c&J . __ __ lif!J ELSEYIER Nuclear Instruments and Methods in Physics Research A 391 (1997) 485-491 zyxwvutsrqponmlkjihgfedcbaZYXW NUCLEAR INSTRUMENTS 8 METHODS IN PHYSICS RESEARCH zyxwvutsrqponm Secmn A Neutron diffraction from sound-excited crystals A. Remhof”, K.-D. Li13b, A. Magerl”.” a Ruhiutlioersith’t Bochttm. D-44780 Bochurn. German? b European $vnchvotron Radiation Facilip, F-381)43 Grenoble Crdex 9. France ’ hstitut lvlas NW Lalre-Paul Lang&n. F-38041 Grenoble. France Received 20 November 1996; revised form received 11 February 1997 Abstract A longitudinal sound wave modulates the regular arrangement of the atomic planes of a crystal in two ways: first, the spacing between the atomic planes is modified in the regions of compression and expansion introducing a macroscopic strain and, second, the lattice planes acquire a velocity in the oscillating strain field. Bragg reflection in a strained crystal maintains the energy of the radiation, whereas Bragg reflection by a moving lattice provokes a Doppler shift of the radiation. In a diffraction experiment both these effects lead to an enlarged bandwidth of the reflection curve. The relative importance of strain and Doppler depends mainly on the radiation used. For thermal neutron scattering the profile of the rocking curve of a Bragg reflection may permit to separate the two effects. Atomic amplitudes of the sound field of 136 A peak to peak in the bulk of the crystal can be deduced from the rocking profile. The enlarged bandwidth of a sound-excited crystal opens a possibility for diffraction-based optical elements where the trade-off between resolution and intensity can be readily modified. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA PACS: 61.12; 63.20.-e; 03.75.B zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Keyovds: Diffraction; Solid state dynamics; Ultrasound; Neutron optics - 1. Introduction Exciting a longitudinal sound wave in a crystal causes an oscillatory modulation of the lattice parameter in space and time. For a standing wave the positions of the antinodes are stationary whereas the amplitude changes with the sound frequency. These modulations vary the lattice parameter introducing strain- and Doppler gradi- ents. In analogy to a mosaic crystal [l], the diffraction behaviour of a vibrating crystal can be understood as- suming that it consists of small volume elements with a perfect structure. Within each of these strain blocks, there is a well-defined value for the lattice spacing with a negligible gradient. Similarly, the crystal may be bro- ken up into velocity blocks, where all the lattice planes within a block have the same velocity. The sizes of the strain blocks and the velocity blocks may be different. They depend on the material and the reflection used, on *Corresponding author. Tel.: f33 76 207383: fax: f33 76 483906; e-mail: magerl@ill.fr. the amplitude and the frequency of the sound wave, and on the particle velocity of the radiation. In any case, the thickness of the blocks must be chosen such that the variation of the parameter in question within a block is smaller than its variation allowed by the Darwin width from dynamical diffraction [2]. A non-excited crystal will be composed out of one block and its thickness may exceed the extinction length of an ideal crystal. In this case, the diffracted intensity will be limited according to the dynamical theory. At large deformations, both the gradient blocks and the velocity blocks may become thin enough such that primary extinction becomes negligible and the diffracted intensity will approach the kinematical limit. Unlike mosaic blocks, which are tilted against each other, the surface normals of both the gradient blocks and the velocity blocks are parallel to each other and they are also parallel to the propagation vector of the sound wave. Thus, in case the diffraction vector is also chosen parallel to the propagation vector of the wave as in the present experiment, a sound-excited crystal represents an oscillating gradient crystal. Such crystals are known to have favourable diffraction properties 13.41. E.g., they maintain the beam divergencies after 016%9002/97/$17.00 Copyright :CI 1997 Elsevier Science B.V. All rights reserved PI1 SO168-900’(97)0041 l-7