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.

The discussion of experimental methods has been deferred until this point, because relevant background material was covered in Chapters 2 and 3. The usual experimental objectives are to determine one or both of the following quantities: 1. The elastic constants or, equivalently, the ultrasonic velocities; and 2. The dissipation or loss. The second property is known by several names, often depending on the method of measurement – ultrasonic attenuation, internal friction, logarithmic decrement, inverse Q, etc. Many different experimental techniques have been developed over the years, and there are various ways to categorize them. One possible approach is to divide the different methods into continuous wave techniques, in which a standing wave resonance is set up in the specimen, and pulse methods in which a short pulse of ultrasound is sent through the specimen, sometimes called the pulse-echo technique. However, when considering the problems of sample preparation and orientation, transducers, and corrections for non-ideal experimental configurations, it seems better to divide the methods into: 1. plane-wave propagation methods; and 2. methods not based on the plane-wave approximation.

In the discussion to follow, a few seminal papers will be cited, but the emphasis will be on modern techniques. It is not possible to cite the many, many scientists who have contributed to the development of the present-day methods.

Plane-Wave Propagation Methods

The plane-wave propagation methods are divided naturally into pulse techniques and continuous wave (resonance) techniques. After a short discussion of common problems, the pulse and resonance methods will be discussed separately below.

Figure 4.1 shows a typical sample-transducer arrangement used in the planewave propagation methods. Configurations with a single transducer, the same one being used for transmitting and receiving, are also used. The specimen is prepared with flat and parallel end faces. The transducers, which convert electrical voltages to mechanical displacements and vice-versa, are usually specially oriented cuts of piezoelectric materials and are commercially available. Single-crystal quartz, PZT, and lithium niobate are common transducer materials. Polyvinylidene fluoride (PVDF) piezoelectric film transducers have also been used [68].

The present chapter will serve as an overview of the material to be presented in the rest of the book. While it is hoped that the material will prove useful to all those involved in or interested in the use of ultrasound as a probe of condensed matter, a special effort is made to present the material in sufficient detail so as to be helpful to dedicated, upper-level undergraduate students and beginning graduate students. Scientists from several different disciplines are nowadays finding ultrasonic spectroscopy a useful tool, thus a strong background in solid-state physics, statistical physics, and quantum mechanics is not assumed of the readers. Brief background material is presented as needed. Several monographs have contributed to the advancement of ultrasonic spectroscopy, among them References [1, 2, 3]. The author is deeply indebted to those who have helped develop the field of ultrasonic studies of materials.

Chapter 2 deals with classical elasticity; the solid is treated as a continuum. The continuum approximation is valid for virtually all ultrasonic experiments. The present treatment of elasticity is more extensive than is usually found in books on ultrasonic techniques, but this more extensive treatment seems important if the researcher is to understand the widest implications of her/his ultrasonic research. Basic physical parameters in this chapter are stress (a two-index tensor), strain (a two-index tensor), and elastic constants (a four-index tensor, which by Hooke's Law connects stress and strain). Thus, many indices and sums over these indices appear frequently. For pedagogical reasons, it was decided not to use the elegant Einstein summation convention. For those new to the field, it seems better to write out the sums explicitly. The relation of elastic constants to thermodynamic potentials is derived. The condensed (Voigt) notation for stress, strain, and elastic constants is explained in detail. Coordinate transformations are treated. The form of the elastic constant matrix for each of the seven crystal systems is derived, as well as the form for the icosahedral quasicrystal.