Degeneracy and the slowness surface

All the intricate focusing effects considered in Chapter 4 occur for crystals with very high (cubic) symmetry. One might rightly wonder what added complexity appears for crystals of lower symmetry. That is the subject of this chapter. We shall see that the phonon-focusing patterns of real crystals show many variations and yet have underlying similarities. We are aided by the fact that the acoustic symmetry – i.e., the symmetry displayed by the slowness surface – is generally higher than the point-group symmetry of the crystal. For example, all crystals possess centrosymmetric slowness surfaces (i.e., they are invariant to the inversion k → −k), but not all crystal structures have a point of inversion.

Further good news is that we have already been introduced to all the basic topological features. At lower symmetries, there are still only three slowness sheets, which can contact each other conically, tangentially, or, in rare cases, along a line. Just how these degeneracies are distributed over the slowness surface, however, will vary from one crystal class to the next.

The accompanying bad news is that, with the exception of cubic and hexagonal crystals, it is virtually impossible to study systematically the multidimensional elastic parameter spaces. Figure 1 shows several crystal types and the elastic parameters required to describe each. While cubic crystals have only two independent ratios of elastic parameters (a = C11/C44 and b = C12/C44), hexagonal crystals have four ratios, implying a four-dimensional elastic-parameter space.

Time, space, and energy. These are the parameters of a heat-pulse experiment. The principal subject of investigation is nonequilibrium acoustic phonons in a crystalline solid. A phonon is the elementary quantum of vibrational energy of the atoms in the crystal. Historically, the thermal properties of solids have been measured by static or steady-state methods such as heat capacity and thermal conductivity. It is not surprising, then, that time- and spaceresolved measurements of nonequilibrium phonons have provided new perspectives on the thermal properties of condensed matter. The introduction of the heat-pulse method in 1964 by von Gutfeld and Nethercot has certainly revolutionized the way we view and understand phonons.

The basic method is quite simple: A nonmetallic crystal is cooled to a few degrees above absolute zero. A heat pulse is produced by passing a short burst of current through a metal strip deposited on the crystal or, alternatively, by optically exciting a metal film that has been deposited on one surface of the crystal. The electrically or optically excited film acts as a Planckian emitter of heat. Figure 1(a) schematically depicts the spatial distribution of heat energy (high-frequency acoustic phonons) at some instant of time after the excitation pulse. Several nonspherical “shells” of thermal energy, corresponding to longitudinal and transverse modes, propagate away from the heat source. The rise in temperature on the opposite face of the crystal is monitored with a superconducting metal film detector, labeled D.

Having examined the theoretical foundations for phonon scattering at an interface, we now turn to a variety of experiments that involve the interface between two solid materials. It is no surprise that the transmission and reflection of phonons at the boundary of two media have great practical significance, ranging from the dissipation of heat generated by an electrical device to the coupling of phonons into a thin-film detector.

As previously discussed, less-than-perfect interfaces can display more, or less, thermal transmission than that expected from the acoustic-mismatch theory. For the crystal/liquid-He and crystal/rare-gas-film cases, the thermal transmission is much greater than predicted. This is because diffuse phonon scattering at the interface increases the chances of transmission across the boundary.

In view of this, one might wonder if the detailed predictions developed in Chapter 11 for specular refraction and mode conversion at a solid/solid interface can be observed in practice. In particular, can one observe critical cones? … total internal reflection? … mode conversion? In this chapter, we examine phonon-imaging experiments that indeed show such effects of specular phonon scattering at a boundary. And, of course, we find that nature contains some surprises that our limited imaginations did not anticipate. Six different experimental systems are considered: (a) refraction at a specially prepared Ge/MgO interface, (b) transmission through an ordinary sapphire/metal-film interface, (c) internal reflection from high-quality sapphire and Si surfaces, with the crystals immersed in liquid He, and (d) two examples of transmission through internal interfaces: semiconductor superlattices and ferroelectric domains.

The previous chapters have dealt mainly with phonon-focusing patterns that are independent of phonon frequency. This nondispersive behavior is expected when the wavelengths of the phonons are long compared to the spacing between atoms in the crystal. For phonons with wavelengths of only a few lattice spacings, large changes are expected in the slowness surface, group velocities, and focusing pattern. In practice, however, it is quite difficult to observe the ballistic propagation of such short wavelength (large-k) phonons. As their wavelength approaches twice the lattice constant (i.e., k ≡ 2π/λ approaches π/a), the phonons become particularly sensitive to defects in the periodicity in the crystal: they scatter.

The crystalline defects can be impurity atoms or even atoms with different isotopic masses. The fractional difference in mass between two isotopes is usually small, but many elements have more than one isotope of large natural abundance. This implies a high density of weak-scattering centers, even in chemically pure crystals. For long-wavelength phonons, the rate of mass-defect scattering increases as the fourth power of the phonon frequency, similar to Raleigh scattering of light from particles in the atmosphere. For example, the mean free paths of 1-THz phonons in otherwise perfect Ge, InSb, and GaAs are limited to a fraction of a millimeter by isotope scattering.

