Exponential fitting using integral equations

The fitting of a function y =\documentclass{article}\pagestyle{empty}\begin{document}$ \sum\nolimits_{i = 1}^n {A_i {\rm{e}}^{\lambda ix}} $\end{document} to experimental data is considered. Integral equations are developed which have the functions \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\nolimits_{i = 1}^n {A_i {\rm{e}}^{\lambda ix}} $\end{document} as their solutions for n = 1, 2, 3, These integral equations are used to find the frequencies λ_i. Examples are worked out to illustrate the method. The method is shown to be capable of extension to other functions.     

Numerical integration of structural dynamics equations including natural damping and periodic forcing terms

This paper shows that it is comparatively simple to analyse algorithms for the numerical integration of the Space discretized equations from structural dynamics when applied to \documentclass{article}\pagestyle{empty}\begin{document}$\ddot x + \mu \dot x\omega ^2 x = p{\rm e}^{ist} $\end{document}, instead of the usual \documentclass{article}\pagestyle{empty}\begin{document}$ \ddot x + \omega ^2 x = 0 $\end{document}, and suggests that this should be done in order to gain some insight into the effect with natural damping and/or a periodic forcing term. The method is illustrated on some three- and four-time-level schemes.     

Probleme der zerstörenden Prüfung von Proben mit sehr geringer Verformung

Problems of Destructive Testing of Samples with Very Low Deformation In the case of materials with very low deformation at break, for example ceramics and graphite, the stiffness of the testing machine has to fulfill requirements which must be clearly defined and capable of being checked, in order to avoid errors in the test result. Besides sensitive test factors there are also some which remain constant regardless of the spring stiffness of the testing apparatus. The effect on the commonly used loading methods (v_o = constant; \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\rm F}\limits^. = {\rm constant} $\end{document}) is investigated. Possibilities for corrections are discussed an appropriate algorithms are given.     

An iterative technique for fitting exponentials

An interative method to fit the function \documentclass{article}\pagestyle{empty}\begin{document}$\mathop \[y = \sum\nolimits_i^n { = 1} a_i e^{lix}\] $\end{document} to data is considered. The technique used is that of inversion of a linear differential operator with constant coefficients. This method reproduces the parameters for mathematically precise data and gives satisfactory results when the data are affected by random errors.     

Interface structure and strain development during compression tests of Al2O3/Nb/Al2O3 sandwiches

The interface structure of an Al₂O₃/Nb/Al₂O₃ sandwich produced by solid-state diffusion bonding was investigated in detail by various transmission electron microscopy (TEM) methods. The joint possessed at one interface a , , and on the other interface a and orientation relationship. At both interfaces, misfit dislocations formed to compensate the lattice mismatch as found by high-resolution transmission electron microscopy (HRTEM). Electron energy-loss near edge structure (ELNES) studies revealed that the interface is terminating with an Al layer resulting in Al–Nb bonds. Identical sandwiches were investigated on the meso- and macroscopic scale by performing compression tests and simultaneously monitoring the strain development at (001)_Nb and crystal faces. The full-field optical strain measurements (FFOM) revealed that the strain is localized at the interfaces when observed at the (001)_Nb face while it is along the maximum shear directions of 36–54° inclined to the interface when observed at the face. The strain localization along a specific maximum shear direction results in the cleavage of Al₂O₃, always initiating from the interface possessing the and orientation relationship.     

Molecular beam epitaxy of semipolar AlN(11\bar{2}2) and GaN(11\bar{2}2) on m-sapphire

We report on the plasma-assisted molecular-beam epitaxy of semipolar $\hbox{AlN}(11\bar{2}2)$ and GaN( $11\bar{2}2$ ) films on $(1\bar{1}00)$ m-plane sapphire. AlN deposited on m-sapphire settles into two main crystalline orientation domains, $\hbox{AlN}(11\bar{2}2)$ and $\hbox{AlN}(10\bar{1}0),$ whose ratio depends on the III/V ratio. Growth under moderate nitrogen-rich conditions enables to isolate the $(11\bar{2}2)$ orientation. The in-plane epitaxial relationships of $\hbox{AlN}(11\bar{2}2)$ on m-plane sapphire are $[11\bar{2}\bar{3}]_{\rm AlN} \vert \vert [0001]_{\rm sapphire}$ and $[1\bar{1}00]_{\rm AlN} \vert \vert [11\bar{2}0]_{\rm sapphire}.$ GaN deposited directly on m-sapphire results in ( $11\bar{2}2$ )-oriented layers with ( $10\bar{1}\bar{3}$ )-oriented inclusions. A ～100 nm-thick AlN( $11\bar{2}2$ ) buffer imposes the ( $11\bar{2}2$ )-orientation for the GaN layer grown on top. By studying the Ga-desorption on GaN( $11\bar{2}2$ ), we conclude that these optimal growth conditions corresponds to a Ga excess of one monolayer on the GaN( $11\bar{2}2$ ) surface.     

