Page 19 - Winter 2008

RESONANCES AND TIME-OF-FLIGHT: ULTRASONIC PROBES OF CONDENSED MATTER GO DIGITAL
Albert Migliori
National High Magnetic Field Laboratory, Los Alamos National Laboratory Los Alamos, New Mexico 87545
Introduction
Perhaps the speed of sound is one of the most fundamental and most often measured attributes of a solid. Acoustic students have been doing such measurements for years, seldom appreci- ating the wealth of knowledge that can be obtained from the solid from this appar- ently simple experiment. But to under- stand exactly what has been measured, what has influenced the measurment, how best to perform the measurement, and how accurately the measurement has been taken has often demanded a life- time of experience. This paper describes some of the latest techniques that con- densed matter physicists now use to probe solids.
The elastic stiffnesses of a solid can
be determined with outstanding preci-
sion. Together with density, the stiffnesses control the speed of propagation of stress waves (sound) and depend on the variation of fundamental thermodynamic quantities—inter- nal energy or free energy—with deformation. Unlike most of the quantities used to characterize condensed matter, the elastic moduli are fourth-rank tensors containing a wealth of detail, directional information, and consistency constraints that provide one of the most revealing probes of solids. Let us review this briefly by expanding on the concepts underlying stiffness, thermodynamics, and deformation in solids, and then examine two advances in measurement techniques.
The microscopic complexity of crystalline solids, as opposed to a gas or a liquid, reveals that many independent elastic moduli are present. This complicates everything. For example, in a triclinic crystal, the shape and structure of a unit cell includes nothing at right angles to anything else. A consequence of the tensor description of elasticity is that there are 21 independent elastic moduli, and there are no elastic waves for which displacements of atoms are either par- allel to (compression) or perpendicular to (shear) the propa-
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gation direction. The situation is worse if the time-reversal
symmetry-breaking of a magnetic field is present. This all occurs because any stress applied to a solid’s surface can be decomposed into two shear components parallel to the sur- face and a compressional one perpendicular in each of the three spatial directions for a total of 9 stress components. With the corresponding nine strain components, the Hooke’s law matrix connecting stress to strain must have 81 compo- nents. However, time reversal symmetry and shear symmetry under interchange of coordinates requires a constrained symmetric matrix. Thus the total in zero magnetic field is
“Recently, full utilization of digitally generated and acquired signals has made possible advances in measurement techniques for both Pulse Echo (PE) and Resonant Ultrasound Spectroscopy (RUS) that are not possible with purely analog systems.”
reduced to 21 (the maximum number of independent entries in a symmetric 6x6 matrix). Even in a simple cubic crystal, three moduli are required, and only in special directions can elastic waves be called shear or compressional. Therefore, a complete measurement of the elastic stiffness tensor of a solid requires a lot of work
In well-behaved solids, the effects of very small strains are reversible and the state of the solid in thermal equilib- rium uniquely relates pressure, volume, and temperature via the equation of state (EOS). Changing both volume and pressure, for example, forces a unique change in temperature. For shear waves, the EOS does not come into play because there are no volume changes during propagation. But for compres-
sional waves, the volume change between the compressed and expanded parts of a single cycle of sound can produce small temperature differences. Rather oddly, these tempera- ture differences disappear at very high frequency. The root of this is that the thermal penetration depth δK, the characteris- tic length for decay of temperature changes, varies inversely as the square root of the frequency, while the wavelength λ is proportional to the inverse of frequency. Thus λ becomes shorter than δK at high frequency, permitting nearly instanta- neous thermal equilibration. This has important conse- quences—there are two longitudinal sound speeds, isother- mal and adiabatic, and the adiabatic sound speed, the one usually measured by both time-of-flight and resonance tech- niques, depends on the thermodynamic internal energy, not the free energy as in isothermal sound.
The adiabatic elastic moduli are often the very first pre- diction of any theory of the electronic structure of a solid because the theorist needs only to change the distance between atoms (the volume) and rerun the code to get the moduli. Thus adiabatic sound speeds provide an important test of electronic structure while all elastic wave speeds pro- vide crucial characterization of phase changes (when the solid changes from one set of attributes to another).
For these and many other reasons, the measurement of the elastic moduli of solids has a long and glorious tradition with many methods used, ranging from neutron scattering to optical spectroscopies. But the two most widely used meth- ods are the time of flight (TOF) of an acoustic pulse2 (pulse- echo (PE)) and frequency of resonances (resonant ultra-
sound spectroscopy (RUS)), both extensively reviewed else-
where.
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Both methods use readily available hardware and
Digital Ultrasonics 17