Analytical Fitting Functions Research Articles

Recommended electron-impact excitation and ionization cross sections for Be II

An overview of the current status of electron-impact excitation and ionization cross sections for Be II is given and the recommended data sets for use in plasma modeling are presented. Accurate cross sections between the lowest 14 atomic terms of 1s2nl(n≤5) configurations are calculated with the convergent close-coupling (CCC) method and compared with the available experimental and theoretical results. The recommended electron-impact excitation and ionization cross sections are represented by analytical fitting functions that preserve correct asymptotic behavior. Uncertainties in the recommended data sets are determined by assessing the accuracy of the target structures, the convergence in the subsequent collisional calculations, and the fitting method. They are well within 15 % for most of the transitions. The fitting coefficients for 91 excitation cross sections and 14 ionization cross sections are provided for use in plasma modeling simulations.

Electron beam impact parameters for the creation of excited species in N2 gas

The number of electron–ion pairs and the distribution of excited species created by the passage of an intense electron beam in a gas are important parameters for many applications. The previously published values for molecular nitrogen rely on a differential ionization cross section that uses a number of fitting parameters and excitation cross sections determined from analytical fitting functions [S. P. Slinker, A. W. Ali, and R. D. Taylor, J. Appl. Phys. 67, 679 (1990)]. Slinker used cross section fits to solve the Boltzmann equation which was then used to compute the important beam-impact parameters. In this work, it is shown that an alternative approach based on the continuous slowing down approximation (CSDA) can be used to compute the energy expended per electron-ion pair and the distribution of excited gas species. This method results in an integral equation that can be solved iteratively and converges rapidly. The binary-encounter Bethe (BEB) differential ionization cross section is used [Y. K. Kim and M. E. Rudd, Phys. Rev. A 50, 3954 (1994); W. Hwang, Y.-K. Kim and M. E. Rudd, J. Chem. Phys. 104, 2956 (1996)]. The BEB model naturally extends to relativistic energies and has no free parameters. This makes the BEB considerably easier to use than previous models based on fitting parameters. The BEB model requires orbital constants obtained from quantum chemistry calculations. To demonstrate the technique, the electron-beam impact parameters are computed for nitrogen gas. The tabulated low-energy excitation cross sections are extended to relativistic energies using Bethe's asymptotic value for the inelastic cross sections [M. Inokuti, Rev. Mod. Phys. 43, 297 (1971)]. It is shown that the results for the energy expended per electron–ion pair as well as the distribution of excited states agree with published experimental values and are similar to previously published theoretical results.

Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues: IV. Vibrationally-resolved ionization of the ground and excited electronic states

We present a comprehensive set of vibrationally-resolved cross sections for electron-impact ionization of molecular hydrogen and its isotopologues (H2, D2, T2, HD, HT, and DT) in both the ground and excited electronic states. We apply the adiabatic-nuclei molecular convergent close-coupling (MCCC) method to calculate cross sections from threshold to 1000 eV for ionization of the ground and excited vibrational levels of the X1Σg+, B1Σu+, C1Πu, EF1Σg+, a3Σg+, and c3Πu electronic states, representing all states with united-atoms-limit principle quantum number n=1–2. The cross sections are presented in graphical form and provided as both numerical values and analytic fit functions in supplementary data files. The data can also be downloaded from the MCCC database at mccc-db.org.

Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues: III. Vibrational excitation via electronic excitation and radiative decay

We present state-resolved cross sections for vibrational excitation via electronic excitation followed by radiative decay (ERD), for electrons scattering on all bound vibrational levels of the ground electronic state (X1Σg+) of molecular hydrogen and its isotopologues (H2, HD, HT, D2, DT and T2). We consider excitation of the singlet n=2–3 states (where n refers to the united-atoms-limit principle quantum number) and account for all possible decay pathways back to the bound vibrational levels of the ground electronic state (X1Σg+) to produce an estimate for the ERD cross sections. A selection of the results are presented in graphical form and a full data set is provided as both numerical values and analytic fit functions in supplementary data files. The uncertainty in the cross sections is estimated to be 11%, except for scattering on the highest bound vibrational level of HD, HT, D2, and DT, where the uncertainty is 21%. The data can be downloaded from the MCCC database at mccc-db.org.