High-field ESR on Light-Induced Transition of Spin Multiplicity in FeCo Complex

Cyanide-bridged Fe-Co complex [Fe(Tp)(CN)₃]₂Co(bpe)?5H₂O (1?5H₂O; Tp = hydro-tris(pyrazolyl)borate; bpe = 1,2-bis(4-pyridyl)ethane) shows temperature- and light- induced metal-to-metal charge transfer (MMCT) involving spin state changes between magnetic $\mathrm{Fe}^{\mathrm{III}}_{\phantom{\mathrm{III}}\mathrm{LS}}\mbox{--}\mathrm{Co}^{\mathrm{II}}_{\phantom{\mathrm{II}}\mathrm{HS}}$ (HS = high spin, LS = low spin) state and nonmagnetic $\mathrm{Fe}^{\mathrm{II}}_{\phantom{\mathrm{II}}\mathrm{LS}}\mbox{--}\mathrm{Co}^{\mathrm{III}}_{\phantom{\mathrm{III}}\mathrm{LS}}$ state, while the dehydrated material 1 does not show any MMCT and holds $\mathrm{Fe}^{\mathrm{III}}_{\phantom{\mathrm{III}}\mathrm{LS}}\mbox{--}\mathrm{Co}^{\mathrm{II}}_{\phantom{\mathrm{II}}\mathrm{HS}}$ state. We have investigated the magnetic properties of each spin state in 1 and 1?5H₂O by means of magnetization and ESR measurement under pulsed high magnetic field. At low temperature below T _N, in both 1 and 1?5H₂O, the saturation magnetization in the induced ferromagnetic phase is well explained by S and g values derived from the magnetic susceptibility study. In the ESR of 1, we observed characteristic modes corresponding to a spin excitation in the induced ferromagnetic phase where its temperature dependence shows an evolution of spin correlation in the $\mathrm{Fe}^{\mathrm{III}}_{\phantom{\mathrm{III}}\mathrm{LS}}\mbox{--}\mathrm{Co}^{\mathrm{II}}_{\phantom{\mathrm{II}}\mathrm{HS}}$ state at low temperature. We further found that the similar ESR modes grow in the light-induced state of 1?5H₂O. The results strongly suggest that the light-induced magnetization in 1?5H₂O is driven by a light-induced MMCT, which involves transition of spin multiplicity from the nonmagnetic $\mathrm{Fe}^{\mathrm{II}}_{\phantom{\mathrm{II}}\mathrm{LS}}\mbox{--}\mathrm{Co}^{\mathrm{III}}_{\phantom{\mathrm{III}}\mathrm{LS}}$ to the magnetic $\mathrm{Fe}^{\mathrm{III}}_{\phantom{\mathrm{III}}\mathrm{LS}}\mbox{--}\mathrm{Co}^{\mathrm{II}}_{\phantom{\mathrm{\mathrm{II}}}\mathrm{HS}}$ pair.     

Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit

The homogeneous Boltzmann equation for inelastic Maxwell mixtures is considered to study the dynamics of tracer particles or impurities (solvent) immersed in a uniform granular gas (solute). The analysis is based on exact results derived for a granular binary mixture in the homogeneous cooling state (HCS) that apply for arbitrary values of the parameters of the mixture (particle masses m _i , mole fractions c _i , and coefficients of restitution ?? _ij ). In the tracer limit (c ₁ ?? 0), it is shown that the HCS supports two distinct phases that are evidenced by the corresponding value of E ₁/E, the relative contribution of the tracer species to the total energy. Defining the mass ratio ${\mu\equiv m_1/m_2}$ , there indeed exist two critical values ${\mu_{\rm HCS}^{(-)}}$ and ${\mu_{\rm HCS}^{(+)}}$ (which depend on the coefficients of restitution), such that E ₁/E?=?0 for ${\mu_{\rm HCS}^{(-)}<\mu<\mu_{\rm HCS}^{(+)}}$ (disordered or normal phase), while ${E_1/E\neq 0}$ for ${\mu<\mu_{\rm HCS}^{(-)}}$ and/or ${\mu>\mu_{\rm HCS}^{(+)}}$ (ordered phase).     

