Theory of the Gorsky effect for low interstitial concentrations

The formalism of the preceding paper is applied to work out the theory of the Gorsky effect, or anelastic relaxation due to the long-range diffusion of interstitials in a host lattice, for non-interacting (low-concentration) interstitials (e.g., H in Nb). It is shown how linear response theory (LRT) provides a number of advantages that simplify the solution of the problem and permit the handling of complications due to specimen geometry and stress inhomogeneity. The multiple-relaxation time creep function of Alefeldet al is first re-derived. Next, the dynamic responseand the short-time behaviour of the creep function are deduced exactly, and theω ^−1/2 fall-off of the internal friction at high frequencies is exhibited. Finally, it is pointed out that the true asymptotic behaviour of the dynamic response must be found by going beyond the diffusion equation model. A two-state random walk analysis is used to predict a cross-over to a trueω ⁻¹ asymptotic behaviour, and the physical reasons for this phenomenon are elucidated.     

Coherency stresses and diffusion in solids: neutron spectroscopic versus Gorsky effect measurements of deuterium diffusion in niobium

Published neutron scattering and Gorsky effect results on the diffusion and spatial fluctuations of D interstitials in NbD _x (x0.5) reveal striking differences. It is shown that these differences are caused by coherency stresses which raise the elastic energy of the short-wavelength D fluctuations studied by coherent neutron scattering. Gorsky effect measurements, on the other hand, probe long-range fluctuations which were coherencystress free in the experiments above. The observed differences can quantitatively be explained within the elasticity-theoretical concepts of Cahn, Krivoglaz, and Wagner and Horner. This represents the first successful application of these concepts to a diffusion coefficient. The correlation factor for D diffusion is, finally, determined from a comparison of the Gorsky effect and neutron scattering data.     

General linear response analysis of anelasticity

Linear response theory is used to express the anelastic response (creep function and generalized compliance) of a system under an applied stress, in terms of the equilibrium strain auto-correlation. These results extend an earlier analysis to cover inhomogeneous stresses and the tensor nature of the variables. For anelasticity due to point defects, we express the strain compactly in terms of the elastic dipole tensor and the probability matrix governing dipole re-orientation and migration. We verify that re-orientations contribute to the deviatoric strain alone (Snoek, Zener, etc. effects), while the dilatory part arises solely from the long-range diffusion of the defects under a stressgradient (the Gorsky effect). Our formulas apply for arbitrary orientational multiplicity, specimen geometry, and stress inhomogeneity. The subsequent development of the theory in any given situation then reduces to the modelling of the probability matrix referred to. In a companion paper, we apply our formalism to work out in detail the theory of the Gorsky effect (anelasticity due to long-range diffusion) for low interstitial concentrations, as an illustration of the advantages of our approach to the problem of anelastic relaxation.     

The effect of traps on the muon spin relaxation function

Summary In summary it is clear that SR in the presence of traps has led to a significant generalization of prior NMR theories of spin relaxation to include the effects of a non-stationary distribution of diffusing particles. In both the high field transverse geometry and in zero field it is possible to extract significant information regarding the concentrations and binding energies of traps, and, in the latter case, to unambiguously measure the temperature dependence of the relaxation rate to traps.At the same time it is also clear that significant problems still remain to be solved. Many of these are concerned with our present inadequate knowledge regarding the diffusion of muons in even pure materials, let alone the disordered systems upon which we have concentrated here. However, given the growth in our understanding of trapping phenomena over the past several years, it is probably not too much to hope that in another few years many of these present questions will also be answered.     