In order to observe the ballistic propagation of large-wavevector phonons in such crystals, the experimenter must use thin samples of high chemical and structural purity.

This book summarizes the results of the fifty-years work of a team of scientists and engineers of the Russian Federal Nuclear Center or All-Russia Research Institute of Experimental Physics (RFNC–VNIIEF) in Sarov (former Arzamas-16), N. Novgorod region. Their effort was concentrated on measuring parameters of compression of condensed materials under intense shock. For the last forty years, the author has participated in this research.

It has been no secret for a long time that the research on shock compression initiated in the USA during World War II and in Russia (then USSR) immediately after the war was directly related to the extremely difficult problem of designing nuclear weapons. No wonder that these investigations remained top secret for a long time, and for ten years after the start of this research only some measurements of shock compression of ‘non-secret’ metals were published (data about radioactive materials have not been published to this day).

The application of new techniques led to a considerable jump in pressure as compared to the parameters of high-pressure static experiments of that time, P = 10 GPa, performed by P. Bridge-man, who was later awarded the Nobel Prize in physics. The first American publications about shock compression at pressures of up to 50 GPa date to 1955, and the first Russian publications at 400 GPa to 1958.

To begin with, the compression parameters of metals, especially those used in structural components of nuclear bombs, were measured. In due time the range of materials studied under shock compression widened rapidly. It included other metals and non-metallic chemical elements, metal alloys, various mixtures of materials. minerals, organic conpmounds, liquids, etc.

Absolute laboratory measurements of kinematic parameters

Shock compression of about sixty metals of the periodic system, i.e., of the majority of metals, has been studied over various ranges of pressure.

The reported maximum pressures achieved in laboratory conditions are 1.8 TPa for iron, 1.0 TPa for aluminum, 2.5 TPa for tantalum, 1.3 TPa for titanium, and 2.1 for molybdenum [30, 31]. More than ten metals have been studied under pressures of up to 1.0 TPa, the rest of them, depending on their initial density, have been tested in a range of 60 GPa (lithium) to 500 GPa (tungsten) [32].

Let us recall that the parameters directly measured in dynamic experiments are the shock velocity D and the particle velocity U behind the shock front. The thermodynamic parameters, such as pressure, compression, and energy are derived using conservation equations, and the temperature is derived using the equation of state. We shall analyze most of the experimental data in terms of the directly measured parameters D and U.

A remarkable feature of Hugoniots plotted in D – U coordinates is the linear or close to linear dependence between these parameters in fairly wide ranges of parameters. This linearity between the shock velocity and particle velocity in certain ranges of these parameters has not been proved theoretically, but was observed in hundreds of experiments with various materials ranging from gases to solids in their initial states. Certainly, this statement is an approximation to the real state of things and, to some extent, depends on the measurement accuracy of shock parameters and density of experimental points on the plots.

The all-out effort undertaken to produce weapons of nuclear deterrence that was started in the USSR after World War II demanded measurements of properties of various materials, including their shock compressibility, under high pressure. This research was prompted primarily by the necessity of rapidly closing the gap between the USA and USSR in nuclear military technology.

Measurements of material shock compression had been started in 1946, when research aimed at deriving equations of state of various materials was initiated as part of the nuclear program. These equations, which establish relations between thermodynamic parameters, such as energy (or shock compression temperature), and pressure or density, were needed to complete the equation system describing the compression of compressed continuous media.

The equations of state are based on experimental data on the shock compression of materials, in which pressure is determined as a function of density and energy called a shock adiabat (sometimes this relationship is named a Hugoniot equation of state). Unlike the USA, more attention in the USSR, especially in the first years of the nuclear program, was focused on the techniques and devices for studying equations of state of materials. The problem of primary importance was measuring parameters of explosives which triggered the nuclear reaction and metals from which the bomb structure was made. The pioneering research in this field was performed by L.V. Al'tshuler, E.I. Zababakhin, Ya.B. Zel'dovich and many other workers of the Russian Nuclear Center (formerly All-Russia Research Institute of Experimental Physics in Arzamas-16, presently Sarov in Nizhnii Novgorod region).

Certainly, it is impossible to measure the compressibility of all natural minerals, especially as pure chemical compounds are in a minority among them, and most natural minerals are composite structures, which makes their total number much larger. Therefore we studied the most typical species of various groups of minerals and generalized our results as far as possible.

Let us recall that inorganic minerals are usually classified into four groups in terms of chemical bonds between their structural units. These are oxides, oxysalts, halides, and native elements (this group includes metals discussed in the previous section). The first three groups are subdivided into smaller subclasses, but we shall consider the compressibility of representatives of large groups only.