Partial, Zombie, and Full Tidal Disruption of Stars by Supermassive Black Holes

Abstract We present long-duration numerical simulations of the tidal disruption of stars modeled with accurate stellar structures and spanning a range of pericenter distances, corresponding to cases where the stars are partially and completely disrupted. We substantiate the prediction that the late-time power-law index of the fallback rate n ∞ ≃ −5/3 for full disruptions, while for partial disruptions—in which the central part of the star survives the tidal encounter intact—we show that n ∞ ≃ −9/4. For the subset of simulations where the pericenter distance is close to that which delineates full from partial disruption, we find that a stellar core can reform after the star has been completely destroyed; for these events the energy of the zombie core is slightly positive, which results in late-time evolution from n ≃ −9/4 to n ≃ −5/3. We find that self-gravity can generate an n(t) that deviates from n ∞ by a small but significant amount for several years post-disruption. In one specific case with the stellar pericenter near the critical value, we find that self-gravity also drives the recollapse of the central regions of the debris stream into a collection of several cores while the rest of the stream remains relatively smooth. We also show that it is possible for the surviving stellar core in a partial disruption to acquire a circumstellar disk that is shed from the rapidly rotating core. Finally, we provide a novel analytical fitting function for the fallback rates that may also be useful in a range of contexts beyond tidal disruption events.

Preliminary Evaluation of the LASSO Method for Prediction of the Relative Power Density Distribution in Mixed Oxide (Pu, DU)O2 Fuel Pellets

Abstract Predicting the power distribution within nuclear fuel is essential for predicting reactor fuel performance, since power distributions can impact pellet temperature distributions and fission product transport and migration. Analytical expressions for radial power distribution in fuel pellets were sought using lattice physics calculations to generate data and a machine learning technique to find representative expressions. Analytical approximations can be useful in nuclear fuel performance codes, such as element simulation and stresses (ELESTRES)/ element simulation code in a loss of coolant accident (ELOCA) for providing very rapid predictions of power distributions with reduced computational effort and memory requirements, relative to using an embedded or coupled neutron transport/burnup reactor physics code. Radial power distributions were calculated a priori using lattice physics codes to model mixed oxide (MOX) 37-element fuel bundles in pressure tube heavy water reactors. Such advanced fuels are of interest for future fuel cycles. Several datasets were generated with different amounts of PuO2 and variable neutron energy spectrum. Results of preliminary studies with the least absolute shrinkage and selection operator (LASSO) regression machine learning method have obtained analytical fitting functions with a mean maximum relative error (MRE) of 0.056 and a maximum MRE of 0.152 on the test set. However, using LASSO to estimate the coefficients of a physically motivated modified Bessel plus an exponential function, results in a lower MRE (mean MRE 0.041 and maximum MRE 0.11) on the same test set. Further potential improvements in both the curve fit and the machine learning methods are discussed.

An analysis of angular directivity for small caliber firearm noise

Accurately describing the sound field generated by an impulsive source is non-trivial. Conventional methods for analyzing impulsive noise of firearms consists of deploying one or more circular microphone arrays centered on the muzzle of the source. Currently, much debate exists around the design of these microphone arrays (radius, quantity of sensors, angular spacing, etc.) and the resulting effect on capturing source directivity of the impulse event. An in-depth investigation of data collected on small caliber firearms operating in outdoor environments is performed with an emphasis on describing source directivity as a function of (1) array radius, (2) quantity of microphones, and (3) angular spacing. Ultimately, given a sufficient circular microphone array, the angular directivity of an impulsive source can be generalized using analytical curve fitting functions.

