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Stopping power of electron liquid for slow quantum projectiles

Vladimir U. Nazarov, E. K. U. Gross

TL;DR

The work develops a quantum-mechanical theory of stopping power by applying Exact Factorization to treat the projectile and target electrons on equal footing, moving beyond classical Ehrenfest dynamics. In a mean-field (TDSCF) approximation reformulated via TDDFT, the authors derive a friction coefficient $Q=Q_1+Q_2$ for slow projectiles, where $Q_1$ captures binary collisions through KS response and $Q_2$ encodes dynamic XC effects. Calculations in a jellium model reveal a striking mass dependence of friction for projectiles with the same charge and velocity, a purely quantum phenomenon arising from differences in projectile wave-packet sizes. The results highlight the significance of quantum projectile dynamics and establish an EF+TDDFT framework for SP in inhomogeneous quantum media, with potential implications for interpreting SP in metallic systems.

Abstract

We revisit the problem of deceleration of a charge moving in a medium. Going beyond the traditional approach, which relies on Ehrenfest dynamics, we treat the projectile fully quantum mechanically, on the same footing as the electrons of the target. In order to separate the dynamics of the projectile from that of the electrons, we employ the Exact Factorization method. We illustrate the resulting theory by applying it to the problem of the stopping power (SP) of a jellium-model metal for slowly moving charges. The quantum mechanical nature of particles manifests itself remarkably in the differences in the SP for projectiles of the same charge moving with the same velocity, but having different masses.

Stopping power of electron liquid for slow quantum projectiles

TL;DR

The work develops a quantum-mechanical theory of stopping power by applying Exact Factorization to treat the projectile and target electrons on equal footing, moving beyond classical Ehrenfest dynamics. In a mean-field (TDSCF) approximation reformulated via TDDFT, the authors derive a friction coefficient for slow projectiles, where captures binary collisions through KS response and encodes dynamic XC effects. Calculations in a jellium model reveal a striking mass dependence of friction for projectiles with the same charge and velocity, a purely quantum phenomenon arising from differences in projectile wave-packet sizes. The results highlight the significance of quantum projectile dynamics and establish an EF+TDDFT framework for SP in inhomogeneous quantum media, with potential implications for interpreting SP in metallic systems.

Abstract

We revisit the problem of deceleration of a charge moving in a medium. Going beyond the traditional approach, which relies on Ehrenfest dynamics, we treat the projectile fully quantum mechanically, on the same footing as the electrons of the target. In order to separate the dynamics of the projectile from that of the electrons, we employ the Exact Factorization method. We illustrate the resulting theory by applying it to the problem of the stopping power (SP) of a jellium-model metal for slowly moving charges. The quantum mechanical nature of particles manifests itself remarkably in the differences in the SP for projectiles of the same charge moving with the same velocity, but having different masses.

Paper Structure

This paper contains 16 sections, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stopping power problem with a quantum projectile: Exact Factorization approach
  3. Mean field approximation
  4. TDDFT reformulation
  5. Ground state
  6. Low-velocity stopping power
  7. Method
  8. Results and discussion
  9. Conclusions
  10. Mean field approximation: Derivation of Eqs. (\ref{['N1']})-(\ref{['E3']})
  11. Derivation of Eq. (\ref{['res']})
  12. 1st order in $\mathbf{E}$
  13. 2nd order in ${\mathbf{E}}$
  14. Density response functions
  15. 1st order
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Self- and mutually consistent ground-state properties of the system of the impurity particle of the charge $Z=+1$ a.u. immersed in the jellium-model electron gas of the density parameter $r_s=2.07$. The cases of the infinite mass particle, proton ($M=1837$ a.u.), antimuon ($M=207$ a.u.), and positron ($M=1$ a.u.) are compared. (a) KS potential of the electronic system; (b) The electron density relative to that of the unperturbed HEG; (c) The potential experienced by the impurity particle and (d) the corresponding impurity particle density distribution.
  • Figure 2: The same as Fig. \ref{['static']} but for negatively charged impurities of $Z=-1$ a.u. The cases of the infinite mass particle, antiproton, muon, and a fictitious distinguishable electron are compared. Some lines are too close to each other to be discerned.
  • Figure 3: Friction coefficient versus the electron gas density parameter $r_s$ for positively (left) and negatively (right) charged projectiles.