Searcharxiv⌕ Search

arXiv subjects

Jean-Christophe Pain

Publications and source records attributed to Jean-Christophe Pain.

At least 55 records · Page 3Linked to original sources
arXiv2025

Helioseismic inference of the solar radiative opacity

The Sun is the most studied of all stars, and thus constitutes a benchmark for stellar models. However, our vision of the Sun is still incomplete, as illustrated by the current debate on its chemical composition. The problem reaches far beyond chemical abundances and is intimately linked to microscopic and macroscopic physical ingredients of solar models such as radiative opacity, for which experimental results have been recently measured that still await theoretical explanations. We present opacity profiles derived from helioseismic inferences and compare them with detailed theoretical computations of individual element contributions using three different opacity computation codes, in a complementary way to experimental results. We find that our seismic opacity is about 10% higher than theoretical values used in current solar models around 2 million degrees, but lower by 35% than some recent available theoretical values. Using the Sun as a laboratory of fundamental physics, we show that quantitative comparisons between various opacity tables are required to understand the origin of the discrepancies between reported helioseismic, theoretical and experimental opacity values.

astro-ph.SR↗
arXiv2025

Electrical conductivities and low frequency opacities in the warm dense matter regime

In this article, we examine different approaches for calculating low frequency opacities in the warm dense matter regime. The relevance of the average-atom approximation and of different models for calculating opacities, such as the Ziman or Ziman-Evans models is discussed and the results compared to \textit{ab initio} simulations. We begin by recalling the derivation of the Ziman-Evans resistivity from Kubo's linear response theory, using the local approximation to the solutions of the Lippmann-Schwinger equation. With the help of this approximation, we explicitly introduce an ionic structure factor into the Ziman formula, without resorting to the Born approximation. Both approaches involve the calculation of scattering phase shifts, which we integrate from Calogero equation with an adaptive step numerical scheme based on a Runge-Kutta-Merson solver. We show that if the atomic number $Z$ is not too large, integrating the phase shifts in this way is more time-efficient than using a classical Numerov-type scheme to solve the radial Schrödinger equation. Various approximations are explored for phase shifts to further improve computation time. For the Born approximation, we show that using Born phase shifts directly in the scattering cross-section gives more accurate results than with the integral formula based on the Fourier transform of the electron-ion potential. We also compare an analytical formula based on a Yukawa fit of the electron-ion potential to a numerical integration. The average-atom results are compared with DFT-based molecular dynamics simulations for aluminum in the dilute regime and for copper, aluminum and gold at solid density and different temperatures.

physics.plasm-ph↗
arXiv2025

On the electrons really contributing to dc conductivity of warm dense matter

Atomic properties of warm dense matter is an active field of research. Understanding transport properties of these states is essential for providing coefficients needed by magneto-radiative hydrodynamics codes for many studies, including hydrodynamic instabilities, energy balances or heating in fusion plasmas, difficult to investigate by experimental means. In this paper, we present an average-atom approach for the calculation of direct-current electric conductivity within Ziman's theory. The mean ion charge $Z^*$, commonly called ionization, is an important input of the Ziman formula, but is not clearly defined within average-atom models. Our study spans a wide range of thermodynamical conditions, i.e., for the densities, from a few $10^{-2}$ to about 4 times the solid's density, and, for the temperatures, typically from 0.1 eV to 700 eV, favorable to large differences in the mean ion charge $Z^*$ according to its definition. We compare and discuss different ways of defining $Z^*$ while trying to figure out which electrons really contribute to electric conduction. We compare our results with experimental data and published theoretical values, in particular from the second transport code comparison workshop, which was held in July 2023 at Lawrence Livermore National Laboratory. These comparisons lead us to propose indicators for the relevance of including different charges predicted by our average-atom model in the definition of $Z^*$.

physics.plasm-ph↗
arXiv2025

Expression of special stretched $9j$ coefficients in terms of $_5F_4$ hypergeometric series

