Analytical Fitting Functions Research Articles

Motional narrowing under diffusion of two spin sub-lattices in LiNaSO4

An expression for the dipolar correlation function characterizing the fluctuations of the Hamiltonian under uncorrelated diffusion of two cationic sub-lattices in a solid has been obtained. A corresponding formula, which can be used for fitting the two-step temperature dependencies of the NMR line width, has been deduced. We also considered the case of a distribution of correlation times and deduced a corresponding analytical fitting function. Elaborated expressions have been applied to the NMR line width analysis of LiNaSO4 in the temperature range 295–788K.

Influence of Femtosecond Laser Parameters on Autler-Townes Splitting of Three-Level Ladder Molecules

We theoretically investigate the Autler-Townes (AT) splitting in the photoelectron spectra of three-level ladder K2 molecule driven by a pump-probe pulse via employing the time-dependent wave packet approach. The dependence of AT splitting on two laser intensities and wavelengths are studied in detail. We firstly quantify these effects on peak shift and AT separation. The photoelectron spectra show double splitting with symmetric profiles, but with asymmetric profiles when the wavelength is changed. The magnitude of AT splitting increases with the pump laser intensity, but does not vary with probe intensity. The shifts of the absorption peaks and the splitting between AT doublet are predicted by using an analytical fitting function when the intensity/wavelength of one of the two fields is changed. These novel results are of importance for the molecular spectroscopy and may further stimulate the first principles theoretical studies analytically.

Wavelength dependence of Autler-Townes splitting in a four-level ladder molecular system

We theoretically investigated the Autler-Townes (AT) splitting in the photoelectron spectra of four-level ladder K2 molecules in the presence of two control pump fields and one probe field by employing the time-dependent wave packet approach. The wavelength dependence of AT splitting on three laser pulses was studied in detail. The photoelectron spectra show triple splitting with symmetric profiles, but with asymmetric profiles when the wavelength is changed. The shifts of the absorption peaks are predicted by using an analytical fitting function when the wavelength of one of the three fields is changed, and the functional dependence appears to be linear. These novel results are of importance for the molecular spectroscopy, and may stimulate further theoretical studies analytically from the first principles.

Electrical conductivity measurements on disk-shaped samples.

We have developed a sample holder design that allows for electrical conductivity measurements on a disk-shaped sample. The sample holder design is based on and compatible with popular measurement systems that are currently restricted to bar-shaped samples. The geometrical correction factors which account for the adjusted measurement configuration were calculated using finite element modeling for a broad range of sample and measurement geometries. We also show that the modeling results can be approximated by a simple analytical fit function with excellent accuracy. The proposed sample holder design is compatible with a concurrent measurement of the Seebeck coefficient. The chosen sample geometry is furthermore compatible with a thermal conductivity measurement using a laser flash apparatus. A complete thermoelectric characterization without cutting the sample is thus possible.

Methods for CT Automatic Exposure Control Protocol Translation Between Scanner Platforms

An imaging facility with a diverse fleet of CT scanners faces considerable challenges when propagating CT protocols with consistent image quality and patient dose across scanner makes and models. Although some protocol parameters can comfortably remain constant among scanners (eg, tube voltage, gantry rotation time), the automatic exposure control (AEC) parameter, which selects the overall mA level during tube current modulation, is difficult to match among scanners, especially from different CT manufacturers. Objective methods for converting tube current modulation protocols among CT scanners were developed. Three CT scanners were investigated, a GE LightSpeed 16 scanner, a GE VCT scanner, and a Siemens Definition AS+ scanner. Translation of the AEC parameters such as noise index and quality reference mAs across CT scanners was specifically investigated. A variable-diameter poly(methyl methacrylate) phantom was imaged on the 3 scanners using a range of AEC parameters for each scanner. The phantom consisted of 5 cylindrical sections with diameters of 13, 16, 20, 25, and 32 cm. The protocol translation scheme was based on matching either the volumetric CT dose index or image noise (in Hounsfield units) between two different CT scanners. A series of analytic fit functions, corresponding to different patient sizes (phantom diameters), were developed from the measured CT data. These functions relate the AEC metric of the reference scanner, the GE LightSpeed 16 in this case, to the AEC metric of a secondary scanner. When translating protocols between different models of CT scanners (from the GE LightSpeed 16 reference scanner to the GE VCT system), the translation functions were linear. However, a power-law function was necessary to convert the AEC functions of the GE LightSpeed 16 reference scanner to the Siemens Definition AS+ secondary scanner, because of differences in the AEC functionality designed by these two companies. Protocol translation on the basis of quantitative metrics (volumetric CT dose index or measured image noise) is feasible. Protocol translation has a dependency on patient size, especially between the GE and Siemens systems. Translation schemes that preserve dose levels may not produce identical image quality.

