Accessing the Single-Particle Structure of the Pygmy Dipole Resonance in ${}^{208}\mathrm{Pb}$

New experimental data on the neutron single-particle character of the Pygmy Dipole Resonance (PDR) in ${}^{208}\mathrm{Pb}$ are presented. They were obtained from $(d,p)$ and resonant proton scattering experiments performed at the Q3D spectrograph of the Maier-Leibnitz Laboratory in Garching, Germany. The new data are compared to the large suite of complementary, experimental data available for ${}^{208}\mathrm{Pb}$ and establish $(d,p)$ as an additional, valuable, experimental probe to study the PDR and its collectivity. Besides the single-particle character of the states, different features of the strength distributions are discussed and compared to large-scale shell model (LSSM) and energy-density functional plus quasiparticle-phonon model theoretical approaches to elucidate the microscopic structure of the PDR in ${}^{208}\mathrm{Pb}$.

Figure 1

(a),(b) ${}^{207}\mathrm{Pb}(d,p){}^{208}\mathrm{Pb}$ spectra taken at $\theta =25°$ for two different magnetic settings. Only a part of the spectrum is shown in panel (b). Contamination from the ${}^{12}\mathrm{C}$$(d,p){}^{13}\mathrm{C}$ reaction is observed (labeled with ${}^{13}\mathrm{C}$) due to the Carbon backing of the target. The kinematic correction with the Q3D multipole element was applied to the ${}^{207}\mathrm{Pb}(d,p)$ reaction causing the peaks observed from ${}^{12}\mathrm{C}(d,p)$ to be significantly broader (compare [84]). Known $J^{\pi }=1^{-}$ states of ${}^{208}\mathrm{Pb}$ [85, 86, 87], which could be resolved, are highlighted with vertical, dashed lines. All other states, seen in the spectra, correspond to excited states of ${}^{208}\mathrm{Pb}$ with a $3p_{1/2}$ neutron-hole component in their wave function. Below $S_n$, many of them were experimentally observed before [85, 86, 88, 89, 90].

Figure 2

Measured $(d,p)$ angular distributions (differential cross sections $d\sigma /d\mathrm{\Omega }$) for selected $J^{\pi }=1^{-}$ states (circles) in comparison to DWBA calculations (lines). Two different 1p-1h configurations, $(3p_{1/2}{)}^{-1}(4s_{1/2}{)}^{+1}$ (blue, longer dashed lines) and $(3p_{1/2}{)}^{-1}(3d_{3/2}{)}^{+1}$ (red, shorter dashed lines), have been assumed to describe the experimental distributions. Black, solid lines correspond to superpositions of these two individual configurations. No multistep transfer was considered. For states above $S_n$, an effective neutron-separation energy of $S_n=8.5\mathrm{MeV}$ had to be used. Otherwise, the shape of the angular distribution would have been heavily distorted. A similar approach had been chosen in [96]. The unique features of the $l=0$ and $l=2$ transfers remain unchanged. For the $1^{-}$ state at 6264 keV, a Carbon contaminant prevented a cross section measurement at $\theta =25°$.

Figure 3

(a) Angle-integrated $(d,p)$ cross sections ${\sigma }_{(d,p)}$, (b) $c_{LJlj}$ amplitudes from $(p,p^{\prime }{)}_{\mathrm{IAR}}$ [71, 72, 75, 76], (c) isovector $B(E1)$ strengths from $(p,p^{\prime })$ [35], and (d) differential cross sections from $({}^{17}\mathrm{O},{}^{17}{\mathrm{O}}^{\prime }\gamma )$ [54]. The latter probe the isoscalar character of the $1^{-}$ states [54]. (e) ${\sigma }_{(d,p)}$ predicted by combining LSSM spectroscopic factors with DWBA calculations. (f) Decomposition of the LSSM wave functions into neutron 1p-1h components relative to the total wave function ${\psi }_{\text{total}}$. (g) 1p-1h and 2p-2h contributions to ${\psi }_{\text{total}}$. LSSM isovector $B(E1)$ strength predicted when (h) including all or (i) excluding the specified contributions. (j)–(m) same as (e)–(h) but for $\mathrm{EDF}+\mathrm{QPM}$. SVS stands for “state-vector structure” [97, 98].

Figure 4

Summed transition densities for the first five $1_{\mathrm{QRPA}}^{-}$ states, which contain the $(3p_{1/2}{)}^{-1}(4s_{1/2}{)}^{+1}$ and $(3p_{1/2}{)}^{-1}(3d_{3/2}{)}^{+1}$ components. All five states are dominant neutron 1p-1h states (compare [98]). For comparison, the summed transition densities for the GDR are shown.