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The newly popular topic of "phonon diodes" is discussed in the context of a broader issue of reciprocity in reflection/transmission (R-T) of waves. We first review a theorem well known in electromagnetism and optics but underappreciated in acoustics and phonon physics, stating that the matrix of R-T coefficients for properly normalized amplitudes is symmetric for linear systems that conform to power conservation and time reversibility for wave fields. It is shown that linear structures proposed for "acoustic diodes" in fact do obey R-T reciprocity, and thus should not strictly be called diodes or isolators. We also review examples of nonlinear designs violating reciprocity, and conclude that an efficient acoustic isolator has not yet been demonstrated. Finally, we consider the relationship between acoustic isolators and "thermal diodes", and show that ballistic phonon transport through a linear structure, whether an acoustic diode or not, is unlikely to form the basis for a thermal diode.

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As an alternative to atomistic calculations of long-wavelength acoustic modes of atomically thin layers, which are known to converge very slowly, we propose a quantitatively predictive and physically intuitive approach based on continuum elasticity theory. We describe a layer, independent of its thickness, by a membrane, characterize its elastic behavior by a (3×3) elastic matrix as well as the flexural rigidity. We present simple quantitative expressions for frequencies of long-wavelength acoustic modes, which we determine using 2D elastic constants calculated by ab initio Density Functional Theory. The calculated spectra accurately reproduce observed and calculated long-wavelength phonon spectra of graphene and phosphorene, the monolayer of black phosphorus. Our approach also correctly describes the observed dependence of the radial breathing mode frequency on the diameter of carbon fullerenes and nanotubes.

The shape of the acoustic ray surface of cubic crystals is investigated with the object of providing a framework within which the results of phonon imaging and other ballistic phonon experiments can be interpreted. This surface is shown to display considerable variability in shape, particularly with regard to the way in which it is folded. The correspondence between these folds and contours of zero Gaussian curvature on the slowness surface is explored, and the bearing this has on the presence of caustics in the anistropic flux of phonons emanating from a localized hear source is discussed. Several of the elementary catastrophes as well as some remarkable types of structural instability ar'e shown to occur in these caustics. Conditions on the elastic constants are established for the existence of various systems of folds in the ray surface.

Numerical methods are in general required foc. the determination of the stable configurations of N point charges on a sphere. The stable configurations for N up to 50 have previously been ascertained and we extend the calculations here for values up to 101. We repon far the first time some remarkable global features of these configurations. We show that the minimum energy accurately follows a simple half-integral power law in 1 / N over the full range we have investigated. This power law is explicable in terms of the idealization of mapping a planar Wigner lattice onto the surface of the unit sphere: the pair distribution functions of the larger-N configurations indicate predominant hexagonal coordination. The coeficients of the observed power law are closely straddled by values calculated on the basis of hexagonal and square Wigner lattices. This highly accurate description of the energy permits us to remark on the detailed deviations of the individual structures from the general trend. For N t 3 0 , we note that structures with N prime are relatively less stable, while structures with N equal to 6, 12, 32, 44, 48 and 60 seem more stable.

We study negative refraction and focusing of elastic waves in a simple mechanical system comprised of a free standing plate with a step change in thickness. A point focused and intensity modulated laser source is used to excite backward propagating Lamb waves on one side of the step, and the displacement field is probed using an optical interferometer. Conversion between forward and backward propagating modes at the interface leads to negative refraction, and we demonstrate for the first time the operation of a flat lens, similar to that predicted by Veselago in negative index media, for guided elastic waves in isotropic media. We propose that guided elastic waves provide a convenient and powerful experimental test bed for the study of negative index physics.2