Ex situ metrology of x-ray diffraction gratings

The idea of measurements of groove density distributions of diffraction gratings suggested and first realized in Proceedings of SPIE 5858, (2005) 58580A consists of determination of the spatial frequency of the first harmonic peak appearing in the power spectral density (PSD) distribution of the grating surface profile observed with a microscope. Using a MicroMap™-570 interferometric microscope, it was experimentally proven that this technique is capable of high precision measurements with x-ray gratings with groove densities of about 250grooves/mm, varying along the grating by ±5%. In the present work, we provide analytical and experimental background for useful application of PSD characterization of groove densities of diffraction gratings. In particular, we analyze the shape of harmonic peaks and derive an analytical fitting function suitable for fitting the PSD peaks obtained with gratings with a variety of groove shapes. We demonstrate the capabilities of the method by application to the groove density distribution measurements with a 300-groove/mm grating suitable for soft x-ray applications.

The halo mass function in interacting dark energy models

We present a detailed investigation of the effects imprinted by a direct interaction between dark energy and cold dark matter particles on the halo mass function of groups and clusters of galaxies. Making use of the public halo catalogues of the codecs simulations, we derive the halo mass function for several different types of coupled dark energy scenarios, based on both the friends-of-friends algorithm and the spherical overdensity halo identification for different values of the overdensity threshold Δc. We compare the computed halo mass functions for coupled dark energy cosmologies with ΛCDM as well as with the predictions of the standard analytic fitting functions. Our results show that the standard fitting functions still reproduce both the friends-of-friends and spherical overdensity halo mass functions of interacting dark energy cosmologies reasonably well at intermediate masses and low redshifts, once rescaled to the characteristic amplitude of linear density perturbations of each specific model as given by σ8. However, we also find that such apparent degeneracy with σ8 is broken by both the high-mass tail and the redshift evolution of our halo mass functions, with deviations beyond ∼10 per cent for most of the models under investigation. Furthermore, the discrepancy with respect to the predictions of standard fitting functions rescaled with the characteristic value of σ8 shows - for some models - a strong dependence on the spherical overdensity threshold Δc used for the halo identification. We find that such an effect is due to a significant increase of halo concentration at low redshifts in these models that is, however, absent in the majority of the cosmological scenarios considered in this work. We can therefore conclude that the universality of the halo mass function is violated by cosmological models that feature a direct interaction between dark energy and cold dark matter.

A Note on the Inverse Problem with LTB Universes

The inverse problem with Lema\^itre-Tolman-Bondi (LTB) universe models is discussed. The LTB solution for the Einstein equations describes the spherically symmetric dust-filled spacetime. The LTB solution has two physical functional degrees of freedom of the radial coordinate. The inverse problem is constructing an LTB model requiring that the LTB model be consistent with selected important observational data. In this paper, we assume that the observer is at the center and consider the distance-redshift relation $\da$ and the redshift-space mass density $\mu$ as the selected important observational data. We give $\da$ and $\mu$ as functions of the redshift $z$. Then, we explicitly show that, for general functional forms of $\da(z)$ and $\mu(z)$, the regular solution does not necessarily exist in the whole redshift domain. We also show that the condition for the existence of the regular solution %in terms of $\da(z)$ and $\mu(z)$ is satisfied by the distance-redshift relation and the redshift-space mass density in $\Lambda$CDM models. Deriving regular differential equations for the inverse problem with the distance-redshift relation and the redshift-space mass density in $\Lambda$CDM models, we numerically solve them for the case $(\Omega_{\rm M0},\Omega_{\Lambda0})=(0.3,0.7)$. A set of analytic fitting functions for the resultant LTB universe model is given. How to solve the inverse problem with the simultaneous big-bang and a given function $\da(z)$ for the distance-redshift relation is provided in the Appendix.