Stochastic model of muon diffusion in the presence of traps: Nonsecular effects on spin depolarization

A stochastic model is presented for calculating the depolarization of the positive muon when the latter undergoes jump diffusion in a solid in the presence of trapping impurities. The theory, restricted to low concentration of traps, is applicable to relaxation studies in both transverse and zero magnetic fields. In the transverse geometry, the dipolar interaction between the μ⁺ and the surrounding nuclear spins can be treated classically, and the analysis is simpler. The results obtained are shown to be identical to those derived before in a ‘two-state’ model, widely used recently in interpreting various experimental studies on trap-limited diffusion of light interstitials (μ⁺,H, etc.) in solids. The zero field technique is more versatile but is also more complicated to analyze as it involves nonsecular terms in the dipolar interaction. Assuming that the local dipolar fields are isotropic with their magnitudes distributed in a Gaussian manner, tractable results can be obtained for the Laplace transform of the zero field relaxation function. The latter needs to be inverted in order to facilitate comparison with experimental data, which are normally recorded in the time space. A scheme to handle this numerical problem is described and results are presented using realistic parameters.     

The simplest non-minimal matter–geometry coupling in the <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) cosmology

f(R, T) gravity is an extended theory of gravity in which the gravitational action contains general terms of both the Ricci scalar R and the trace of the energy-momentum tensor T. In this way, f(R, T) models are capable of describing a non-minimal coupling between geometry (through terms in R) and matter (through terms in T). In this article we construct a cosmological model from the simplest non-minimal matter–geometry coupling within the f(R, T) gravity formalism, by means of an effective energy-momentum tensor, given by the sum of the usual matter energy-momentum tensor with a dark energy contribution, with the latter coming from the matter–geometry coupling terms. We apply the energy conditions to our solutions in order to obtain a range of values for the free parameters of the model which yield a healthy and well-behaved scenario. For some values of the free parameters which are submissive to the energy conditions application, it is possible to predict a transition from a decelerated period of the expansion of the universe to a period of acceleration (dark energy era). We also propose further applications of this particular case of the f(R, T) formalism in order to check its reliability in other fields, rather than cosmology.     

Electronic and diffusion-induced spin-lattice relaxation and the lattice expansion about interstitial12B in Al

The spin-lattice relaxation of¹²B in single-crystal aluminum was measured by NMR in the temperature range 290 to 700 K. The total relaxation rate was separated into two well understood contributions caused by the interaction with conduction electrons and by¹²B diffusion, and into a third unexplained part at high temperatures. From the comparison of the diffusional relaxation rate with theoretical calculations it was concluded that the nearest neighbours of¹²B, trapped in the octahedral interstitial site, are displaced away by 8.8 (30)%.     

Diffusion and coupled fluxes in concentrated alloys under irradiation: a self-consistent mean-field approach

When an alloy is irradiated, atomic transport can occur through the two types of defects which are created: vacancies and interstitials. Recent developments of the self-consistent mean field (SCMF) kinetic theory could treat within the same formalism diffusion due to vacancies and interstitials in a multi-component alloy. It starts from a microscopic model of the atomic transport via vacancies and interstitials and yields the fluxes with a complete Onsager matrix of the phenomenological coefficients. The jump frequencies depend on the local environment through a ‘broken bond model’ such that the large range of frequencies involved in concentrated alloys is produced by a small number of thermodynamic and kinetic parameters. Kinetic correlations are accounted for through a set of time-dependent effective interactions within a non-equilibrium distribution function of the system. The different approximations of the SCMF theory recover most of the previous diffusion models. Recent improvements of the theory were to extend the multi-frequency approach usually restricted to dilute alloys to diffusion in concentrated alloys with jump frequencies depending on local concentrations and to generalize the formalism first developed for the vacancy diffusion mechanism to the more complex diffusion mechanism of the interstitial in the dumbbell configuration. To cite this article: M. Nastar, C. R. Physique 9 (2008).     

Random walks in a random field of decaying traps

We study random walks on ^d (d 1) containing traps subject to decay. The initial trap distribution is random. In the course of time, traps decay independently according to a given lifetime distribution. We derive a necessary and sufficient condition under which the walk eventually gets trapped with probability 1. We prove bounds and asymptotic estimates for the survival probability as a function of time and for the average trapping time. These are compared with some well-known results for nondecaying traps.     

