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Giant Dipole Resonance and Related Spin-dependent Excitations

E. B. Balbutsev, I. V. Molodtsova

TL;DR

This paper develops and applies a spin‑aware time‑dependent Hartree–Fock framework, implemented via the Wigner Function Moments method, to study the Giant Dipole Resonance (GDR) and related spin‑dipole excitations in heavy deformed nuclei, exemplified by $^{164}$Dy. By solving the TDHF equations with a harmonic oscillator mean field plus spin–orbit coupling and separable dipole–dipole, quadrupole–quadrupole, and spin‑dipole interactions, the authors derive and linearize a comprehensive set of dipole dynamical equations, separating isovector/isoscalar and spin channels. They examine both $oldsymbol{η=0}$ (no spin–orbit) and $oldsymbol{η eq 0}$ cases to quantify the deformation‑induced GDR splitting, the emergence of an electric spin dipole resonance (ESDR), and the distribution of $B(E1)$ and $B(M2)$ strengths, noting a strong isovector dominance in spin‑M2 modes and enabling predictions for ESDR positions and strength exhaustion. The results align with global GDR trends, reveal a modest deformation splitting for the spin M2 sector, and provide a detailed map of how spin dynamics couple to electromagnetic responses in heavy nuclei, with implications for interpreting spin‑dependent resonances in experiments.

Abstract

The time-dependent Hartree-Fock equation is solved by the Wigner Function Moments method taking into account spin degrees of freedom. Energies and reduced transition probabilities of $K^π=0^-$, $1^-$ and $2^-$ excitations are calculated taking $^{164}$Dy as an example. The spin degrees of freedom give rise to the electric Spin Dipole Resonance. Its properties and interplay with the Giant Dipole Resonance are investigated. The deformation-induced splitting of the spin $M2$ resonance is discussed. The results of calculations are compared with the experimental data and other theoretical studies.

Giant Dipole Resonance and Related Spin-dependent Excitations

TL;DR

This paper develops and applies a spin‑aware time‑dependent Hartree–Fock framework, implemented via the Wigner Function Moments method, to study the Giant Dipole Resonance (GDR) and related spin‑dipole excitations in heavy deformed nuclei, exemplified by Dy. By solving the TDHF equations with a harmonic oscillator mean field plus spin–orbit coupling and separable dipole–dipole, quadrupole–quadrupole, and spin‑dipole interactions, the authors derive and linearize a comprehensive set of dipole dynamical equations, separating isovector/isoscalar and spin channels. They examine both (no spin–orbit) and cases to quantify the deformation‑induced GDR splitting, the emergence of an electric spin dipole resonance (ESDR), and the distribution of and strengths, noting a strong isovector dominance in spin‑M2 modes and enabling predictions for ESDR positions and strength exhaustion. The results align with global GDR trends, reveal a modest deformation splitting for the spin M2 sector, and provide a detailed map of how spin dynamics couple to electromagnetic responses in heavy nuclei, with implications for interpreting spin‑dependent resonances in experiments.

Abstract

The time-dependent Hartree-Fock equation is solved by the Wigner Function Moments method taking into account spin degrees of freedom. Energies and reduced transition probabilities of , and excitations are calculated taking Dy as an example. The spin degrees of freedom give rise to the electric Spin Dipole Resonance. Its properties and interplay with the Giant Dipole Resonance are investigated. The deformation-induced splitting of the spin resonance is discussed. The results of calculations are compared with the experimental data and other theoretical studies.

Paper Structure

This paper contains 27 sections, 79 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. TDHF equation and WFM equations of motion
  3. Dipoles
  4. Results of calculations
  5. The Giant Dipole Resonance ($\eta=0$ limit)
  6. GDR ($K^\pi=1^-$)
  7. GDR ($K^\pi=0^-$)
  8. GDR sum rule
  9. Spin-dependent excitations ($\eta=0$ limit)
  10. $K^\pi=1^-$
  11. $K^\pi=0^-$
  12. Exact calculations ($\eta\neq 0$)
  13. $K^\pi=0^-$
  14. $K^\pi=1^-$
  15. $K^\pi=2^-$
  16. ...and 12 more sections

Figures (6)

  • Figure 1: Energies of $1^-$ excitations as a function of the mass number calculated in the frame of WFM metod with SkM$^*$ force for nuclei on the beta-stability line. Solid points corresponds to experimental GDR centroids. Reproduced from Ref. Piper.
  • Figure 2: The $1^-$ excitations in the GDR energy region. Calculations taking into account the moments of the first and third ranks. The dashed line indicates the energy centroid. The dotted line shows the results of calculations taking into account only first-rank tensors. Reproduced from Ref. Piper.
  • Figure 3: Energies $E_{1K}$ (a, b, d, e) and values of $S(E1K)=E_{1K}B(E1K)$ (c, f) vs. $\xi_0^{K=\mu}/\tilde{\xi}_0^{K=\mu}$ ratio. The calculations are performed for $^{164}$Dy without the spin-orbit interaction ($\eta=0$). The solid blue line corresponds to the GDR, and the dotted red line to the CMM. The black short dashed lines in the panels (c) and (f) indicate the EWSR (\ref{['SRE1']}) values.
  • Figure 4: Energies (a) and $E1$ strengths (b) of $K^{\pi}=0^-$ GDR (red line) and ESDR (blue dashed line) vs. deformation $\delta$. The energy centroid $\bar{E}_{10}$ and summed $B(E10)$ value are shown by black dot-dashed lines. Calculations are performed for $^{164}$Dy.
  • Figure 5: Energies of IV (a) and IS (b) spin-dipole excitations for $\delta=0$ (see Table \ref{['tab8']}) and for $\delta=0.26$ (see Tables \ref{['tab4']}, \ref{['tab5']}, \ref{['tab6']} and \ref{['tab7']}). Calculations are performed for $^{164}$Dy.
  • ...and 1 more figures