SU-GG-T-597: General Analytical Solution for Photon Beam Profile Deconvolution

Purpose: To develop a general analytical solution for extracting “true” photon beam profiles with minimal volume averaging effect Method and Materials: We have derived several analytical functions suitable for photon beam profile fitting by examining the physics process of photon dose deposition. Beam profile measured with an ionization chamber can be described by a convolution equation with a detector response function of that chamber. A general analytical solution that minimizes the volume averaging effect of large volume chambers can be developed by linking the photon dose deposition process and beam profile measurement This work demonstrates that when dose deposition kernel and detector response function are carefully selected, photon beam profiles with minimal volume averaging effect can be easily recovered by a simple change of parameters in the analytical fitting function. Specific analytical fitting functions such as Gaussian, sum of multiple Gaussian and Lorentz functions are identified in this work along with solutions to extract the “true” beam profile. Results: Our analysis shows that detector response can be accurately modeled by a Gaussian or Lorentz function, any of three afore‐mentioned analytical fitting functions can accurately extract a “true” photon beam profile from profiles measured with a 3 mm radius ion chamber. The extracted profile perfectly matches profiles measured with a 0.8×0.8 mm2 diode detector.Conclusion: A general solution for minimizing volume averaging effect in a photon beam profile measurement with large size detectors is developed. It is found that three specific fitting functions and corresponding solutions presented in this work are ideal for recovering “true” photon beam profiles.

The size distribution of void filaments in a ΛCDM cosmology

The size distribution of mini-filaments in voids has been derived from the Millennium Run halo catalogs at redshifts z=0,0.5,1 and 2. It is assumed that the primordial tidal field originated the presence of filamentary substructures in voids and that the void filaments have evolved only little, keeping the initial memory of the primordial tidal field. Applying the filament-finding algorithm based on the minimal spanning tree (MST) technique to the Millennium voids, we identify the mini-filaments running through voids and measure their sizes at each redshift. Then, we calculate the comoving number density of void filaments as a function of their sizes in the logarithmic interval and determine an analytic fitting function for it. It is found that the size distribution of void mini-filaments in the logarithmic interval has an almost universal shape, insensitive to the redshift: In the short-size section it is well approximated as a power-law, while in the long-size section it decreases exponentially. We expect that the universal size distribution of void filaments may provide a useful cosmological probe without resorting to the rms density fluctuations.

Homogeneous nucleation of nitrogen

We investigated the homogeneous nucleation of nitrogen in a cryogenic expansion chamber [A. Fladerer and R. Strey, J. Chem. Phys. 124, 164710 (2006)]. Gas mixtures of nitrogen and helium as carrier gas were adiabatically expanded and cooled down from an initial temperature of 83 K until nucleation occurred. This onset was detected by constant angle light scattering at nitrogen vapor pressures of 1.3-14.2 kPa and temperatures of 42-54 K. An analytical fit function well describes the experimental onset pressures with an error of +/-15%. We estimate the size of the critical nucleus with the Gibbs-Thomson equation yielding critical sizes of about 50 molecules at the lowest and 70 molecules at the highest temperature. In addition, we estimate the nucleation rate and compare it with nucleation theories. The predictions of classical nucleation theory (CNT) are 9 to 19 orders of magnitude below the experimental results and show a stronger temperature dependence. The Reguera-Reiss theory [Phys. Rev. Lett. 93, 165701 (2004)] predicts the correct temperature dependence at low temperatures and decreases the absolute deviation to 7-13 orders of magnitude. We present an empirical correction function to CNT describing our experimental results. These correction parameters are remarkably close to the ones of argon [Iland et al., J. Chem. Phys. 127, 154506 (2007)] and even those of water [J. Wolk and R. Strey, J. Phys. Chem. B 105, 11683 (2001)].

Electron-impact excitation and ionization cross sections for ground state and excited helium atoms

Comprehensive and critically assessed cross sections for the electron-impact excitation and ionization of ground state and excited helium atoms are presented. All states (atomic terms) with n ⩽ 4 are treated individually, while the states with n ⩾ 5 are considered degenerate. For the processes involving transitions to and from n ⩾ 5 levels, suitable cross section scaling relations are presented. For a large number of transitions, from both ground and excited states, convergent close coupling calculations were performed to achieve a high accuracy of the data. The evaluated/recommended cross section data are presented by analytic fit functions, which preserve the correct asymptotic behavior of the cross sections. The cross sections are also displayed in graphical form.

Single impact crater functions for ion bombardment of silicon

The average effect of a single 500eV incident argon ion on a silicon surface is studied using molecular dynamics simulations. More than 103 ion impacts at random surface points are averaged for each of seven incidence angles, from 0° to 28° off normal, to determine a local surface height change function, or a crater function. The crater shapes are mostly determined by mass rearrangement; sputtering has a relatively small effect. Analytical fitting functions are provided for several cases, and may serve as input into kinetic Monte Carlo calculations or stability analyses for surfaces subjected to ion bombardment.

