International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 88-89
Section 1.3.4.6. Experimental determination of elastic constants
a
Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France |
As mentioned in Section 1.3.4.1, the elastic constants of a material can be obtained by the elastic response of the material to particular static forces; however, such measurements are not precise and the most often used approach nowadays consists of determining the velocity of ultrasonic waves propagating along different directions of the crystal and calculating the elastic constants from the Christoffel determinants (1.3.4.8). The experimental values are often accurate enough to justify the distinction between static and dynamic values of the elastic constants and between phase and group velocities, and the careful consideration of the frequency range of the experiments.
The use of the resonance technique is a well established approach for determining the velocity of sound in a gas by observing nodes and antinodes of a system of standing waves produced in the so-called Kund tube. In the case of transparent solids, optical means allow us to visualize the standing waves and to measure the wavelength directly (Zarembowitch, 1965). An easier procedure can be used: let us consider a transparent crystal in the shape of a parallelepiped (Fig. 1.3.4.1). A piezoelectric transducer is glued to the crystal and excited at varying frequencies. If the bonding between the transducer and the crystal is loose enough, the crystal can be considered as free from stress and the sequence of its resonance frequencies is given bywhere n is an integer, V the phase velocity of the wave in the direction orthogonal to the parallel faces and l the distance between these faces.
The looseness of the bonding can be checked by the regularity of the arithmetic ratio, . On account of the elasto-optic coupling, a phase grating is associated with the elastic standing-wave system and a light beam can be diffracted by this grating. The intensity of the diffraction pattern is maximum when resonance occurs. A large number of resonance frequencies can be detected, usually more than 100, sometimes 1000 for non-attenuating materials. Consequently, in favourable cases the absolute value of the ultrasonic velocity can be determined with an uncertainty less than .
Pulse-echo techniques are valid for transparent and opaque materials. They are currently used for measuring ultrasonic velocities in solids and can be used in very simple as well as in sophisticated versions according to the required precision (McSkimmin, 1964). In the simplest version (Fig. 1.3.4.2), an electronic pulse generator excites the mechanical vibrations of a piezo-electric transducer glued to one of two plane-parallel faces of a specimen. An ultrasonic pulse whose duration is of the order of a microsecond is generated and transmitted through the specimen. After reflection at the opposite face, it returns and, when it arrives back at the transducer, it gives rise to an electronic signal, or echo. The whole sequence of such echos is displayed on the screen of an oscilloscope and it is possible to measure from them the time interval for transit. Usually, X-cut quartz crystals or ferroelectric ceramics are used to excite longitudinal waves and Y-cut quartz is used to excite transverse waves. In many cases, a circulator, or gate, is used to protect the receiver from saturation following the main `bang'. This method is rough because the beginning and the end of a pulse are not well characterized. Several improvements have therefore been made, mainly based on interferometric techniques (pulse-superposition method, `sing around' method etc.). Nevertheless, if the absolute value of the ultrasonic velocity is not determined with a high accuracy by using pulse-echo techniques, this approach has proved valuable when relative values of ultrasonic velocities are needed, e.g. temperature and pressure dependences of ultrasonic velocities.
References
Brillouin, L. (1932). Propagation des ondes électromagnétiques dans les milieux matériels. Congrès International d'Électricité, Vol. 2, Section 1, pp. 739–788. Paris: Gauthier-Villars.Google ScholarMcSkimmin, H. J. (1964). Ultrasonic methods for measuring the mechanical properties of solids and fluids. Physical acoustics, Vol. IA, edited by W. P. Mason, pp. 271–334. New York: Academic Press.Google Scholar
Michard, F., Zarembowitch, A., Vacher, R. & Boyer, L. (1971). Premier son et son zéro dans les nitrates de strontium, barium et plomb. Phonons, edited by M. A. Nusimovici, pp. 321–325. Paris: Flammarion. Google Scholar
Zarembowitch, A. (1965). Etude théorique et détermination optique des constantes élastiques de monocristaux. Bull. Soc. Fr. Minéral. Cristallogr. 28, 17–49.Google Scholar