The mechanism of Landau-Rumer relaxation of thermal and high-frequency phonons in cubic crystals of germanium,silicon, and diamond

Transverse phonon relaxation according to the Landau-Rumer mechanism is considered for an isotropic medium and crystals of germanium, silicon, and diamond possessing a cubic symmetry. The energy of elastic deformation caused by the anharmonicity of vibrations of the cubic crystal lattice is expressed via the second-and third-order moduli of elasticity. Using the known values of these elastic moduli, parameters determining the frequencies of the transverse phonon relaxation in the Landau-Rumer mechanism are evaluated for the germanium, silicon, and diamond crystals. It is shown that the dependence of the relaxation frequency on the wavevector of thermal and high-frequency phonons sharply differs from the classical Landau-Rumer relationship both in the isotropic medium and in the cubic crystals. It is established that the observed peculiarities in the relaxation frequency are related to the angular dependence of the probability of anharmonic scattering and the anisotropy of elastic properties of the germanium, silicon, and diamond crystals. A new method is proposed for the experimental determination of the relaxation frequency of high-frequency phonons as a function of the wavevector using the temperature dependence of the coefficient of absorption of high-frequency ultrasound.     

具有立方晶系结构的多晶体材料的弹性常数——Y弹性常数

提出了一个新的物理参量“Ｙ弹性常数”,并阐述了其物理含义.并将其应用于具有立方晶系结构的多晶体材料,推导了立方晶系结构的多晶体材料的Ｙ弹性常数,通过算例与具有立方晶系结构的多晶体材料的X射线弹性常数进行了比较.运用这个Ｙ弹性常数进一步推导出的多晶体材料整体的机械弹性常数的表达式与Krner的研究结果完全符合. 关键词： Y弹性常数 立方晶系 多晶体材料     

Anharmonic processes of scattering and relaxation of slow quasi-transverse phonons in cubic crystals

Relaxation of slow quasi-transverse phonons in anharmonic processes of scattering in cubic crystals with positive (Ge, Si, diamond) and negative (KCl, NaCl) anisotropies of the second-order elastic moduli has been considered. The dependences of the relaxation rates on the direction of the wave vector of phonons in scattering processes with the participation of three quasi-transverse phonons (the TTT relaxation mechanisms) are analyzed within the anisotropic continuum model. It is shown that the TTT relaxation mechanisms in crystals are associated with their cubic anisotropy, which is responsible for the interaction between noncollinear phonons. The dominant contribution to the phonon relaxation comes from large-angle scattering. For crystals with significant anisotropy of the elastic energy (Ge, Si, KCl, NaCl), the total contribution of the TTT relaxation mechanisms to the total relaxation rate exceeds the contribution of the Landau-Rumer mechanism either by several factors or by one to two orders of magnitude depending on the direction. The dominant role of the TTT relaxation mechanisms as compared to the Landau-Rumer mechanism is governed, to a considerable extent, by the second-order elastic moduli. The total relaxation rates of slow quasi-transverse phonons are determined. It is demonstrated that, when the anharmonic processes of scattering play the dominant role, the inclusion of one of the relaxation mechanisms (the Landau-Rumer mechanism or the mechanisms of relaxation of the slow quasi-transverse mode by two slow or two fast modes) is insufficient for describing the anisotropy of the total relaxation rates in cubic crystals.     