Void statistics and void galaxies in the 2dF Galaxy Redshift Survey

In the 2dF Galaxy Redshift Survey, we study the properties of voids and of fainter galaxies within voids that are defined by brighter galaxies. Our results are compared with simulated galaxy catalogues from the Millennium simulation coupled with a semi-analytical galaxy formation recipe. We derive the void size distribution and discuss its dependence on the faint magnitude limit of the galaxies defining the voids. While voids among faint galaxies are typically smaller than those among bright galaxies, the ratio of the void sizes to the mean galaxy separation reaches larger values. This is well reproduced in the mock galaxy samples studied. We provide analytic fitting functions for the void size distribution. Furthermore, we study the galaxy population inside voids defined by galaxies with BJ− 5 log h < −20 and diameter larger than 10 h−1Mpc. We find a clear bimodality of galaxies inside voids and in the average field but with different characteristics. The abundance of blue cloud galaxies inside voids is enhanced. There is an indication of a slight blueshift of the blue cloud. Furthermore, galaxies in void centres have slightly higher specific star formation rates as measured by the η parameter. We determine the radial distribution of the ratio of early- and late-type galaxies through the voids. We find and discuss some differences between observations and the Millennium catalogues.

Excitation, ionization, and electron capture cross sections for collisions of Li 3+ with ground state and excited hydrogen atoms

Using the available experimental and theoretical data, as well as the established cross section scaling relationships, a comprehensive cross section database for excitation, ionization and electron capture in collisions of Li 3+ ions with ground state and excited hydrogen atoms has been generated. The critically assessed cross sections are represented by analytic fit functions that have the correct asymptotic behavior both at low and high collision energies. The derived cross sections are also presented in graphical form.

Influence of the interaction potential and of the temperature on the thermodiffusion (Soret) coefficient in a model system

In this paper the thermodiffusive behavior of an equimolar binary mixture subject to repulsive potentials of the form (sigma/r)(n) is investigated by using nonequilibrium molecular dynamics (NEMD) and the thermodiffusion (Soret) coefficient, S(T), is computed in a wide range of temperatures. With the aim to contribute to the study of the dependence of the Soret coefficient on the interaction potential, the exponent n of the potential is varied from 1 to 12, that is from a pseudocoulombian to a pseudohard-sphere interaction. The steady state equation is integrated for the composition function under reasonable assumptions and it is shown that in some cases the request for it to be linear cannot be satisfied. For this reason nonlinear functions are used to fit the NEMD composition data. The simulations indicate a negligible dependence of S(T) on the composition (in the composition range here considered) while the dependence on the temperature is more marked. The computed values of S(T) as a function of the temperature are fitted with analytical functions. It is found that with n> or =3 (medium and short range interaction) the model system behaves like a dilute gas mixture with the Soret coefficient varying with the temperature almost like 1/T. In the case of n=1 (long range interaction), S(T) has a more complex dependence on T: in particular it shows a change of sign. The analytical fitting functions, S(T)(T), are used in the integrated steady state equation thus obtaining the steady state composition profile and its comparison with the NEMD results indicates the grounding of the approach here proposed.

Argon nucleation in a cryogenic nucleation pulse chamber

Homogeneous nucleation of argon droplets has been measured with a newly designed cryogenic nucleation pulse chamber presented already in a previous paper [Fladerer and Strey, J. Chem. Phys. 124, 16 (2006)]. Here we present the first systematic nucleation onset data for argon measured in a temperature range from 42 to 58 K and for vapor pressures from 0.3 to 10 kPa. For these data we provide an analytical fit function. From the geometry of the optical detection system and the time of nucleation the experimental nucleation-rate range can be estimated. This allows a comparison of the data with the predictions of classical nucleation theory. We found 16-26 orders of magnitude difference between theory and experiment, and a too strong theoretical dependence of the nucleation rate on temperature. A comparison with the self-consistent theory of Girshick and Chiu [J. Chem. Phys. 93, 1273 (1990)] showed improved temperature dependence but still discrepancies of 11-17 orders of magnitude compared to experimental data. The thermodynamically consistent theory of Kashchiev [J. Chem. Phys. 118, 1837 (2003)] was found to agree rather well with experiment in respect to the temperature dependence and to predict rates about 5-7 orders of magnitude below the experimental ones. With the help of the Gibbs-Thomson equation we were able to evaluate the size of the critical nucleus to be 40-80 argon atoms.

