Eur. Phys. J. Plus (2025) 140: 612
https://doi.org/10.1140/epjp/s13360-025-06556-x

Regular Article

Thermodynamic properties of harmonically trapped sodium ²³Na gas in one dimension

¹ Department of Physics, Faculty of Science, The Hashemite University, 13133, Zarqa, Jordan
² Department of Applied Physics, Tafila Technical University, 66110, Tafila, Jordan
³ Department of Physics, School of Science, The University of Jordan, 11942, Amman, Jordan

^a msugh@hu.edu.jo

Received: 12 January 2025
Accepted: 16 June 2025
Published online: 1 July 2025

Abstract

The static fluctuation approximation (SFA) is used to calculate the condensate fraction and thermodynamic properties of a harmonically trapped, one-dimensional interacting, ²³Na gas. The interatomic interaction is a repulsive contact potential. A closed set of coupled nonlinear integral equations is derived and solved numerically as a first step for calculating the following thermodynamic properties: energy spectrum, particle distribution, correlations, and fluctuations. Based on these quantities, the condensate fraction $\frac{N_0}{N}$, chemical potential $\mu$, energy per particle U/N (N being the total number of particles in the system), specific heat capacity c_v, pressure P, and entropy S are calculated. The effects of the temperature T, longitudinal frequency $\omega$, and transverse frequency ${\omega }_{\text{per}}$ on these properties are explored. Our findings indicate that at very low T, $\frac{N_0}{N}$ is close to 0.55; at a specific T, it increases with $\omega$, but decreases with ${\omega }_{\text{per}}$. The $\mu$ results show that the system is in the quantum regime at ‘low’ T and in the classical regime above the transition temperature T_c. The behavior of $\mu$ with $\omega$ and ${\omega }_{\text{per}}$ shows that both $\mu$ and T_c increase with $\omega$ and ${\omega }_{\text{per}}$. U/N is parabolic with T and $\omega$, but is nearly linear with respect to ${\omega }_{\text{per}}$. P reaches saturation above T_c; it is parabolic with T and decreases with $\omega$, but shows linear behavior at ‘high’ values of ${\omega }_{\text{per}}$ [(${\omega }_{\text{per}}/\left(2\pi \right)>1200$ Hz]. S increases with T and behaves more like P with $\omega$ and ${\omega }_{\text{per}}$.