Internal friction in alloys due to plasticity of the diffusion phase transformation

An analytic expression is obtained for the time dependence Q ^?1(t) of internal friction associated with plasticity of a phase transformation. Time dependences Q ^?1(t) of internal friction of the Pb-62Sn and Pb-1.9Sn alloys (wt.%) alloys were studied in the regime of continuous excitation of resonant flexural vibrations. The measurements of the Q ^?1(t) dependences for 1 h at room temperature and a fixed strain amplitude ε₀ ≈ 7 and 19 min) for the Pb-62Sn alloy. For the Pb-1.9Sn alloy under the same conditions, an exponential decrease followed by an internal friction peak (at t _m ≈ 7 min) is observed. It is shown numerically that the above singularities of internal friction are formed by processes of intermittent phase decomposition of Pb-Sn alloys in the cyclic stress field produced by an external load. Experimental data on Q ^?1(t) are used for reconstructing the kinetic curves describing the decomposition (conversion) ratio as a function of time and for calculating the corresponding values of parameters K and n of the Avrami kinetic equation for the Pb-62Sn alloy.     

On the nature of the liquid-to-glass transition equation

Within the model of delocalized atoms, it is shown that the parameter δT_g, which enters the glasstransition equation qτ_g = δT_g and characterizes the temperature interval in which the structure of a liquid is frozen, is determined by the fluctuation volume fraction \({f_g} = {\left( {{{\Delta {V_e}} \mathord{\left/ {\vphantom {{\Delta {V_e}} V}} \right. \kern-\nulldelimiterspace} V}} \right)_{T = {T_g}}}\) frozen at the glass-transition temperature T_g and the temperature T_g itself. The parameter δT_g is estimated by data on f_g and T_g. The results obtained are in agreement with the values of δT_g calculated by the Williams–Landel–Ferry (WLF) equation, as well as with the product qτ_g—the left-hand side of the glass-transition equation (q is the cooling rate of the melt, and τ_g is the structural relaxation time at the glass-transition temperature). Glasses of the same class with f_g ≈ const exhibit a linear correlation between δT_g and T_g. It is established that the currently used methods of Bartenev and Nemilov for calculating δT_g yield overestimated values, which is associated with the assumption, made during deriving the calculation formulas, that the activation energy of the glass-transition process is constant. A generalized Bartenev equation is derived for the dependence of the glass-transition temperature on the cooling rate of the melt with regard to the temperature dependence of the activation energy of the glasstransition process. A modified version of the kinetic glass-transition criterion is proposed. A conception is developed that the fluctuation volume fraction f = ΔV_e/V can be interpreted as an internal structural parameter analogous to the parameter ξ in the Mandelstam–Leontovich theory, and a conjecture is put forward that the delocalization of an active atom—its critical displacement from the equilibrium position—can be considered as one of possible variants of excitation of a particle in the Vol’kenshtein–Ptitsyn theory. The experimental data used in the study refer to a constant cooling rate of q = 0.05 K/s (3 K/min).     

Forced snaking: Localized structures in the real Ginzburg-Landau equation with spatially periodic parametric forcing

We study spatial localization in the real subcritical Ginzburg-Landau equation u _t = m ₀ u + Q(x)u + u _xx + d|u|² u ?|u|⁴ u with spatially periodic forcing Q(x). When d>0 and Q ≡ 0 this equation exhibits bistability between the trivial state u = 0 and a homogeneous nontrivial state u = u ₀ with stationary localized structures which accumulate at the Maxwell point m ₀ = ?3d ²/16. When spatial forcing is included its wavelength is imprinted on u ₀ creating conditions favorable to front pinning and hence spatial localization. We use numerical continuation to show that under appropriate conditions such forcing generates a sequence of localized states organized within a snakes-and-ladders structure centered on the Maxwell point, and refer to this phenomenon as forced snaking. We determine the stability properties of these states and show that longer lengthscale forcing leads to stationary trains consisting of a finite number of strongly localized, weakly interacting pulses exhibiting foliated snaking.     