Oxides

Silicon dioxide

Of all oxide compounds whose shock compression has been measured, quartz (or, more precisely, various modifications of SiO2) is the best studied material as concerns both the range of applied pressure, of initial states and phases, and the diversity of effects produced by shock in samples. The shock compression of α-quartz or its polycrystalline phase quartzite, coesite (one of the high-density phases of silicon dioxide), crystoballite (a less dense phase), and amorphous quartz was studied at different times for different purposes. Porous Hugoniots, double-compression Hugo- niots, and expansion Hugoniots of SiO2 have been measured. In fact, this is the only oxide whose compression was measured not only in laboratory, but also in full-scale experiments. In the latter measurements, the pressure was up to 2 TPa, i.e., nearly an order of magnitude higher than in laboratory experiments performed with such materials.

To end this review, let us summarize the results of decades of research on the shock compression of condensed matter carried out in Russia.

Metals and numerous metallic compounds have been the subjects of the most intense research. The only group of these materials not studied under shock pressure is phosphides, which are relatively rare.

The number of experiments with minerals and oxides has been fairly large. Since the number of classes of these materials is enormous, the experimental data have been generalized to predict Hugoniots of similar materials in a wide range of shock parameters.

Research on shock compression of liquid condensed materials was started much later, but has been intense over recent years. However, the most interesting discoveries in this field related, in particular, to fast chemical reactions are expected in the late 1990s.

In conclusion let us consider some urgent tasks in the field of the shock compression of materials with a view to applying the results to the derivation of equations of state.

Primarily, this is further research into the compression of various materials at maximum pressures available in laboratory experiments. The aim is to obtain Hugoniots of solid (i.e., with a density ρ0 = ρ0,cr), porous (ρ0 < ρ0,cr), and superporous (ρ0 ≪ ρ0,cr) materials.

Another important task is to develop a technique for measuring the compressibility of heated and precompressed vapors of materials, first of all, metals, with an initial density ρ0 < 0.1 g/cm3. The behavior of melted materials with initial temperatures T > 1500 – 1700° C should be also studied.

Since the number of liquids is practically unlimited, and they should be classified into numerous groups, which are often homologous series, include various solutions, mixtures, etc., it is impossible to study their properties individually. It is advisable to study representatives of various groups and extrapolate the measurements to other liquids of each group. A priori it was clear that many groups of liquids should behave similarly, and their properties would be described by similar Hugoniots. On the other hand, it was natural to admit that some liquids might have individual features and should be studied in dedicated experiments. A massive effort to study properties of liquids has been undertaken in the past ten to fifteen years, although on a lesser scale than the research on solids. Nonetheless, dozens of different liquids have been investigated, and some common features can be derived from these results.

Naturally, attention was primarily focused on water, which is the most abundant and important liquid on the Earth. The first measurements of its compression date back to 1957–58 [119, 120]. Measurements of its parameters were repeated with the aim of refining the results and extending the range of measured parameters of water under shock compression [8, 17, 28, 46, 47, 72].

The other groups of liquids subjected to shock compression were:

1. – saturated aqueous solutions of salts–halides, sulphates, and thiosulphates [121];

2. – organic liquids [28, 46, 47, 115, 122, 123];

3. – saturated hydrocarbons (alkanes) CnH2n+2 (n = 6, 7, 10, 13, 14, 16);

4. – nonsaturated hydrocarbons of the ethylene series (alkenes) CnH2n (n=6–8) and of the acetylene series (alkynes) CnH2n−2 (n=6, 8);

5. – aromatic hydrocarbons, namely, benzene and its derivatives, such as toluene, nitrobenzene, aniline, and styrene;

6. […]

The interest in organic materials is driven by several factors, one of which is their extensive application to devices operating under extreme conditions of high pressure and temperature, and information about their thermodynamic parameters in this region is essential. The second field of their application, which is no less important, is high-pressure chemistry. It is known that shock triggers various chemical transformations in many organic materials. Their nature is not clear, and investigation of physical processes in these reactions is an urgent problem addressed by researchers.

The primary topic is a classification of Hugoniots in accordance with classes of organic materials aiming to separate their common and distinctive features. Besides, it is virtually impossible to study all types of organic materials, therefore some classification of their properties is of great importance.

Generally speaking, few organic materials have been studied under shock compression in comparison to other solids, some of which were reviewed in previous sections. Therefore any new research on them is very interesting.

We measured the shock compressibility of six organic acids and three anhydrides [112] and also four polymers and one saturated hydrocarbon [46, 113–117]. They belong to the following groups: the succinic HOOC(CH2)2COOH and glutaric HOOC(CH2)3COOH acids are saturated dibasic acids; palmitic acid C13H27COOH is monobasic; maleic acid HOOCCH2COOH is a nonsaturated diacid; tartaric or dihydroxysuccinic acid is HOOC(CHOH)2COOH; and phthalic acid is orthophenylenecar-boxylic diacid. The anhydrides were phthalic, succinic (CH2CO)2O, and myristic (C13H28CO)2O. Tetracosane has the formula C24H50 (CnH2n+2) n = 24), polyethylene (-CH2-CH2-), polytetrafluo-rethylene (-CF2-CF2-)n, polymethylmethacrylate (C5H8O2)n, and paraffin is a mixture of heavy saturated hydrocarbons, CnH2n+2, with n > 20.