Mixed pronton–electron conducting properties of Yb doped strontium cerate

The electrical conductivity and hydrogen permeation properties of membranes were studied as a function of temperature and gradient. The bulk conductivity of was an order of magnitude higher than the grain boundary conductivity over the temperature range 100–250 °C in feed gas of 4% H₂/balance He (pH₂O = 0.03 atm). The significantly lower grain boundary conductivity indicates that larger-grained materials might be more suitable for proton transport. The hydrogen flux through the membranes is proportional to thickness down to 0.7 mm. The hydrogen permeation flux increases with an increase in gradient where the increase in hydrogen flux was explained by an increase in electron conduction as a function of temperature. The ambipolar conductivity calculated from hydrogen permeation fluxes shows the same and dependence as electron concentrations. The hydrogen and oxygen potential dependence of the ambipolar conductivity (, ) was understood from the defect structure. From this, it was confirmed that hydrogen permeation might be limited by electron transport at wet reducing atmosphere. From the temperature dependence of the electronic conductivity, the activation energy calculated at wet reducing conditions is 0.63 eV.     

The Transport Properties of Sodium Atoms and the Heat Capacity of Sodium Dimers at High Temperatures

Including the contribution of excited state atoms can improve calculations of dilute gaseous transport properties at high temperatures. For sodium, experimental and/or theoretical information is available about the potential energy curves associated with each of ten low-lying states of the sodium dimer. These include the ${{\rm X}^{1}\Sigma_{\rm g}{}^{+}}$ and ${^{3}\Sigma_{\rm u}{}{}^{+}}$ states that dissociate to two ground state ²S sodium atoms and the four ${^{3}\Sigma_{\rm g,u}{}^{+},\,^{1}\Sigma _{\rm g,u}{}^{+},\,^{1}\Pi _{\rm g,u},\,^{3}\Pi _{\rm g,u}}$ gerade/ungerade pairs of states that dissociate to a ground state ²S atom and an excited state ²P atom. Nine of these are bound states and have been fitted with the Hulburt–Hirschfelder potential, a very good general purpose atom–atom potential. The ³Π_g state is not bound and has been fitted with the exponential repulsive potential. We have used these potentials to calculate viscosity collision integrals as a function of temperature, and employed degeneracy-weighted averaging to determine the viscosity and translational contribution to the thermal conductivity of the sodium atoms. These same potentials have been used to calculate the heat capacity, ${C_{p}^{\rm o}}$ , of the sodium dimer using an approach that depends on the second virial coefficient and its first two temperature derivatives. Again, the inclusion of molecular states that dissociate to an excited state atom allows ${C_{p}^{\rm o}}$ to be determined with improved accuracy at higher temperatures. Thus, thermophysical property calculations for sodium have been extended to 25,000 K. These results are compared with previous results, including heat capacities given in the NIST-JANAF Thermochemical Tables.     

Dislocation structure at a \{ {\overline{1} 2\overline{1} 0} \}/ {\langle} 10\overline{1} 0 {\rangle} low-angle tilt grain boundary in LiNbO3

LiNbO₃ is a ferroelectric material with a rhombohedral R3c structure at room temperature. A LiNbO₃ bicrystal with a $ \{ {\overline{1} 2\overline{1} 0} \}/ {\langle}10\overline{1} 0{\rangle}$ 1° low-angle tilt grain boundary was successfully fabricated by diffusion bonding. The resultant boundary was then investigated using high-resolution TEM. The boundary composed a periodic array of dislocations with $ b = { 1}/ 3{\langle} \overline{1} 2\overline{1} 0{\rangle} $ . They dissociated into two partial dislocations by climb. A crystallographic consideration suggests that the Burgers vectors of the partial dislocations should be $ 1/ 3{\langle}01\overline{1} 0{\rangle} $ and $ 1/ 3{\langle}\overline{1} 100{\rangle} $ , and a stacking fault on $ \{ {\overline{1} 2\overline{1} 0} \} $ is formed between the two partial dislocations. From the separation distance of a partial dislocation pair, a stacking fault energy on $ \{ {\overline{1} 2\overline{1} 0} \} $ was estimated to be 0.25?J/m² on the basis of isotropic elasticity theory.     