Kinetic Theory and Lax Equations for Shock Clustering and Burgers Turbulence

We study shock statistics in the scalar conservation law ? _t u+? _x f(u)=0, x∈?, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈?. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Differential-geometric structures associated with Lagrangians corresponding to scalar physical fields

The paper is devoted to the investigation, using the method of Cartan–Laptev, of the differential-geometric structure associated with a Lagrangian L, depending on a function z of the variables t, x ¹,...,x ⁿ and its partial derivatives. Lagrangians of this kind are considered in theoretical physics (in field theory). Here t is interpreted as time, and x ¹,...,x ⁿ as spatial variables. The state of the field is characterized by a function z(t, x ¹,..., x ⁿ ) (a field function) satisfying the Euler equation, which corresponds to the variational problem for the action integral. In the present paper, the variables z(t, x ¹,..., x ⁿ are regarded as adapted local coordinates of a bundle of general type M with n-dimensional fibers and 1-dimensional base (here the variable t is simultaneously a local coordinate on the base). If we agree to call t time, and a typical fiber an n-dimensional space, then M can be called the spatiotemporal bundle manifold. We consider the variables t, x ¹,...,x ⁿ , z (i.e., the variables t, x ¹,...,x ⁿ with the added variable z) as adapted local coordinates in the bundle H over the fibered base M. The Lagrangian L, which is a coefficient in the differential form of the variational action integral in the integrand, is a relative invariant given on the manifold J ¹ _H (the manifold of 1-jets of the bundle H). In the present paper, we construct a tensor with components Λ⁰⁰, Λ⁰ⁱ , Λ ^ij (Λ ^ij = Λ ^ji ) which is generated by the fundamental object of the structure associated with the Lagrangian. This tensor is an invariant (with respect to admissible transformations the variables t, x ¹,...,x ⁿ , z) analog of the energy-momentum tensor of the classical theory of physical fields. We construct an invariant I, a vector G ⁱ , and a bivalent tensor G ^jk generated by the Lagrangian. We also construct a relative invariant of E (in the paper, we call it the Euler relative invariant) such that the equation E = 0 is an invariant form of the Euler equation for the variational action integral. For this reason, a nonvariational interpretation of the Euler equation becomes possible. Moreover, we construct a connection in the principal bundle with base J ² H (the variety of 2-jets of the bundle H) and with the structure group GL(n) generated by the structure associated with the Lagrangian.     

Screening of strongly charged macroparticles in liquid electrolyte solutions

A modified Poisson-Boltzmann model has been proposed which makes it possible to describe the screening of strongly charged macroparticles in liquid electrolyte Z: Z solutions in the case when parameter B= ZeQ₀/εRT?1(Q₀ is the surface electric charge, T is the temperature, ε is the solution permittivity, and Z is the valence of ions) provided that the solution is dilute: κR ≡ (8πZ²e²n_i0/εT)^1/2R?1 (n_i0 is the equilibrium number density of ions). It is assumed that the charge Q₀ of a macroparticle appears as a result of adsorption of ions of a certain polarity on its surface. Quantitative criteria of division of dissolved ions into capable and incapable of adsorption are formulated. For aqueous solutions, the adsorption mechanism always leads to values of B ? 1. It is shown that the charge inversion effect predicted by other authors on the basis of different models must be observed for such solutions for all Z ≥ 1. The effect of Brownian movement of macroparticles on their screening is considered. It is shown that viscous forces emerging during such movement lead to peripheral destruction (“washing out”) of the screening ionic shell of macroparticles and, as a result, to violation of their electroneutrality. This results in the emergence of two types of oppositely charged compound particles with small radii close to R and with radii much larger than R, the charge polarity of the latter being opposite to the polarity of Q₀. It is found that both types of ions of compound particles obey the “law of distribution” of the mean energy of their electric field, expressed by formula (29). The problem of ionic screening of gas bubbles accompanied by the formation of bubstons (bubbles stabilized by ions) is considered separately. It is shown that the bubston radius R in pure water and in aqueous solutions of electrolytes is equal to 14 nm irrespective of the ion number density n_i0. The value of n_i0 determines the number density n _b of bubstons themselves, which are formed spontaneously under equilibrium conditions.     

Percolation and the electron–electron interaction in an array of antidots

A square lattice of microcontacts with a period of 1 μm in a dense low-mobility two-dimensional electron gas is studied experimentally and numerically. At the variation of the gate voltage V _g , the conductivity of the array varies by five orders of magnitude in the temperature range T from 1.4 to 77 K in good agreement with the formula σ(V _g ) = (V _g ?V _g ^* (T))^β with β = 4. The saturation of σ(T) at low temperatures is absent because of the electron–electron interaction. A random-lattice model with a phenomenological potential in microcontacts reproduces the dependence σ(T, V _g ) and makes it possible to determine the fraction of microcontacts x(V _g , T) with conductances higher than σ. It is found that the dependence x(V _g ) is nonlinear and the critical exponent in the formula σ ∝ ? (x - 1/2) ^t in the range 1.3 < t(T, V _g ) < β.     

