The interactions among the atoms in a solid enable the propagation of waves. In contrast to electromagnetic waves, which involve oscillating electric and magnetic fields, the waves of interest here involve mechanical movements of the atoms or molecules. Also in contrast to electromagnetic waves, which are transverse in nature, the mechanical waves in a solid have both transverse and longitudinal character. Liquids and gases also support mechanical waves, but only the longitudinal polarizations are possible due to the absence of shear moduli in those states of matter. Such waves in gases (air) are, of course, sound waves. This term is extended to solids (for both longitudinal and transverse polarizations) where the waves are called “infrasound,” “sound,” or “ultrasound” depending on whether the frequency is respectively below, in, or above the audible range. In common usage the term “acoustics” seems to refer to all types of mechanical waves in solids, liquids, and gases, and that is the sense in which the term is used in the present work.

The treatment of acoustic waves in solids falls naturally into two regimes, depending on wavelength. If the wavelength is much greater than the interatomic spacing, the material is treated as a continuum and classical elasticity applies. For shorter wavelengths the discrete nature of the lattice must be taken into account and the approach of lattice dynamics is used.

AcousticWaves in the Classical Elasticity Limit

Acoustic waves described by classical elasticity find many uses. Just to mention a few, these uses involve basic problems in condensed matter physics, material inspections, device operations, and seismology. The basic equation of motion was developed previously, Equation 2.29, but will be repeated here for convenience, Using Equations 2.9, 2.42, and 2.46, and making an exchange of dummy indices within a sum produces,

where, as usual, i, j, k run from 1 to 3. Combining the previous equations gives,

for the new equation of motion in terms of the displacements ui.

The use of ultrasonic methods for the study of materials continues to flourish and evolve. These methods find uses in many areas including fundamental condensed matter physics, materials science, various branches of engineering, geophysics, and applied studies of device-related material parameters. Advancements in experimental methods, especially resonant ultrasound spectroscopy, have enabled quantitative measurements on dramatically reduced specimen sizes, thereby vastly expanding the possibilities for the study of novel materials. The title of the book “Ultrasonic Spectroscopy” is taken here to mean simply the investigation of material properties by the use of ultrasonic waves.

A major purpose of this book is to present an in-depth coverage of the main issues underlying the planning and interpretation of ultrasonic investigations of materials. It is intended that the level of the presentation be accessible to dedicated upper-level undergraduate students, but at the same time achieve a depth of coverage useful to graduate students and other researchers. The approach is to present in careful detail a number of topics, with two objectives in mind. One objective, of course, is to educate the reader about basic concepts in the field – concepts that should become familiar to any researcher in this area. A second objective, perhaps more important, is to illustrate theoretical ideas that can be applied to a wide variety of problems. The emphasis is on basic concepts, not specific materials. The goal is to provide a fundamental background for beginning researchers so that – with the help of a good scientific Internet search engine to obtain more focused information – they are able to attack any of the gamut of interesting problems amenable to ultrasonic methods.

The mathematical methods used should be familiar to upper-level undergraduate students. Some knowledge of thermodynamics, statistical mechanics, and solidstate physics is required, but an effort is made to present the key concepts from these subjects as needed, and provide references to more detailed sources.

The book includes one chapter on experimental methods. Both continuous wave and pulse techniques are discussed.

In this Appendix the quasistatic approximation will be tested by applying Equations 5.72 to two specific cases: silver and diamond. For these examples the. values, i.e., initial values, will be taken to be the experimental 0 K results. The expansion due to the zero point motion is included in these parameters.

Silver

Table E.1 gives the third-order elastic constants for silver and also diamond, which will be discussed in the follwing example. For Ag, the experimental second-order elastic constants are found in Refs. [242] and [27]. Also needed to carry out the calculations is, the fractional change in length due to thermal expansion [243]. (The inset figure on p. 299 of Ref. [243] has an error. The 10−6 on the vertical axis should be instead 10−5). Figure E.1 is based on the data of Ref. [243].

The third-order elastic constants, the 0 K second-order elastic constants, and the fractional change in length due to thermal expansion will now be used in Equations 5.72 to calculate the temperature-dependent, second-order elastic constants of Ag. The calculated results for c11 are shown in Figure E.2 along with the experimental results.

Figure E.3 shows similar results for c12 and c44. Figures E.2 and E.3 show exceptionally close agreement between the experimental values and those calculated based on thermal expansion and third-order elastic for all three independent elastic constants for silver. Error bars resulting from errors in the measurements of the third-order elastic constants were calculated for Figures E.2 and E.3. These are barely visible in the figures. The calculated and experimental results are in good agreement, and provide strong support for the quasistatic approximation, at least for silver.

Diamond

As a second example, a very different material is considered: single-crystal diamond. The third-order elastic constants [245], the thermal expansion [246], and the temperature-dependent second-order elastic constants have all been measured [151]. Figure E.4 shows the fractional change in length for diamond over the temperature range of 0–300 K. The data for this graph were obtained by a numerical integration of figure 5 of Ref. [246]. Comparison of Figures E.1 and E.4 show that diamond has a much smaller change in length than silver, and the temperature dependence is quite different.