Dynamical scheme for computing the mass parameter of a system in a medium

TL;DR

This work develops a time-dependent density functional theory framework to extract inertia (mass) parameters $M_Q$ for collective degrees of freedom in a medium, with application to impurities in neutron-star crusts. By adding a tunable harmonic term to the energy functional and analyzing damped oscillations, the method determines $M_Q$ from the $\omega(k)$ dependence, linking microscopic dynamics to an effective Hamiltonian $H_Q \approx \frac{M_Q \dot{Q}^2}{2} + V(Q)$. Applied to center-of-mass and quadrupole modes of proton-rich impurities in superfluid neutron matter, the study reveals density-dependent trends: $M_{\text{c.m.}}$ and $M_{Q_{20}}$ decrease with $\bar{\rho}$, with quadrupole modes showing substantial damping and a transition toward crust-core behavior around $\bar{\rho} \sim 0.09$ fm$^{-3}$. The results support using TDDFT-derived parameters to build microscopic, effective models of neutron-star crust dynamics, including lattice vibrations and transport properties, while highlighting the method’s complementarity to static analyses and its current limitations regarding vortices.

Abstract

We present a new method for extracting a mass parameter using time-dependent density functional theory for an arbitrary physical system, provided the adiabatic limit is achievable. This approach works for collective variables also in the presence of a medium, in particular for the nuclei interacting with a neutron background. We apply the method to extract mass parameters of impurities in the neutron star crust, like their inertial masses and quadrupole mass parameters. The extracted mass parameters at various depths of the inner crust are compared with other methods, including the hydrodynamic approach. The presented method opens avenues for the construction of an effective model of neutron star crust grounded in microscopic calculations.

Figures (7)

• Figure 1: Scenario considered in this work: a nucleus ${}_{40}$Zr immersed in a superfluid neutron matter. The simplest types of collective modes that nuclei can execute are: (a) center of mass and (b) quadrupole moment oscillations. From the time evolution of the collective degrees of freedom associated with the mode, we can fit the parameters of interest, such as the mass parameter and dissipation coefficients.
• Figure 2: a) Position of the center of mass of protons as a function of time for different values of the $k$ parameter. The considered system has average nuclear density $\bar{\rho} = 0.015 \textrm{~fm}^{-3}$. b) The extracted frequency of oscillations as a function of the $k$ parameter. The dashed line is the fit of Eq. (\ref{['eq:omegak']}), $\omega^2 = 7\cdot 10^{-6} k$.
• Figure 3: The effective mass $M_\textrm{c.m.}$ (in units of neutron mass $m_n$) as a function of mean nuclear density $\bar{\rho}$ extracted for small amplitude $A=1$ fm (orangle triangles) and large amplitude $A=1$ fm (green squares) oscillations. We compare the results of the proposed method (orange triangles) with the effective mass obtained recently using a different dynamical scheme (blue diamonds) pecak2024WBSkMeff.
• Figure 4: a) The damped oscillations of the center of mass of protons for selected density $\bar{\rho} = 0.019 \textrm{~fm}^{-3}$ (we show that point in panel c), and b) associated absolute value of velocity as a function of time. The velocity is compared to Landau's velocity. In panel c), we show the extracted damping coefficient $\gamma_\textrm{c.m.}$ as a function of average nuclear density.
• Figure 5: a) Oscillations of the quadrupole moment $Q_{20}$ as a function of time for different values of the $k$ parameter. The considered system has average nuclear density $\bar{\rho} = 0.015 \textrm{~fm}^{-3}$. b) The extracted frequency of oscillations as a function of the $k$ parameter. The dashed line is the fit of Eq. (\ref{['eq:omegak']}), $\omega^2(k)=(0.1023)^2 + \frac{k}{0.0973}$.