Response of Dark Matter Halos to Condensation of Baryons: Cosmological Simulations and Improved Adiabatic Contraction Model

The cooling of gas in the centers of dark matter halos is expected to lead to a more concentrated dark matter distribution. The response of dark matter to the condensation of baryons is usually calculated using the model of adiabatic contraction, which assumes spherical symmetry and circular orbits. In contrast, halos in the hierarchical structure formation scenarios grow via multiple violent mergers and accretion along filaments, and particle orbits in the halos are highly eccentric. We study the effects of the cooling of gas in the inner regions of halos using high-resolution cosmological simulations which include gas dynamics, radiative cooling, and star formation. We find that the dissipation of gas indeed increases the density of dark matter and steepens its radial profile in the inner regions of halos compared to the case without cooling. For the first time, we test the adiabatic contraction model in cosmological simulations and find that the standard model systematically overpredicts the increase of dark matter density in the inner 5% of the virial radius. We show that the model can be improved by a simple modification of the assumed invariant from M(r)r to M(r_av)r, where r and r_av are the current and orbit-averaged particle positions. This modification approximately accounts for orbital eccentricities of particles and reproduces simulation profiles to within 10-20%. We present analytical fitting functions that accurately describe the transformation of the dark matter profile in the modified model and can be used for interpretation of observations.

H 1 multiple-quantum nuclear magnetic resonance investigations of molecular order distributions in poly(dimethylsiloxane) networks: Evidence for a linear mixing law in bimodal systems

We present proton–proton multiple-quantum investigations on a series of monomodal and strongly bimodal end-linked poly(dimethylsiloxane) model networks. A robust pulse sequence characterized by a well-defined double-quantum Hamiltonian along with a specific normalization approach is used to obtain double-quantum build-up curves. These curves are analyzed in terms of the spin dynamics of a local subsystem of monomer-fixed spins, where analytical fitting functions yielding residual dipole–dipole coupling constants are derived on the basis of exact solutions provided by simulations. Further employing the novel experimental strategy of double-quantum preselection of elastically active network chains, it is shown that the network response is purely heterogeneous, and that the data can be analyzed in terms of distributions of local dynamic order parameters using different models. The results yield consistent proof that local chain order in bimodal networks obeys a linear mixing law of short- and long-chain components. The order parameter distribution in a long-chain monomodal network is found to be surprisingly narrow, with a rather high average order parameter. Implications on the validity of present theories used to explain order and dynamics in networks are discussed.

An analytical model for the history of cosmic star formation

We use simple analytic reasoning to identify physical processes that drive the evolution of the cosmic star formation rate, , in cold dark matter universes. Based on our analysis, we formulate a model to characterize the redshift dependence of and compare it with results obtained from a set of hydrodynamic simulations that include star formation and feedback. We find that the cosmic star formation rate is described by two regimes. At early times, densities are sufficiently high and cooling times sufficiently short that abundant quantities of star-forming gas are present in all dark matter haloes that can cool by atomic processes. Consequently, generically rises exponentially as z decreases, independent of the details of the physical model for star formation, but dependent on the normalization and shape of the cosmological power spectrum. This part of the evolution is dominated by gravitationally driven growth of the halo mass function. At low redshifts, densities decline as the universe expands to the point that cooling is inhibited, limiting the amount of star-forming gas available. We find that in this regime the star formation rate scales approximately as , in proportion to the cooling rate within haloes. We demonstrate that the existence of these two regimes leads to a peak in the star formation rate at an intermediate redshift z=zpeak. We discuss how the location of this peak depends on our model parameters, and show that the peak cannot occur above a limiting redshift of z≈ 8.7. For the star formation efficiency adopted in our numerical simulations, zpeak≈ 5–6, with half of all stars forming at redshifts larger than z≃ 2.2. We derive analytic expressions for the full star formation history and show that they match our simulation results to better than ≃10 per cent. Using various approximations, we reduce the expressions to a simple analytic fitting function for that can be used to compute global cosmological quantities that are directly related to the star formation history. As examples, we consider the integrated stellar density, the supernova and gamma-ray burst rates observable on Earth, the metal enrichment history of the Universe, and the density of compact objects. We also briefly discuss the expected dependence of the star formation history on cosmological parameters and the physics of the gas.