The Clebsch-Gordan coefficients or Wigner $3j$ symbols are known to be proportional to a $_3F_2(1)$ hypergeometric series, and Racah $6j$ coefficients to a $_4F_3(1)$. In general, however, non-trivial $9j$ symbols can not be expressed as a $_5F_4$. In this letter, we show, using the Dougall-Ramanujan identity, that special stretched $9j$ symbols can be reformulated as $_5F_4(1)$ hypergeometric series.

math.CA↗
arXiv2025

Congruence properties of prime sums and Bernoulli polynomials

In this article, we derive a congruence property of particular sum rules involving prime numbers. The resulting expression involves Bernoulli numbers and polynomials, for which we obtain, as a consequence, a general congruence relation as well.

math.HO↗
arXiv2025

A new approach to include electron interaction effects in super transition array opacity theory

A method is presented for the improved calculation of super-shell partition functions which include the repulsive electron-electron interaction energy terms in the Boltzmann factor. Heretofore these interaction terms were approximately treated via the use of Feynman-Jensen inequalities. Such investigations are of particular interest for the super-transition-array approach of hot-plasma radiative opacity.

physics.atom-ph↗
arXiv2025

From the Schaar and Lösch-Schoblick integrals to representations of the Glaisher-Kinkelin constant

In this article, we present two integral representations of the logarithm of the Glaisher-Kinkelin constant, relying on two different integral formulations of the so-called Binet function $μ(x)$. The first one is attributed to Schaar (and also often referred to as ``the second Binet formula''), and the second one is due to Lösch and Schoblik. It seems that the two new expressions (formulas (28) and (33) of the present article) of the Glaisher-Kinkelin constant, can not be easily deduced from know ones.

math.GM↗
arXiv2025

On the number of atomic configurations in hot plasmas

We propose approximate and accurate formulas for the number of electron configurations in hot plasmas. Such a quantity is an ingredient of algorithms devoted to the generation of configurations or superconfigurations, which is a pre-requisite of opacity calculations. One of the main formulas involves Bessel functions of fractional order and the procedure for improving the accuracy through a series expansion is explained.

physics.atom-ph↗
arXiv2024

A family of polylogarithmic integrals

In this paper we investigate a class of integrals that were encountered in the course of a work on statistical plasma physics, in the so-called Sommerfeld temperature-expansion of the electronic entropy. We show that such integrals, involving some parameters, can be fully described in closed form represented by special functions.

math.CA↗
arXiv2024

Electron-impact ionization of Si IV-VIII relevant for inertial confinement fusion

In this work, we investigate the ionization of silicon by electron impacts in hot plasmas. Our calculations of the cross sections and rates rely on the Coulomb-Born-Exchange, Binary-Encounter-Dipole and Distorted-Wave methods implemented in the Flexible Atomic Code (FAC), and are compared with measurements and other theoretical values. We use a semi-empirical formula for the cross section, which involves a small set of adjustable parameters. Configuration interaction is taken into account and is shown to affect the cross section at low energy, in particular for Si$^{3+}$. The rate coefficient is then expressed in terms of these parameters and is represented in a large temperature interval, up to 10$^8$ K. As expected, the agreement with measurements improves for increasing ion charges, confirming the applicability of our approach to hot plasma studies such as inertial-confinement fusion, and its reliability.

physics.atom-ph↗
arXiv2024

Expression of Farhi's integral in terms of known mathematical constants

In an interesting article entitled "A curious formula related to the Euler Gamma function", Bakir Farhi posed the open question of whether it was possible to obtain an expression of $$ \boldsymbolη=2\int_0^1\logΓ(x)\,\cdot\sin(2πx)\,\mathrm{d}x=0.7687478924\dots $$ in terms of the known mathematical constants as $π$, $\logπ$, $\log 2$, $Γ(1/4)$, $e$, etc. In the present work, we show that $$ \boldsymbolη=\frac{1}π\left[γ+\log(2π)\right], $$ where $γ$ is the usual Euler-Mascheroni constant, and provide two different proofs, the first one involving the Glaisher-Kinkelin constant, and the second one based on the Malmstén integral representation of $\logΓ(x)$. The resulting formula can also be obtained directly from the knowledge of the Fourier series expansion of $\logΓ(x)$.