The Schrödinger Equation,the Zero-Point Electromagnetic Radiation,and the Photoelectric Effect

A Schrödinger type equation for a mathematical probability amplitude Ψ(x,t) is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M and charge e, moving in a potential V(x). The particle phase space probability density is denoted Q(x,p,t), and the entire system is immersed in the “vacuum” zero-point electromagnetic radiation. We show, in the first part of the paper, that the generalized Liouville equation is reduced to a simpler Liouville equation in the equilibrium limit where the small radiative corrections cancel each other approximately. This leads us to a simpler Liouville equation that will facilitate the calculations in the second part of the paper. Within this second part, we address ourselves to the following task: Since the Schrödinger equation depends on \(\hbar \), and the zero-point electromagnetic spectral distribution, given by \(\rho _{0}{(\omega )} = \hbar \omega ^{3}/2 \pi ^{2} c^{3}\), also depends on \(\hbar \), it is interesting to verify the possible dynamical connection between ρ₀(ω) and the Schrödinger equation. We shall prove that the Planck’s constant, present in the momentum operator of the Schrödinger equation, is deeply related with the ubiquitous zero-point electromagnetic radiation with spectral distribution ρ₀(ω). For simplicity, we do not use the hypothesis of the existence of the L. de Broglie matter-waves. The implications of our study for the standard interpretation of the photoelectric effect are discussed by considering the main characteristics of the phenomenon. We also mention, briefly, the effects of the zero-point radiation in the tunneling phenomenon and the Compton’s effect.     

Temperature range of the liquid–glass transition

It has been shown that the currently used method for calculating the temperature range of δT_g in the glass transition equation qτ_g = δT_g as the difference δT_g = (T₁₂–T₁₃) results in overestimated values, which is explained by the assumption of a constant activation energy of glass transition in deriving the calculation equation (T₁₂ and T₁₃ are the temperatures corresponding to the logarithmic viscosity values of logη = 12 and logη = 13). The methods for the evaluation of δT_g using the Williams–Landel–Ferry equation and the model of delocalized atoms are considered, the results of which are in satisfactory agreement with the product qτ_g (q is the cooling rate of the melt and τ_g is the structural relaxation time at the glass transition temperature). The calculation of τ_g for inorganic glasses and amorphous organic polymers is proposed.     

Schrödinger’s Equation with Gauge Coupling Derived from a Continuity Equation

A quantization procedure without Hamiltonian is reported which starts from a statistical ensemble of particles of mass m and an associated continuity equation. The basic variables of this theory are a probability density ρ, and a scalar field S which defines a probability current j=ρ ? S/m. A first equation for ρ and S is given by the continuity equation. We further assume that this system may be described by a linear differential equation for a complex-valued state variable χ. Using these assumptions and the simplest possible Ansatz χ(ρ,S), for the relation between χ and ρ,S, Schrödinger’s equation for a particle of mass m in a mechanical potential V(q,t) is deduced. For simplicity the calculations are performed for a single spatial dimension (variable q). Using a second Ansatz χ(ρ,S,q,t), which allows for an explicit q,t-dependence of χ, one obtains a generalized Schrödinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrödinger’ equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for a non-unique external q,t-dependence of χ, one obtains Schrödinger’s equation with electrodynamic potentials A,φ in the familiar gauge coupling form. This derivation shows a deep connection between non-uniqueness, quantum mechanics and the form of the gauge coupling. A possible source of the non-uniqueness is pointed out.     

Applications of the Stroboscopic Tomography to Selected 2-Level Decoherence Models

In the paper we discuss possible applications of the so-called stroboscopic tomography (stroboscopic observability) to selected decoherence models of 2-level quantum systems. The main assumption behind our reasoning claims that the time evolution of the analyzed system is given by a master equation of the form \(\dot {\rho } = \mathbb {L} \rho \) and the macroscopic information about the system is provided by the mean values m _i (t _j ) = T r(Q _i ρ(t _j )) of certain observables \(\{Q_{i}\}_{i=1}^{r} \) measured at different time instants \(\{t_{j}\}_{j=1}^{p}\). The goal of the stroboscopic tomography is to establish the optimal criteria for observability of a quantum system, i.e. minimal value of r and p as well as the properties of the observables \(\{Q_{i}\}_{i=1}^{r} \).     

Deutung der optischen Absorptionsspektren der Übergangselemente der Eisenreihe,insbesondere der des Rubins

Crystal field, ligand field and molecular orbital theories have been used to explain the optical spectra of transition elements. It is shown an additional possibility to explain the absorption bands of the transition elements, especially those of ruby, as transitions not between the nonbondingt _2g and the antibondinge _g ^* but between the bondinga _1g and the nonbondingt _2g.     