On the possibility of delocalization of interstitials in rare gas solids

Using the previously developed statistical-mechanical formalism, we have obtained the expressions for single-particle distribution functions, which point to the possibility of delocalization of interstitials in rare gas solids.     

Proton conductivity in silica gels

Commercial silica gel drying pearls were used in impedance measurements between ambient temperature and 300 °C in dry and humid atmospheres. A pronounced difference was seen supporting previous conclusions that proton conduction was operative in these amorphous samples. The same conclusion was reached on the basis of a fuel cell experiment at ambient temperature. A Nernst voltage close to the theoretical one was observed. Mechanical tests were carried out with silica gel plates made from the above material. Such plates displayed in the bending mode a time-dependent anelastic relaxation, i.e., a Gorsky effect, which arose from the long-range diffusion of protonic defects from the compressed to the dilated part of the plate. The chemical diffusivity and the density of the protonic defects could be determined. The same Gorsky effect may be expected in crystalline proton conductors, and also in Li-containing ceramics.     

Microscopic study of a lattice defect trapped at in impurities in plastically deformed Al

¹¹¹In doped Al specimes (c _In1 ppm) were cold worked by extension at 77 K to several amounts of strain. The PAC method was applied to study the trapping of lattice defects at the indium impurities between 77 K and 293 K. From electron irradiation experiments it is known that the In probes are suitable traps for self-interstitials in Al. Since in the present experiment no such effect is observed, it is concluded that by extension at 77 K in Al essentially no freely migrating interstitials are created. Between 110 K and 210 K an increasing fraction of the indium impurities traps a well defined defect. The symmetry axis of the resulting indium-defect-complex is found to point along 111 crystallographic directions. It is proposed that the In impurity traps a divacancy at nearest neighbour sites and relaxes into the open space, thus forming an equilateral triangle of three vacancies in a {111} plane with the In atom in its centre. Reasons for the formation of this complex at rather low annealing temperatures are discussed.     

Mechanical damping by intercrystalline diffusion of hydrogen in metallic polycrystals

Anelastic relaxation by intercrystalline atomic diffusion-analogous to thermal diffusion (i.e., thermoelastic) effects but never observed hitherto-was recently suggested as a new mechanism to explain part of the hydrogen damping spectra in intermetallic compounds. A critical experimental test of this model is now presented using a Zr65Cu17.5Ni10Al7.5 alloy, which allows for a quantitative kinetic analysis in comparison to a closely related reorientation process. The results are in full agreement with the predictions of the model and clearly corroborate the proposed "intercrystalline Gorsky effect" as a new type of mechanical damping in sufficiently fine-grained polycrystals.     

Atomistic Modeling of Point Defects and Diffusion in Copper Grain Boundaries

The atomic structure of several symmetrical tilt grain boundaries (GBs) in Cu and their interaction with vacancies and interstitials as well as self-diffusion are studied by molecular statics, molecular dynamics, kinetic Monte Carlo (KMC), and other atomistic simulation methods. Point defect formation energy in the GBs is on average lower than in the lattice but variations from site to site within the GB core are very significant. The formation energies of vacancies and interstitials are close to one another, which makes the defects equally important for GB diffusion. Vacancies show interesting effects such as delocalization and instability at certain GB sites. They move in GBs by simple vacancy-atom exchanges or by long jumps involving several atoms. Interstitial atoms can occupy relatively open positions between atoms, form split dumbbell configurations, or form highly delocalized displacement zones. They diffuse by direct jumps or by the indirect mechanism involving a collective displacement of several atoms. Diffusion coefficients in the GBs have been calculated by KMC simulations using defect jump rates determined within the transition state theory. GB diffusion can be dominated by vacancies or interstitials, depending on the GB structure. The diffusion anisotropy also depends on the GB structure, with diffusion along the tilt axis being either faster or slower than diffusion normal to the tilt axis. In agreement with Borisov's correlation, the activation energy of GB diffusion tends to decrease with the GB energy.     

