2. Production of synchrotron radiation^{a}ESRF, 71 Avenue des Martyrs, CS40220, 38043 Grenoble Cedex 9, France 
Synchrotron radiation is emitted by charged particles travelling at relativistic speeds when they are accelerated to move in a curved trajectory. In a modern synchrotron facility dedicated to the production of Xray beams for scientific experiments, electrons are circulated in a closed horizontal orbit in a storage ring at an energy of several GeV, steered by magnetic fields from bending magnets. The overall circumference of the orbit can be several hundred metres depending on the design and specifications of an individual ring. The synchrotron ring is built up of cells (Fig. 2.2.1) comprising a straight section and a bending magnet by which the electrons are guided into the following straight section. Beamlines emerge tangentially from the bending magnets where synchrotron radiation is emitted by the electrons as they curve from one straight section into the next. Beamlines are also constructed on the straight sections where insertion devices, arrays of magnets providing an alternating magnetic field, are placed to cause the path of the electrons to oscillate and so also emit synchrotron radiation. By choosing the period of the magnetic array and by varying the strength of the magnetic field, the wavelength distribution and divergence of the Xrays emitted from an insertion device can be controlled. A straight section may accommodate more than one insertion device in series, allowing greater intensity or flexibility in the emitted radiation for the associated beamline. In the storage ring, the energy that the electrons lose by emitting synchrotron radiation is replaced by coupling the electrons to radiofrequency radiation supplied from klystrons or solidstate devices. Thus the synchrotron facility converts electrical energy, via radio waves and relativistic electrons, into powerful beams of electromagnetic radiation.
One key parameter of a storage ring is the energy of the circulating electrons. The energy of an electron moving with speed v iswhere m_{e} is the rest mass of the electron, 9.10938356 (11) × 10^{−31} kg. The term is referred to as γ and is the factor by which the mass of the electron increases from its rest mass because of its relativistic speed. Expressed in eV (the conversion factor from kg to eV is c^{2}/e), the electron rest mass is 5.109989461 (31) × 10^{5} eV, so thatwhen E_{e} is given in the customary units of GeV. Thus for a 3GeV machine, a common energy for a synchrotronradiation source, γ has the value of 5871. The mass of an electron with energy 3 GeV is therefore 3.22 atomic mass units, so around 7% more massive than a stationary atom of ^{3}H or ^{3}He.
Electrons do not circulate individually in the storage ring but in a series of bunches that are in phase with the accelerating radio frequency. Radiation is therefore emitted in pulses as each bunch passes through a bending magnet or insertion device. Thus the number and distribution of the electron bunches around the orbit determine the time structure of the emitted radiation. For most powderdiffraction applications using synchrotron radiation, the pulsed nature of the source can be neglected and the radiation can be regarded as continuous, although attention should also be paid to the performance of detectors that are more susceptible to pulse pileup problems when the radiation arrives at very high average rates or in concentrated bursts (Cousins, 1994; Laundy & Collins, 2003; Honkimäki & Suortti, 2007), which can happen with certain bunchfilling modes. Certain specialized experiments requiring very fast time resolution can exploit the time structure of the source. In such experiments the longitudinal dimension of the bunches controls the pulse duration, which is usually a few tens of picoseconds.
In discussing the performance of different beamlines, the spectral brightness (Mills et al., 2005) is often quoted for the source and is defined aswhere `0.1% bandwidth' represents δλ/λ = 0.001, the mrad^{2} term expresses the solidangle of the emission of the Xrays from the source and the mm^{2} term relates to the crosssectional area of the source. Thus a source of high spectral brightness emits many photons per second of the specified energy, into a narrow solid angle, with a small source size. The source size, which may well differ in the horizontal and vertical directions, is an important consideration as source size and beam divergence ultimately limit the performance of the beamline optical system in terms of collimation, energy resolution and focal spot size.
A bending magnet provides a vertical magnetic field to deflect the electrons laterally in the horizontal plane from a straightline trajectory, and thereby causes the emission of synchrotron radiation (see Fig. 2.2.2). The lateral Lorentz force, F, acting on an electron travelling at velocity v in a magnetic field B is mutually perpendicular to both the magnetic field and the direction of travel of the electron, and is given by

Emission of a fan of radiation by the electron beam as it curves in a bending magnet from one straight section of the ring to the next. 
In a bending magnet the magnetic field is applied over an extended distance leading to a curved path of radius ρ. The centripetal acceleration is F/γm_{e}, which for circular motion is equal to v^{2}/ρ. Since v ≃ c,so the radius of curvature decreases with magnetic field strength and increases with machine energy via increased γ. With the electron energy expressed in GeV, this can be approximated to ρ ≃ 3.34E_{e} [GeV]/B (where 10^{9}/c ≃ 3.34).
Synchrotron radiation is emitted in a forward cone tangential to the direction of the electrons' motion (Fig. 2.2.3) with a nominal Gaussian distribution and an opening angle of the order of 1/γ. Thus the radiation is highly collimated in the vertical plane. In the horizontal plane, synchrotron radiation is emitted in a broad fan, tangential to the curved trajectory of the electrons as they sweep through the bending magnet. Only a fraction of the radiation emitted by a bending magnet enters the associated beamline via a cooled aperture defining a horizontal acceptance angle of a few mrad. The radiation is polarized in the plane of the synchrotron orbit. Sometimes, more than one beamline can be built on a bending magnet with a suitable angular separation between them.

Synchrotron radiation is emitted in a cone of opening angle of the order of 1/γ tangential to the electrons as they follow a curved trajectory through the bending magnet. 
