research papers 888 Capitan and Alvarez  Quasicrystal analyser J. Appl. Cryst. (2000). 33, 888±892 Journal of Applied Crystallography ISSN 0021-8898 Received 14 April 1999 Accepted 4 February 2000 # 2000 International Union of Crystallography Printed in Great Britain ± all rights reserved Icosahedral quasicrystals applied as a harmonic-free crystal analyser M. J. Capitan* and J. Alvarez Dpto. Fisica de la Materia Condensada, Universidad Autonoma de Madrid, 28049-Madrid, Spain. Correspondence e-mail: capitan@hobbes.fmc.uam.es The use of quasicrystals as crystal analysers has been studied in monochromatic X-ray synchrotron radiation applications. The quasiperiodicity of their structures gives a theoretical third-harmonic rejection factor of up to 5  10 4 when the icosahedral (20,32) Bragg peak is used. This ratio is compared with the intrinsic value for silicon perfect crystals in Bragg con®guration, namely 2  10 2 . This rejection factor may be increased for some well de®ned con®gurations, as reported elsewhere for the case of periodic crystals. 1. Introduction The application of synchrotron radiation in the characteriza- tion of materials by means of X-ray techniques has progressed greatly in recent years. This expansion has been possible thanks to the advent of synchrotron radiation sources and their speci®c properties. In particular, two characteristics of synchrotron sources, namely the large range of energy emis- sion available for tuning and the high brilliance, have neces- sitated reviews of the beamline optical elements (Malgrange, 1992; Matsushita & Hashizume, 1992; Freund et al., 1990; Paquin & Howells, 1997; Freund, 1998). The large energy spectrum of synchrotron radiation implies the necessity of monochromatization for experiments in which only one particular wavelength with a given spread is required at a time. The energy tunability is extensively achieved by Bragg optics and more speci®cally by means of perfect single crystals. The energy selection by the monochromator is made according to Bragg's law. Thus, for a given monochromator con®guration, the wavelengths /n (n = 1, 2, 3, ... ) are all allowed, provided that the diffraction peaks are not forbidden by the crystal- lographic structure. Usually, the harmonic rejection is controlled by mirrors which act as high-energy cut-off ®lters. Nowadays, for the third-generation synchrotron sources, mirrors are used for energies up to 30 keV, or higher. This imposes considerable restrictions on the materials and roughness of the mirrors (Freund et al., 1990; Paquin & Howells, 1997). Apart from mirrors, two different possibilities have been studied for the harmonic rejection: either making an off-tuning of the second crystal in a double-crystal mono- chromator or using a three-axis con®guration with a crystal analyser, different to that used for the monochromator (Bonse et al., 1976; Matsushita & Hashizume, 1992; Freund, 1998). After the discovery of quasicrystals (Shechtman et al., 1984), a large number of studies were performed with the aim of determining the atomic structures of these intriguing materials. These alloys are characterized by ®vefold or tenfold axis symmetry in their structure. This symmetry implies that the atomic structure is non-periodic in three-dimensional space. However, quasicrystals have well de®ned Bragg-peak diagrams. Thus, stable icosahedral phases like i-AlPdMn and i- AlCuFe have proved to be of very high structural quality compared to the usual intermetallic structures (Kycia et al., 1993; Janot & Dubois, 1995). The quasiperiodic order induces marked changes of intensity between the nq-related Bragg peaks. Based on this property of quasiperiodicity, Sasaki et al. (1995) proposed the use of a quasiperiodic undulator in order to change the intensity ratio between the harmonic peaks; this quasiperiodic undulator has already been applied in some beamline designs (Chavanne et al. , 1998). The harmonic intensity distribution of the undulator spectrum is markedly changed by its one-dimensional quasiperiodicity. This quasi- undulator allows the third-harmonic intensity to be decreased by a factor of 8 with respect to a periodic undulator (Chavanne et al., 1998). The improvement of the harmonic rejection by these devices is limited by the presence of a background of continuous energy distribution after the undulator. However, this solution is not applicable to beamlines that have wigglers as insertion devices, or to bending-magnet beamlines. This article proposes another solution based on the same principle. The aim of this article is to show that, pro®ting from the quasiperiodicity of icosahedral quasicrystals, a new kind of crystal analyser with a high harmonic rejection ef®ciency can be proposed. 2. Experimental details A single grain (of size 18 mm) of the icosahedral phase of the ternary alloy AlPdMn was grown from the melt by the Czochralski method using a twofold-axis seed quasicrystal. The sample was cut, perpendicularly to the growth direction, from the mother single grain, which had been previously annealed at 1073 K. The disc was mechanically polished and subsequently annealed at 873 K in high vacuum in order to relax the internal stresses produced during the polishing. The