Bending Stiffness of Damaged Cross-Ply Laminates

The bending stiffness of carbon/epoxy and glass/epoxy cross-ply laminates with intralaminar cracks in the surface 90° plies and local delaminations were studied experimentally using 4-point bending tests. The bending stiffness is defined as the slope of the relation between the applied bending moment and the corresponding midplane curvature. To measure the midplane curvature of laminates with different damage states, the digital image correlation system was used. The reduction in the bending stiffness with increasing density of transverse cracks and delamination length was also analyzed using a 3-D FEM model. The analysis and optical microscopy observations showed that, in the initial stage of damage evolution, local delaminations were small, but with increasing load, the delaminations grew rapidly from the tips of existing and newly created cracks, enhancing the bending stiffness degradation. As an alternative to the 3-D FEM modelling of the test, analytical approaches in conjunction with the classical laminate theory were suggested. The analytical approach was based on the concept of the “effective stiffness of damaged ply,” where the initial stiffness of the damaged ply was replaced by the effective stiffness depending on the damage state. In the present work, two routines were used to determinate the effective stiffness of the damaged ply: a) back-calculation from the difference in the stiffnesses of damaged and undamaged laminates employing the FEM model of the representative volume element; b) simple analytical fitting functions. It is shown that the analytical approach suggested is accurate and convenient for predicting the degradation of bending stiffness of a laminate with an evolving microdamage.

Impacts of random filling on spin squeezing via Rydberg dressing in optical clocks

We analyze spin squeezing via Rydberg dressing in optical lattice clocks with random fractional filling. We compare the achievable clock stability in different lattice geometries, including unity-filled tweezer clock arrays and fractionally filled lattice clocks with varying dimensionality. We provide practical considerations and useful tools in the form of approximate analytical expressions and fitting functions to aid in the experimental implementation of Rydberg-dressed spin squeezing. We demonstrate that spin squeezing via Rydberg dressing in one-, two-, and three-dimensional optical lattices can provide significant improvements in stability in the presence of random fractional filling.

Universality and thermoelectric transport properties of quantum dot systems

We discuss the temperature-dependent thermoelectric transport properties of semiconductor nanostructures comprising a quantum dot coupled to quantum wires: the thermal dependence of the electrical conductance, thermal conductance, and thermopower. We explore the universality of the thermoelectric properties in the temperature range associated with the Kondo crossover. In this thermal range, general arguments indicate that any equilibrium property's temperature dependence should be a universal function of the ratio ${T}^{*}=T/{T}_{K}$, where ${T}_{K}$ is the Kondo temperature. Considering the particle-hole symmetric, spin-degenerate Anderson model, the zero-bias electrical conductance has already been shown to map linearly onto a universal conductance through a quantum dot embedded or side-coupled to a quantum wire. Employing rigorous renormalization-group arguments, we calculate universal thermoelectric transport coefficients that allow us to extend this result to the thermopower and the thermal conductance. We present numerical renormalization-group results to illustrate the physics in our findings. Applying the universal thermoelectric coefficients to recent experimental results of the electrical conductance and thermovoltages versus ${V}_{\mathrm{gate}}$, at different temperatures in the Kondo regime, we calculate all the thermoelectric properties and obtain simple analytical fitting functions that can be used to predict the experimental results of these properties. However, we cannot check all of them, due to the lack of available experimental results over a broad temperature range.

Statistical modelling of the cosmological dispersion measure

ABSTRACT We have investigated the basic statistics of the cosmological dispersion measure (DM)—such as its mean, variance, probability distribution, angular power spectrum, and correlation function—using the state-of-the-art hydrodynamic simulations, IllustrisTNG300, for the fast radio burst cosmology. To model the DM statistics, we first measured the free-electron abundance and the power spectrum of its spatial fluctuations. The free-electron power spectrum turns out to be consistent with the dark matter power spectrum at large scales, but it is strongly damped at small scales (≲ Mpc) owing to the stellar and active galactic nucleus feedback. The free-electron power spectrum is well modelled using a scale-dependent bias factor (the ratio of its fluctuation amplitude to that of the dark matter). We provide analytical fitting functions for the free-electron abundance and its bias factor. We next constructed mock sky maps of the DM by performing standard ray-tracing simulations with the TNG300 data. The DM statistics are calculated analytically from the fitting functions of the free-electron distribution, which agree well with the simulation results measured from the mock maps. We have also obtained the probability distribution of source redshift for a given DM, which helps in identifying the host galaxies of FRBs from the measured DMs. The angular two-point correlation function of the DM is described by a simple power law, $\xi (\theta) \approx 2400 (\theta /{\rm deg})^{-1} \, {\rm pc}^2 \, {\rm cm}^{-6}$, which we anticipate will be confirmed by future observations when thousands of FRBs are available.

Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues: II. Fully vibrationally-resolved electronic excitation of the isotopologues of H2 ([formula omitted

We present a comprehensive set of vibrationally-resolved cross sections for electron-impact electronic excitation of the isotopologues of molecular hydrogen (D2, T2, HD, HT, and DT) initially in the ground electronic state. We apply the adiabatic-nuclei molecular convergent close-coupling (MCCC) method to calculate cross sections from threshold to 500 eV for excitation of all bound vibrational levels and dissociative excitation of the B1Σu+, C1Πu, EF1Σg+, B′1Σu+, GK1Σg+, I1Πg, J1Δg, D1Πu, H1Σg+, b3Σu+, c3Πu, a3Σg+, e3Σu+, d3Πu, h3Σg+, g3Σg+, i3Πg, and j3Δg electronic states from all bound vibrational levels of the ground electronic (X1Σg+) state. Including the previously-published MCCC e-H2 cross sections (Scarlett et al., Atom. Data Nucl. Data Tables 137 (2021) 101361) the data set contains cross sections for over 60,000 electronic and vibrational transitions. The cross sections are presented in graphical form and provided as both numerical values and analytic fit functions in supplementary data files. The data can also be downloaded from the MCCC database at mccc-db.org.

Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues: I. Fully vibrationally-resolved electronic excitation of H[formula omitted

We present a comprehensive set of vibrationally-resolved cross sections for electron-impact electronic excitation of molecular hydrogen suitable for implementation in collisional-radiative models. The adiabatic-nuclei molecular convergent close-coupling method is used to calculate cross sections for excitation of all bound vibrational levels and dissociative excitation of the B1Σu+, C1Πu, EF1Σg+, B′1Σu+, GK1Σg+, I1Πg, J1Δg, D1Πu, H1Σg+, b3Σu+, c3Πu, a3Σg+, e3Σu+, d3Πu, h3Σg+, g3Σg+, i3Πg, and j3Δg electronic states from all vi=0–14 bound vibrational levels of the ground electronic (X1Σg+) state. The data set consists of cross sections from threshold to 500 eV for over 5000 transitions, representing all possible electronic and vibrational transitions between the X1Σg+ state and the n=2–3 singlet and triplet states (where n refers to the united-atoms-limit principle quantum number). The cross sections are presented in graphical form and provided as both numerical values and analytic fit functions in supplementary data files. The data can also be downloaded from the MCCC database at http://mccc-db.org.

