International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.2, pp. 53-54

Section 2.2.2.2.2. Undulators

A. Fitcha*

aESRF, 71 Avenue des Martyrs, CS40220, 38043 Grenoble Cedex 9, France
Correspondence e-mail: fitch@esrf.fr

2.2.2.2.2. Undulators

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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 on-axis cone (fundamental and odd harmonics), surrounded by rings from higher-order even harmonics. The flux density arriving on a small sample from this central cone is therefore very high. With high on-axis intensity, it is therefore the undulators that provide the beams with the highest spectral brightness at any synchrotron-radiation 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 by[\lambda_n = {1 + (K^2/2) + \gamma^2\theta^2\over 2n\gamma^2}\lambda_u,]which can be simplified on axis (θ = 0) to[\lambda_n\ [{\rm \AA}] = 1.3056{1+K^2/2 \over nE_e^2\ [{\rm GeV}]}\lambda_u \ [{\rm mm}]]or[\varepsilon_n\ [{\rm keV}] = 9.50 {nE_e^2\ [{\rm GeV}]\over \lambda_u\ [{\rm mm}](1 + K^2/2)}.]

On axis, only odd-numbered harmonics are emitted and it is these that are usually employed in a powder-diffraction experiment. The horizontal and vertical divergence of the radiation emerging from an undulator is of the order of [1/[(nN)^{1/2}\gamma]], where N is the number of magnetic periods making up the device. The spectrum of an undulator at a 6-GeV source with a 35-mm magnetic period is shown in Fig. 2.2.6[link]. By carefully shimming the magnetic lattice so that it is highly regular, the higher-order harmonics persist, allowing the undulator to be a powerful source of high-energy X-rays. Any imperfections in the magnetic periodicity cause the higher-order harmonics to broaden and fade away, reducing the utility of the device at higher energies.

[Figure 2.2.6]

Figure 2.2.6 | top | pdf |

Photon flux versus energy through a 1-mm2 aperture 30 m from the source, 0.1% bandwidth, for an ESRF u35 undulator (magnetic periodicity 35 mm, 1.6 m long, magnetic gap of 11 mm, peak magnetic field B0 = 0.71 T, electron energy 6 GeV, K = 2.31, storage-ring current 200 mA). Odd-numbered harmonics are labelled, which are those usually employed for powder-diffraction experiments as they have maximum intensity on axis.








































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