间隙原子非线性应力感生扩散的简化弹性偶极子模型

为了阐明体心立方晶体中外应力、位错应力和八面体间隙原子相互作用的本质,从理论上解释S-K弛豫和位错对Snoek弛豫的影响,提出一个简化的含有可动位错的一维弹性偶极子格点模型,讨论位错应力场中间隙原子的非线性应力感生扩散,为进一步对实际晶体的从头数值计算奠定基础。对一维模型的计算机模拟计算,表明位错应力场使得间隙原子形成具有非线性扩散特征的缺陷Fermi-Dirac分布,并增强了Snoek效应,在Snoek峰高温侧出现一个非线性扩散Snoek型内耗峰。 关键词：     

Sensitivity of multi-parametric MRI to the compressive state of the isolated intervertebral discs

Objective

Magnetic resonance imaging (MRI) offers great potential as a sensitive and noninvasive technique for describing the alterations in mechanical properties, as shown in vitro on intervertebral disc (IVD) or cartilage tissues. However, in vivo, the IVD is submitted to complex loading stimuli. Thus, the present question focuses on the influence of the mechanical loading during an MRI acquisition on the relaxation times, magnetization transfer and diffusion parameters within the IVD.

Methods

An apparatus allowing the compression of isolated IVDs was designed and manufactured in acrylonitrile butadiene styrene. IVDs were dissected from fresh young bovine tail, measured for their thickness and submitted to compression just before the MRI acquisition. Six discs received 0% (platen positioned at the initial disc thickness), 5% (platen positioned at 95% of the initial disc thickness), 10%, 20% and 40% deformation. The MRI parameters were compared between the loading states using mean and standard deviation for T1 and T2, and matrix subtraction for Magnetization Transfer, fractional anisotropy and apparent diffusion coefficient.

Results

The compression of the IVD did not lead to any significant change of the MRI parameters, except for the diffusion that decreased in the direction of the compressive stress.

Discussion

This experimental in vitro study shows that multi-parametric MRI on isolated discs in vitro is not sensitive to compression or to the partial confined relaxation that followed the compression.     

Spin relaxation times in disordered graphene

We consider two mechanisms of spin relaxation in disordered graphene. i) Spin relaxation due to curvature spin orbit coupling caused by ripples. ii) Spin relaxation due to the interaction of the electronic spin with localized magnetic moments at the edges. We obtain analytical expressions for the spin relaxation times τ_SO and τ_J due to both mechanisms and estimate their values for realistic parameters of graphene samples. We obtain that spin relaxation originating from these mechanisms is very weak and spin coherence is expected in disordered graphene up to samples of length .     

Network with lumped RC parameters as an electro-analog model of diffusion processes II. Simulation of the bending method

In this part of paper there are explained the conception and way of connection of the electroanalog model which enables to simulate diffusion processes manifested by macroscopic deformation (bending) of sample caused by the propagating wave of diffusing interstitials in a sample saturated on one side only. Simple examples of simulation are shown.Concluding, it is my pleasure to thank Dr. J. ermák for his help and stimulating remarks.     

Tunneling of hydrogen in Nb(CH)0.0002

From neutron spectroscopic measurements, we determine a value of (162±4) eV for the tunneling matrix element of trapped H in Nb(CH)_0.0002. The matrix element is about 30% smaller than that of (226±4) eV for H trapped by O. The results demonstrate a significant influence of the trapping impurity on the lattice potential of the tunneling H interstitials.     

Diffusion of hydrogen and deuterium in Nb and Ta at high concentrations

The concentration dependence of the diffusion coefficient of H and D in Nb and Ta in the and phases has been measured between 0 and 40% H or D/Nb or Ta by the Gorsky effect. Since this method allows to determine simultaneously the thermodynamic factor, the tracer-diffusion coefficient can be extracted from the data. An increase of the activation energy as a function of concentration has been found for all four systems. Whereas for Ta the isotope dependence of the diffusion coefficients continues to exist at high concentrations it surprisingly disappears for Nb.