Symmetry energy of the nucleus in the relativistic Thomas–Fermi approach with density-dependent parameters

We introduce an inhomogeneous term, f(t,x), into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions f(t,x) which depend nontrivially on both t and x, we find that there is just one symmetry. If f is a function of only x, there are three symmetries with the algebra s l(2,R). When f is a function of only t, there are five symmetries with the algebra s l(2,R) ⊕ _s 2A ₁. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.     

Momentum distribution of charged particles in jets in dijet events and comparison to perturbative QCD predictions

Inclusive momentum distributions of charged particles are measured in dijet events. Events were produced at the AMY detector with a centre of mass energy of 60 GeV. Our results were compared, on the one hand to those obtained from other e⁺e^?, ep as well as CDF data, and on the other hand to the perturbative QCD calculations carried out in the framework of the modified leading log approximation (MLLA) and assuming local parton–hadron duality (LPHD). A fit of the shape of the distributions yields Q_eff = 263±13 MeV for the AMY data. In addition, a fit to the evolution of the peak position with dijet mass using all data from different experiments gives Q_eff = 226±18 MeV. Next, α_s was extracted using the shape of the distribution at the Z⁰ scale, with a value of 0.118 ± 0.013. This is consistent, within the statistical errors, with many accurate measurements. We conclude that it is the success of LPHD + MLLA that the extracted value of α_s is correct. Possible explanations for all these features will be presented in this paper.     

Self-similar distribution in “giant” magnetic flux creep

The problem of magnetic field penetration into the half-space is considered in parallel geometry in an external magnetic field increasing with time in accordance with the law B(0, t, τ₀ = B _{c 1} (1 + t/τ₀) ^m , m ≥ 0, t ≥ 0 (τ ₀ is the time of magnetic flux redistribution and B _{c 1} is the lower critical field). It is assumed that the flow of vortices is thermally activated in the “giant” creep mode (i.e., for weak pinning creep and high temperatures). A model equation is derived for describing the magnetic flux evolution. Analytic formulas are obtained for the depth and velocity of magnetic field penetration. It is shown that the giant creep regime is stable for 0 ≤ m ≤ 1/2.     

One-dimensional magnetophotonic crystals with magnetooptical double layers

One-dimensional magnetophotonic microcavity crystals with nongarnet dielectric mirrors are created and investigated. The defect layers in the magnetophotonic crystals are represented by two bismuth-substituted yttrium iron garnet Bi:YIG layers with various bismuth contents in order to achieve a high magnetooptical response of the crystals. The parameters of the magnetophotonic crystal layers are optimized by numerical solution of the Maxwell equations by the transfer matrix method to achieve high values of Faraday rotation angle Θ _F and magnetooptical Q factor. The calculated and experimental data agree well with each other. The maximum values of Θ _F =–20.6°, Q = 8.1° at a gain t = 16 are obtained for magnetophotonic crystals with m = 7 pairs of layers in Bragg mirrors, and the parameters obtained for crystals with m = 4 and t = 8.5 are Θ _F =–12.5° and Q = 14.3°. It is shown that, together with all-garnet and multimicrocavities magnetophotonic crystals, such structures have high magnetooptical characteristics.     

Inflation of the early cold Universe filled with a nonlinear scalar field and a nonideal relativistic Fermi gas

We consider a possible scenario for the evolution of the early cold Universe born from a fairly large quantum fluctuation in a vacuum with a size a ₀ ? l _P (where l _P is the Planck length) and filled with both a nonlinear scalar field φ, whose potential energy density U(φ) determines the vacuum energy density λ, and a nonideal Fermi gas with short-range repulsion between particles, whose equation of state is characterized by the ratio of pressure P(n _F ) to energy density ε(n _F ) dependent on the number density of fermions n _F . As the early Universe expands, the dimensionless quantity ν(n _F ) = P(n _F )/ε(n _F ) decreases with decreasing n _F from its maximum value ν_max = 1 for n _F → ∞ to zero for n _F → 0. The interaction of the scalar and gravitational fields, which is characterized by a dimensionless constant ξ, is proportional to the scalar curvature of four-dimensional space R = κ[3P(n _F )–ε(n _F )–4λ] (where κ is Einstein’s gravitational constant), and contains terms both quadratic and linear in φ. As a result, the expanding early Universe reaches the point of first-order phase transition in a finite time interval at critical values of the scalar curvature R = R _c =–μ²/ξ and radius a _c ? a ₀. Thereafter, the early closed Universe “rolls down” from the flat inflection point of the potential U(φ) to the zero potential minimum in a finite time. The release of the total potential energy of the scalar field in the entire volume of the expanding Universe as it “rolls down” must be accompanied by the production of a large number of massive particles and antiparticles of various kinds, whose annihilation plays the role of the Big Bang. We also discuss the fundamental nature of Newton’ gravitational constant G _N .