math.NT↗
arXiv2024

Estimates of the minimum of the Gamma function using the Lagrange inversion theorem and the Faà di Bruno formula

In this article we derive, using the Lagrange inversion theorem and applying twice the Faà di Bruno formula, an expression of the minimum of the Gamma function $Γ$ as an expansion in powers of the Euler-Mascheroni constant $γ$. The result can be expressed in terms of values the Riemann zeta function $ζ$ of integer arguments, since the multiple derivative of the digamma function $ψ$ evaluated in $1$ is precisely proportional to the zeta function. The first terms (up to $γ^6$) were provided in order to address the convergence of the series. Applying the Lagrange inversion theorem at the value $3/2$ yields more accurate results, although less elegant formulas, in particular because the digamma function evaluated in $3/2$ does not simplify.

math.NT↗
arXiv2024

A family of integrals related to values of the Riemann zeta function

We propose a relation between values of the Riemann zeta function $ζ$ and a family of integrals. This results in an integral representation for $ζ(2p)$, where $p$ is a positive integer, and an expression of $ζ(2p+1)$ involving one of the above mentioned integrals together with a harmonic-number sum. Simplification of the latter eventually leads to an integral representation of $ζ(2p + 1)$.

math.NT↗
arXiv2024

An integral representation of Catalan numbers using the Féaux formula

We present an integral expression of the Catalan numbers, based on Féaux' integral representation of $\log\left[Γ(x)\right]$, $Γ$ being the usual Gamma function. The obtained formula may be the starting point of the derivation of new relations involving central binomial coefficients or Catalan numbers.

math.CO↗
arXiv2024

Integral representations of the Riemann zeta function of odd argument

In this article we obtain, using an expression of the digamma function $ψ(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $ζ(2p+1)$ for any positive value of $p$. The integrand consists of the product of a polynomial by one or two elementary trigonometric functions. Examples for the first values of the argument are given. Some of them were already derived by other methods.

math.NT↗
arXiv2024

An integral representation of Catalan numbers using Malmstén's formula

In this article, we propose an integral expression of the Catalan numbers, based on Malmstén's definite-integral representation of $\ln\left[Γ(x)\right]$, $Γ$ being the usual Gamma function. The obtained expression is likely to yield new summations involving Catalan numbers or central binomial coefficients.

math.CO↗
arXiv2024

Distribution of angular momenta $M_L$ and $M_S$ in non-relativistic configurations: statistical analysis using cumulants and Gram-Charlier series

The distributions $P(M_L,M_S)$ of the total magnetic quantum numbers $M_L$ and $M_S$ for $N$ electrons of angular momentum $\ell$, as well as the enumeration of $LS$ spectroscopic terms and spectral lines, are crucial for the calculation of atomic structure and spectra, in particular for the modeling of emission or absorption properties of hot plasmas. However, no explicit formula for $P(M_L,M_S)$ is known yet. In the present work, we show that the generating function for the cumulants, which characterize the distribution, obeys a recurrence relation, similar to the Newton-Girard identities relating elementary symmetric polynomials to power sums. This enables us to provide an explicit formula for the generating function. We also analyze the possibility of representing the $P(M_L,M_S)$ distribution by a bi-variate Gram-Charlier series, which coefficients are obtained from the knowledge of the exact moments of $P(M_L,M_S)$. It is shown that a simple approximation is obtained by truncating this series to the first few terms, though it is not convergent.

physics.atom-ph↗