Improving spectroscopic lithium abundances

Context. Accurate spectroscopic lithium abundances are essential in addressing a variety of open questions, such as the origin of a uniform lithium content in the atmospheres of metal-poor stars (Spite plateau) or the existence of a correlation between the properties of extrasolar planetary systems and the lithium abundance in the atmosphere of their host stars. Aims. We have developed a tool that allows the user to improve the accuracy of standard lithium abundance determinations based on 1D model atmospheres and the assumption of local thermodynamic equilibrium (LTE) by applying corrections that account for hydrodynamic (3D) and non-LTE (NLTE) effects in FGK stars of different metallicity. Methods. Based on a grid of CO5BOLD 3D models and associated 1D hydrostatic atmospheres, we computed three libraries of synthetic spectra of the lithium λ 670.8 nm line for a wide range of lithium abundances, accounting for detailed line formation in 3D NLTE, 1D NLTE, and 1D LTE, respectively. The resulting curves-of-growth were then used to derive 3D NLTE and 1D NLTE lithium abundance corrections. Results. For all metallicities, the largest corrections are found at the coolest effective temperature, Teff = 5000 K. They are mostly positive, up to + 0.2 dex, for the weakest lines (lithium abundance A(Li)1DLTE = 1.0), whereas they become more negative towards lower metallicities, where they can reach − 0.4 dex for the strongest lines (A(Li)1DLTE = 3.0) at [Fe/H] = − 2.0. We demonstrate that 3D and NLTE effects are small for metal-poor stars on the Spite plateau, leading to errors of at most ± 0.05 dex if ignored. We present analytical functions evaluating the 3D NLTE and 1D NLTE corrections as a function of Teff [5000…6500 K], log g [3.5…4.5], and LTE lithium abundance A(Li) [1.0…3.0] for a fixed grid of metallicities [Fe∕H] [ − 3.0…0.0]. In addition, we also provide analytical fitting functions for directly converting a given lithium abundance into an equivalent width, or vice versa, a given equivalent width (EW) into a lithium abundance. For convenience, a Python script is made available that evaluates all fitting functions for given Teff, log g, [Fe∕H], and A(Li) or EW. Conclusions. By means of the fitting functions developed in this work, the results of complex 3D and NLTE calculations are made readily accessible and quickly applicable to large samples of stars across a wide range of metallicities. Improving the accuracy of spectroscopic lithium abundance determinations will contribute to a better understanding of the open questions related to the lithium content in metal-poor and solar-like stellar atmospheres.

A New Fitting Method for Ambipolar Diffusion Length Extraction in Thin Film Structures Using Photoluminescence Measurement with Scanning Excitation

A new simple method is proposed to extract the ambipolar diffusion length for two-dimensional (2D) electronic transport in thin film structures using a scanning photoluminescence microscopy (SPLM) setup. No spatially-resolved photoluminescence detection methods are required. By measuring the excitation-position-dependent PL intensity across the edge of a semiconductor, ambipolar diffusion length can be extracted from the SPLM profile through a simple analytic fitting function. Numerical simulation was first used to verify the fitting method. Then the fitting method was applied to extract the ambipolar diffusion length from the measured SPLM profile of a GaAs thin film structure. Carrier lifetime was obtained in an accompanying time-resolved photoluminescence measurement under the same excitation condition, and thus the ambipolar diffusion coefficient can be determined simultaneously. The new fitting method provides a simple way to evaluate carrier transport properties in 2D electronic transport structures such as thin films or quantum wells.

Small-Signal Analysis of High-Performance VCSELs

In this paper, a comprehensive model to describe the small-signal modulation response of ultra-high performance single- and multi-mode vertical-cavity surface-emitting lasers (VCSELs), with modulation bandwidths exceeding 30 GHz, is presented. Traditionally, utmost simplified dynamic models are used to extract dynamic figures of merit from single-mode edge-emitting lasers. These methods are later on also applied to evaluate the dynamic performance of VCSELs, even though these devices have a very different geometrical layout and modal confinement. However, to understand the dynamic performance of high-speed VCSELs, a model supporting the transverse and longitudinal mode profile, and the driving current inhomogeneity in the active region, is needed. Therefore, multi-mode VCSEL rate equations are established here. Moreover, to access the dynamic figures of merit of these devices, a comprehensive analytical fitting function based on our carrier reservoir splitting approach is derived. Thus, because of the high carrier and photon densities inside these optimized VCSELs, the common carrier reservoir splits up as a result of numerous effects such as mode competition, carrier diffusion and spatial hole burning. These and other effects have a tremendous impact on the small signal modulation response shape and bandwidth, and also on the current distribution profile in the carrier reservoirs. Compared with our recently reported work, this novel model presented includes the effects of gain compression and inhomogeneous current injection between the different lasing modes. Consequently, it is found that the further tuning of our multi-mode VCSEL dynamic model, to include these effects, yields a more physical and consistent figures of merit of high-performance VCSELs.