Activation kinetics of the As acceptor in HgCdTe

The amphoteric model of As in HgCdTe is the basis of an investigation into how the transfer is achieved under Hg saturation and so obtain a method for calculating optimum annealing times for the transfer. It is concluded that Schaake’s assumption that the transfer, or activation, process is diffusion limited, rather than reaction-rate limited, is a better fit to experimental data. The identification of the Te self-diffusivity in HgCdTe as due to Te _i defects is considered to be incorrect. As a result, Schaake’s activation model can only underestimate the optimum anneal times for activation if the experimentally observed Te self-diffusivity is used. An equation based on experimental activation data is given that permits an estimate of the optimum anneal time to be obtained in HgCdTe layers.     

Mixed-mode crack-tip stress fields for orthotropic functionally graded materials

Quasi-static mixed mode stress fields for a crack in orthotropic inhomogeneous medium are developed using asymptotic analysis coupled with Westergaard stress function approach. In the problem formulation, the elastic constants E ₁₁, E ₂₂, G ₁₂, ν ₁₂ are replaced by an effective stiffness ${E=\sqrt {E_{11} E_{22}}}$ , a stiffness ratio ${\delta =\left({{E_{11}}\mathord{\left/ {\vphantom {{E_{11}} {E_{22}}}}\right. \kern-0em} {E_{22}}} \right)}$ , an effective Poisson’s ratio ${\nu =\sqrt {\nu_{12}\nu _{21}} }$ and a shear parameter ${k=\left({E \mathord{\left/ {\vphantom {E {2G_{12}}}}\right. \kern-0em} {2G_{12}}}\right)-\nu }$ . An assumption is made to vary the effective stiffness exponentially along one of the principal axes of orthotropy. The mode-mixity due to the crack orientation with respect to the property gradient is accommodated in the analysis through superposition of opening and shear modes. The expansion of stress fields consisting of the first four terms are derived to explicitly bring out the influence of nonhomogeneity on the structure of the mixed-mode stress field equations. Using the derived mixed-mode stress field equations, the isochromatic fringe contours are developed to understand the variation of stress field around the crack tip as a function of both orthotropic stiffness ratio and non-homogeneous coefficient.     

Surface Recombination Velocity Dependence on Morphological Properties of CdTe Thin Films Prepared by Close-Spaced Sublimation

Cadmium telluride (CdTe) thin films were prepared on glass substrates by employing the close-spaced sublimation technique. Different source ( $T_\mathrm{sou}$ ) and substrate temperatures ( $T_\mathrm{sub}$ ) were used in order to change the structural properties of layers. The ranges chosen were: $550\,^{\circ }\hbox {C} \le T_\mathrm{sou} \le 650\,^{\circ }\hbox {C}$ and $400\,^{\circ }\hbox {C} \le T_\mathrm{sub} \le 600\,^{\circ }\hbox {C}$ . The environment in the growing chamber was also changed with the purpose to study its influence on the crystalline properties of the surface and volume of the material. Three different surroundings were used: vacuum, high-purity argon, and high-purity oxygen. The surface recombination velocity (SRV) was calculated from photoacoustic (PA) measurements by employing the open PA cell configuration. The behavior of the experimental results was analyzed as a function of the structural characteristics of the films: texture and grain size. Scanning electron microscopy, optical absorption, X-ray diffraction, and dark resistivity measurements were also employed to analyze the properties of the CdTe films. The minimum value for the SRV was found for $T_\mathrm{sou} = 650\,^{\circ }\hbox {C},\, T_\mathrm{sub} = 600\,^{\circ }\hbox {C}$ in an oxygen ambient.     