Negative Poisson’s ratio for cubic crystals and nano/microtubes

The paper systemizes numerous cubic crystals which can have both positive and negative Poisson’s ratios (the so-called partial auxetics) depending on the specimen orientation in tension. Several complete cubic auxetics whose Poisson’s ratio is always negative are indicated. The partial cubic auxetics are classified with the use of two dimensionless elastic parameters. For one of the parameters, a critical value is found at which the orientation behavior of the crystals changes qualitatively. The behavior of mesotubes obtained by rolling up plates of cubic crystals (crystals with rectilinear anisotropy) is considered in detail. Such mesotubes with curvilinear cubic anisotropy can have micron and nanometer lateral dimensions. It is shown that uniform tension of nano/microtubes of cubic crystals is possible only in the particular case of zero chirality angle (the angle between the crystallographic axis and the axis of a stretched tube). It is demonstrated by the semi-inverse Saint-Venant method that solution of the axial tension problem for cylindrically anisotropic nano/microtubes of cubic crystals with a non-zero chirality angle is possible with radially inhomogeneous fields of three normal stresses and one shear stress. In the examples considered, the cylindrically anisotropic nano/microtubes of cubic crystals are auxetics even if they are initially non-auxetics with rectilinear anisotropy.     

Theoretical melting curves for cubic solids at high pressure

The Lindemann law, the fourth-order anharmonic equation of state and a model for the volume dependence of the Grüneisen coefficient in cubic crystals are used to derive temperature-strain-pressure relations at melting. The relations depend for their application on knowing the values of fourth-order elastic parameters which are not measured experimentally. It is shown that these values can be estimated from melting data at low pressure. A comparative study of the results with previous determinations obtained from shock-wave data is given for copper, silver and gold. Finally, the theoretical fourth-order melting curves of the three selected metals are calculated using both the Lagrangian and Eulerian strain measures.     

Split interstitials in computer models of single-crystal and amorphous copper

The effect of interstitial atoms in a dumbbell configuration on elastic moduli of single crystal copper has been investigated using molecular dynamics simulation. It has been shown that shear components of the dipole tensor and λ-tensor increase when the concentration of split interstitials exceeds 0.6–0.7%. This is associated with their interaction responsible for a significant change in the distribution of orientations of split interstitials and, hence, for a change in the type of the induced local symmetry breakings of the face-centered cubic structure. It has been found that, in the model of amorphous copper, there are “defects” (elastic dipoles) with properties similar to those of split interstitials in the single crystal. The deviatoric component of their λ-tensor is more than an order of magnitude greater than the dilatation component and is responsible for the decreased value of the shear modulus and thermal effects in noncrystalline metallic materials.     

Relaxation of quasi-transverse phonons in the Herring mechanism and ultrasound absorption in cubic crystals with positive and negative anisotropies of the second-order elastic moduli

The phonon relaxation and quasi-transverse ultrasound absorption in the course of Herring and Landau-Rumer anharmonic scattering processes in cubic crystals with positive (Ge, Si, diamond, InSb, LiF, MgO) and negative (KCl, NaCl, CaF₂) anisotropies of the second-order elastic moduli have been investigated. A new mechanism of transverse phonon relaxation, according to which the fusion of a transverse (slow or fast) phonon with a slow phonon generates a fast transverse phonon, has been considered in the long-wave-length approximation. This mechanism is similar to the Herring relaxation mechanism for longitudinal phonons. It has been demonstrated that, for crystals of the first group with a considerable anisotropy of the elastic energy (Ge, Si, InSb, LiF, MgO), “anomalous” relaxation processes in which the fusion of a slow transverse phonon with a fast phonon generates a slow transverse phonon are possible in contrast to the Herring relaxation mechanism for longitudinal phonons. These relaxation processes appear to be impossible for all crystals of the second group (KCl, NaCl, CaF₂), as well as for crystals of the first group with a small anisotropy of the elastic energy, such as diamond. The angular dependences of the ultrasound absorption coefficient for the Herring and Landau-Rumer mechanisms have been analyzed using the anisotropic-continuum model. It has been shown that, for the crystals of the first group under consideration, the contribution of the Herring mechanism to the long-wavelength ultrasound absorption is small compared to the contribution of the Landau-Rumer mechanism. However, for the KCl and NaCl crystals of the second group in directions of the [001] type, the contribution of the Herring mechanism can significantly exceed the contribution of the Landau-Rumer mechanism.     