Low-energy collisions between electrons and BeD+

Multichannel quantum defect theory is applied in the treatment of the dissociative recombination and vibrational excitation processes for the BeD+ ion in the 24 vibrational levels of its ground electronic state (). Three electronic symmetries of BeD** states (, , and ) are considered in the calculation of cross sections and the corresponding rate coefficients. The incident electron energy range is 10−5–2.7 eV and the electron temperature range is 100–5000 K. The vibrational dependence of these collisional processes is highlighted. The resulting data are useful in magnetic confinement fusion edge plasma modeling and spectroscopy, in devices with beryllium based main chamber materials, such as ITER and JET, and operating with the deuterium–tritium fuel mix. An extensive rate coefficients database is presented in graphical form and also by analytic fit functions whose parameters are tabulated in the supplementary material.

Contact stiffness depth-sensing indentation: Understanding of material properties of thin films attached to substrates

The classic version of the depth-sensing indentation techniques assumes the estimation of the elastic contact modulus of a material sample by measuring the slope (the contact stiffness) of the initial part of the unloading branch of the force-displacement curve. This approach assumes that the curve at loading reflects both elastic and plastic deformations of the material, while the unloading is taking place elastically. Therefore, neglecting the plastic deformations, one can assume that the structure of the material is the same at both branches and the assumptions of the Hertz-type contact theory are valid for the unloading branch. However, the contact problem for an elastic film attached to a substrate depends on the properties of the substrate. Hence, the film contact modulus is usually estimated by measuring the slopes of the initial unloading force-displacement curves obtained for different maximal values of indentation depth, and fitting the experimental points by various empirical analytical dimensionless functions of the ratio between the contact radius, a, and the layer thickness, t. Here, analytical analysis of contact problems for coated materials is performed. Both re-scaling and asymptotic techniques are employed. Asymptotic analysis of the contact at the small-scale indentation range (the ratioa/t is small) shows that the formula of the contact stiffness derived for an elastic half-space, has to be multiplied by the so-called indentation scaling factor that is a function of a/t. Thus, the asymptotic approach allows us to take into account analytically the substrate effect. The analytical fitting function obtained agrees with both some known semi-empirical functional forms and the published experimental results on depth-sensing nanoindentation of thin metallic layers, while the function is in a disagreement with results obtained for inhomogeneous films of brittle materials such as coals. It is argued that the disagreement is caused by structural transformations (crushing) of the coals during loading.

ON THE CONTRIBUTION OF FLUORESCENCE TO LYα HALOS AROUND STAR-FORMING GALAXIES

ABSTRACT We quantify the contribution of Lyα fluorescence to observed spatially extended Lyα halos around Lyα emitters at redshift z = 3.1. The key physical quantities that describe the fluorescent signal include (i) the distribution of cold gas in the circumgalactic medium (CGM); we explore simple analytic models and fitting functions to recent hydrodynamical simulations; and (ii) local variations in the ionizing background due to ionizing sources that cluster around the central galaxy. We account for clustering by boosting the observationally inferred volumetric production rate of ionizing photons, , by a factor of , in which quantifies the clustering of ionizing sources around the central galaxy. We compute by assigning an “effective” bias parameter to the ionizing sources. This novel approach allows us to quantify our ignorance of the population of ionizing sources in a simple parametrized form. We find a maximum enhancement in the local ionizing background in the range 50–200 at r ∼ 10 physical kpc. For spatially uncorrelated ionizing sources and fluorescing clouds we find that fluorescence can contribute up to ∼ 50%–60% of the observed spatially extended Lyα emission. We briefly discuss how future observations can shed light on the nature of Lyα halos around star-forming galaxies.

Analytical fitting functions of finite sample discrete entropies of white Gaussian noise

In order to find the convergence rate of finite sample discrete entropies of a white Gaussian noise(WGN), Brown entropy algorithm is numerically tested.With the increase of sample size, the curves of these finite sample discrete entropies are asymptotically close to their theoretical values.The confidence intervals of the sample Brown entropy are narrower than those of the sample discrete entropy calculated from its differential entropy, which is valid only in the case of a small sample size of WGN. The differences between sample Brown entropies and their theoretical values are fitted by two rational functions exactly, and the revised Brown entropies are more efficient. The application to the prediction of wind speed indicates that the variances of resampled time series increase almost exponentially with the increase of resampling period.