Polling systems with batch service

Motivated by applications in production and computer-communication systems, we study an N-queue polling system, consisting of an inner part and an outer part, and where products receive service in batches. Type-i products arrive at the outer system according to a renewal process and accumulate into a type-i batch. As soon as D _i products have accumulated, the batch is forwarded to the inner system where the batch is processed. The service requirement of a type-i batch is independent of its size D _i . For this model, we study the problem of determining the combination of batch sizes ${\vec{D}^{({\rm opt})} }$ that minimizes a weighted sum of the mean waiting times. This model does not allow for an exact analysis. Therefore, we propose a simple closed-form approximation for ${\vec{D}^{({\rm opt})}}$ , and present a numerical approach, based on the recently proposed mean waiting-time approximation in Boon et?al. (Perform Eval 68, 290–306, 2011). Extensive numerical experimentation shows that the numerical approach is slightly more accurate than the closed-form solution, while the latter provides explicit insights into the dependence of the optimal batch sizes on the system parameters and into the behavior of the system. As a by-product, we observe near-insensitivity properties of ${\vec{D}^{({\rm opt})}}$ , e.g. to higher moments of the interarrival and switch-over time distributions.     

Two-Body Correlations and the Superfluid Fraction for Nonuniform Systems

We extend the one-body phase function upper bound on the superfluid fraction f _s in a periodic solid (a spatially ordered supersolid) to include two-body phase correlations. The one-body current density is no longer proportional to the gradient of the one-body phase times the one-body density, but rather it becomes $\vec{j}(\vec{r}_{1})=\rho_{1}(\vec{r}_{1})\frac{\hbar}{m}\vec{\nabla }_{1}\phi_{1}(\vec{r}_{1})+\frac{1}{N}\int d\vec{r}_{2}\rho_{2}(\vec{r}_{1},\vec{r}_{2})\frac{\hbar }{m}\vec{\nabla}_{1}\phi_{2}(\vec{r}_{1},\vec{r}_{2})$ . This expression therefore depends also on two-body correlation functions. The equations that simultaneously determine the one-body and two-body phase functions require a knowledge of one-, two-, and three-body correlation functions. The approach can also be extended to disordered solids. Fluids, with two-body densities and two-body phase functions that are translationally invariant, cannot take advantage of this additional degree of freedom to lower their energy.     

3D shear-mode fatigue crack growth in maraging steel and Ti-6Al-4V

Fatigue crack growth tests in mixed-mode II + III were performed on maraging steel and Ti-6Al-4V. The 3D evolutions of the crack fronts -measured by SEM after interrupted tests- were analyzed, taking into account the reduction in effective crack driving force by the interlocking and friction of the asperities of the crack surface. Under small-scale yielding conditions, the mixed-mode crack growth rates were found to correlate best with \({\sqrt{{\Delta {\rm K}}_{\rm II}^{{\rm eff}^{2}}+1.2\Delta {\rm K}_{\rm III}^{{\rm eff}^{2}}}}\) in maraging steel, while for Ti-6Al-4V, \({\sqrt{\Delta {\rm K}_{\rm II}^{{\rm eff}^{2}}+0.9\Delta {\rm K}_{\rm III}^{{\rm eff}^{2}}}}\) appeared suitable. For extended plasticity, a crack growth prediction method is proposed and validated for Ti-6Al-4V. This method is based on elastic-plastic F.E. computations and application, ahead of each node of the crack front, of a shear-dominated fatigue criterion.     

Is There a Unified Description of Conductivity of Layered Cuprates ?

We present a novel approach to the analysis of the normal state in-plane $\sigma _{ab} $ and out-of-plane σ_c conductivities of anisotropic layered crystals such as oxygen deficient YBa ₂ Cu ₃ O _x . It can be shown that the resistive anisotropy is determined by the ratio of the phase coherence lengths in the respective directions; i.e., $\sigma _{ab} /\sigma _c = \ell _{ab}^2 /\ell _c^2 $ . From the idea that at all doping levels and temperatures T the out-of-plane transport in these crystals is incoherent, follows that $\ell _c $ is T-independent, equal to the spacing $\ell _0 $ between the neighboring bilayers. Thus, the T-dependence of $\ell _{ab} $ is given by the measured anisotropy, and $\sigma _{ab} (\ell _{ab} )$ dependence is obtained by plotting $\sigma _{ab} {\text{ }}vs{\text{ }}\ell = {\text{ (}}\sigma _{ab} /\sigma _c )^{1/2} \ell _0 $ .The analysis of several single crystals of YBa ₂ Cu ₃ O _x (6.35 < x < 6.93) shows that for all of them $\sigma _{ab} (\ell ) $ is described by a universal dependence $\sigma _{ab} /\overline \sigma = f(\ell /\overline \ell ) $ with doping dependent parameters $\overline \sigma {\text{ }}and{\text{ }}\overline \ell $ .