各向异性立方晶体的弹性格林函数及其应用

本文计算了各向异性立方晶体的弹性格林函数的级数展开式,给出了直到二级近似的展开式的系数。将所得结果应用于弹性偶极子模型,给出了对称中心所产生的位移场及两对称中心间的弹性相互作用的表示式。应用于Cu,K等强各向异性立方晶体,虽然级数的收敛较慢,但所得关于对称中心的位移场,及二对称中心间的互作用能的数值结果,竟与基于点阵的不连续性作出的点阵静力学计算所得的结果基本一致。从而表明,本文给出的直到二级近似的弹性格林函数的解析表示提供了一个可以普遍应用的简便的方法。它可以较准确地描述立方晶体的某些力学行为。 关键词：     

Anharmonic interactions of elastic and orientational waves in one-dimensional crystals

Planar oscillations of a chain of dumbbell-shaped particles possessing three degrees of freedom are studied. This system models the dynamics of quasi-one-dimensional crystals consisting of elongated anisotropic molecules. A system of nonlinear differential equations describing the anharmonic interaction of the elastic and orientational waves in the lattice, corresponding to different degrees of freedom of the particles, is constructed assuming a cubic interparticle interaction potential. It is shown that in the low-frequency approximation the system obtained is equivalent to the equations of the moment theory of elasticity, widely employed for describing nonlinear and dispersion properties of layered crystals and phase transformations in alloys. Some types of three-wave collinear interactions are investigated, suggesting the possibility of exciting orientational waves in organic crystals because of their nonlinear interaction with acoustic waves. Fiz. Tverd. Tela (St. Petersburg) 39, 137–144 (January 1997)     

Effect of anisotropy on the propagation of concentration-elastic waves in crystals with the generation of nonequilibrium local disorder carriers

The effect anisotropy has on the propagation of concentration elastic waves in crystals subjected to the action of laser pulses generating carriers of disorders in the crystalline structure is studied. The existence of four dispersion wave modes—one quasi-longitudinal, two quasi-transverse, and one quasi-concentration—is established. For transversely isotropic and cubic crystals, phase velocities and attenuation (or amplification) increments of waves are determined for different directions of their propagation.     

Anisotropic attenuation of transverse ultrasound in cubic crystals of Ge,Si, and diamond with various isotopic compositions

The attenuation of transverse ultrasound in germanium, silicon, and diamond crystals is considered with allowance for competing isotopic and anharmonic scattering processes. The dependence of the attenuation of transverse ultrasound on the direction of the wave vector of quasi-transverse phonons is analyzed within an anisotropic continuum model. The Landau—Rumer mechanism is considered for anharmonic scattering processes. Given the second-and third-order elastic moduli, the parameters are found determining ultrasonic absorption in the above crystals with various degrees of isotopic disorder. The attenuation coefficients of transverse ultrasound associated with isotopic and anharmonic scattering processes are shown to have qualitatively different angular dependences. Therefore, from studying the anisotropic attenuation of ultrasound in cubic crystals, one can determine the dominant mechanism of ultrasonic absorption in isotopically modified crystals.     

Theory of nuclear and magnetic neutron small angle scattering at dislocations in anisotropic cubic crystals

An analytic expression for the dilatation and the stress tensor in reciprocal space due to dislocations in anisotropic cubic crystals is deduced. From this absolute values of nuclear and magnetic small angle scattering cross sections are calculated. For Cu and Ni they differ considerably from the values calculated with the assumption of elastic isotropy. The new presentation of scattering theory allows further the discussion of the interference in scattering between dislocations in a dislocation network.     

Inversionless amplification of light by a two-level medium as a result of mechanical-to-electromagnetic atomic energy conversion

Nonreciprocal effects in the acousto-optical interaction in gyrotropic cubic crystals with electroinduced anisotropy are studied. It is shown that the presence of optical gyrotropy leads to the doubling of the number of peaks of amplitude nonreciprocity, whereas an external electric field causes their shift and enhancement. It is found that the direction of lasing in a ring laser can be reversed by changing the direction of the control field. The results obtained can be used for the development and optimization of polarization-and electro-controlled acousto-optical nonreciprocal elements on the basis of cubic gyrotropic crystals.     