Photons are emitted over a broad spectral range. The critical photon energy, _{c}, divides the emitted power into equal halves and is given byor, with photon and electron energies in keV and GeV, respectively,
The higher the critical energy, the greater the number of photons produced with short Xray wavelengths. As an example, consider a bending magnet at the ESRF in Grenoble, France, which has a 6GeV storage ring and bending magnets with a field of 0.85 T. The bending radius is 23.5 m and the critical photon energy is 20.3 keV (equivalent to a wavelength of 0.61 Å). The spectrum of such a device is shown in Fig. 2.2.4.
The vertical collimation of the radiation varies with photon energy in a nonlinear manner (Kim, 2001). Nevertheless, the divergence decreases with increased photon energy, so beams with the shortest wavelengths are the most vertically collimated. Various approximations can be written to describe the variation, such as for a single electron (Margaritondo, 1988),where σ_{v}() is the standard deviation of the verticaldivergence distribution of photons of energy . For a population of electrons circulating in a storage ring, the distribution of the trajectories with respect to the plane of the orbit (of the order µrad) must also be considered, as these add to the vertical emission distribution. An approximation such aswill often be adequate to estimate the vertical divergence Ψ_{v} in the vicinity of _{c}. Thus for the bending magnet illustrated in Fig. 2.2.4, photons at the critical energy of 20.3 keV will have a vertical divergence of ∼100 µrad. A beamline would probably accept less than this, e.g. a 1.5mmhigh slit at 25 m from the source defining the beam onto a monochromator crystal defines an angle of ∼60 µrad.
Insertion devices can be classified into two main types, termed `wigglers' and `undulators', illustrated in Fig. 2.2.5. A wiggler has a relatively long magnetic period and the radiation from each oscillation is emitted like a series of powerful bending magnets, summing together to provide increased intensity. An undulator has a relatively short magnetic period and the radiation from sequential oscillations interferes coherently to give modified beam characteristics.
For insertion devices the magnetic field acting on the electrons varies sinusoidally along the device,where B_{0} is the peak magnetic field, z is the distance along the insertiondevice axis and λ_{u} is the magnetic period. With a vertical field, the alternating magnetic field causes the electron path to oscillate in the horizontal plane. Note that the radiation is emitted mainly towards the outsides of the oscillations where the electrons change transverse direction, and where the magnetic field and beampath curvature are highest. The maximum angular deflection of an electron from the axis of the insertion device is K/γ, where the deflection parameter K is given bywhich simplifies to K = 0.0934B_{0}λ_{u} [mm] with λ_{u} expressed in mm. K is a crucial parameter that determines the behaviour of the insertion device.
If K is large (10 or above), the insertion device is a wiggler and the electrons oscillate with an amplitude significantly greater than the emitted radiation's natural opening angle 1/γ. Every oscillation along the device produces a burst of synchrotron radiation and these add together incoherently so increasing the flux in proportion to the number of magnetic periods. The radiation emerges from the wiggler in a horizontal fan with a horizontal opening angle ∼2K/γ. The intensity of a wigglerbased beamline can be very high because each oscillation produces synchrotron radiation, and this radiation is directed close to the axis of the device. Like a bending magnet, wigglers produce a continuous spectrum but with the critical energy shifted to harder energies because the magnetic field is (usually) greater. Thus for a wiggler at a 6GeV source, with a magnetic field of 1.2 T and a magnetic period of 125 mm, K is 14, the maximum deflection of the electrons from the straightline path is 1.2 mrad and the critical photon energy is 28.7 keV. Magnetic fields of several tesla can be exploited using superconducting magnets to obtain even higher critical photon energies.
If the value of K is 2 or less, the insertion device is an undulator. The deflection of the electrons is comparable to the natural opening angle of the emitted radiation 1/γ. Radiation emitted from sequential oscillations interferes coherently, and the beam becomes highly collimated in the horizontal and vertical directions. Thus, the radiation from an undulator is concentrated into a central onaxis cone (fundamental and odd harmonics), surrounded by rings from higherorder even harmonics. The flux density arriving on a small sample from this central cone is therefore very high. With high onaxis intensity, it is therefore the undulators that provide the beams with the highest spectral brightness at any synchrotronradiation source. The interference also modifies the spectrum of the device, which has a series of harmonics derived from a fundamental energy. At a horizontal angle θ to the axis of the insertion device, the wavelength of harmonic n is given bywhich can be simplified on axis (θ = 0) toor
On axis, only oddnumbered harmonics are emitted and it is these that are usually employed in a powderdiffraction experiment. The horizontal and vertical divergence of the radiation emerging from an undulator is of the order of , where N is the number of magnetic periods making up the device. The spectrum of an undulator at a 6GeV source with a 35mm magnetic period is shown in Fig. 2.2.6. By carefully shimming the magnetic lattice so that it is highly regular, the higherorder harmonics persist, allowing the undulator to be a powerful source of highenergy Xrays. Any imperfections in the magnetic periodicity cause the higherorder harmonics to broaden and fade away, reducing the utility of the device at higher energies.
For insertion devices the magnetic field can be modified by changing the vertical distance between the magnetic poles. By opening the gap, the magnetic field and K decrease followingwhere B_{r} is proportional to the remanent magnetic field, which depends upon the nature of the magnets used in the insertion device, and G is the magnetic gap. Decreasing K for an undulator means that the energy of the fundamental harmonic increases; however, this is at the expense of the intensities of the higher harmonics. Thus the insertion device can be tuned to produce high intensity at the wavelength most suitable for a particular measurement. The smallest gap possible for a device depends on the design of the storagering vacuum vessel in which the electrons circulate. It is difficult to have a vessel smaller than about 10 mm high, and hence for an externally applied field a minimum magnetic gap of about 11 mm is to be expected. For smaller gaps, the magnets must be taken into the vacuum of the storage ring, a socalled `invacuum' insertion device.
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