Ferroelectrics of homogeneously deformed rare-earth garnet crystals,excited by elastic wave propagation

The effect of deformation on the electric properties in rare-earth garnet compounds is elucidated. It is shown that inhomogeneous deformation causes the emergence of electric polarization in garnet crystals on account of nonequivalent low-symmetry sites of rare-earth ions in the cubic structure of these crystals, whose symmetry of the environment has no inversion center. The polarization of a rare-earth ion subsystem in garnet crystals is studied upon the elastic wave propagation therein.

     

Effect of annealing under tensile loading on the structure of nanocrystals in the Finemet alloy

The effect of nanocrystallization annealing under tensile loading on the structure of nanocrystals in the soft magnetic alloy Fe-Si-Nb-B-Cu (Finemet) has been investigated. It has been shown that the body-centered cubic (bcc) lattice of α-FeSi nanocrystals is extended along the direction of the application of the load upon annealing and is compressed in the transverse direction. Nanocrystals in the Finemet alloy have a higher degree of anisotropy of mechanical properties as compared to bulk crystals of α-FeSi, so that agreement between the measured and calculated values of the elongation is achieved only with a significant increase in the elastic moduli. Substantial changes in mechanical properties of the crystals with a decrease in their size to the nanometer scale are caused by the influence of the rigid amorphous matrix of the Fe(Nb)-B phase surrounding the nanocrystals.     

Low-temperature anisotropy of the thermal conductivity of single-crystal nanofilms and nanowires

The effect of phonon focusing on the phonon transport in single-crystal nanofilms and nanowires is studied in the boundary scattering regime. The dependences of the thermal conductivity and the free path of phonons on the geometric parameters of nanostructures with various elastic energy anisotropies are analyzed for diffuse phonon scattering by boundaries. It is shown that the anisotropies of thermal conductivity for nanostructures made of cubic crystals with positive (LiF, GaAs, Ge, Si, diamond, YAG) and negative (CaF₂, NaCl, YIG) anisotropies of the second-order elastic moduli are qualitatively different for both nanofilms and nanowires. The single-crystal film plane orientations and the heat flow directions that ensure the maximum or minimum thermal conductivity in a film plane are determined for the crystals of both types. The thermal conductivity of nanowires with a square cross section mainly depends on a heat flow direction, and the thermal conductivity of sufficiently wide nanofilms is substantially determined by a film plane orientation.     

Wave velocities in shock-compressed cubic and hexagonal single crystals above the elastic limit

When solids are subjected to high-pressure shock-wave loading, multiple stress waves propagate with velocities dependent upon the elastic and inelastic compressibilities of the solid. The present paper shows that the inelastic or plastic waves in cubic and hexagonal single crystals do not necessarily propagate with the bulk sound speed as they do in isotropic elastic-plastic solids. This result is a consequence of anisotropy in the plastic deformation which depends on the slip plane orientation in the crystal and has important consequences with regard to the determination of compressibilities from shock-wave data. In particular, for wave propagation in the <110> directions of cubic crystals the departure from the bulk velocity can be significant (5–25 per cent). For wave propagation normal to the c-axis in hexagonal crystals, the plastic wave velocity also differs from the bulk sound speed (10–25 per cent). Plastic wave velocities are tabulated for a number of cubic crystals on the basis of the various slip systems common to these materials. The calculated velocities are then compared with experimental data on shock-loaded single-crystal aluminum and sodium chloride.     

Dielectric behaviour of anisotropic ionic crystals

In the case of cubic ionic crystals Havinga has shown that the temperature variation of dielectric constant could be described in terms of volume and temperature effects. By extending his formalism to anisotropic, ionic crystals it has been shown that unlike in cubic ionic crystals where the volume effect consists of a change in the number per unit volume of the polarizable particles and their polarizability with volume, in the case of anisotropic ionic crystals, in addition to these, a variation in the anisotropy of polarizability due to uneven thermal expansion also has to be taken into account. This method of analysis has been examined by taking rutile as an example.