Fine structure of the isoscalar giant monopole resonance in ${}^{58}\mathrm{Ni}, {}^{90}\mathrm{Zr}, {}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$

Background: Over the past two decades high energy-resolution inelastic proton scattering studies were used to gain an understanding of the origin of fine structure observed in the isoscalar giant quadrupole resonance (ISGQR) and the isovector giant dipole resonance (IVGDR). Recently, the isoscalar giant monopole resonance (ISGMR) in ${}^{58}\mathrm{Ni}, {}^{90}\mathrm{Zr}, {}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ was studied at the iThemba Laboratory for Accelerator Based Sciences (iThemba LABS) by means of inelastic $\alpha$-particle scattering at very forward scattering angles (including $0^{\circ }$). The good energy resolution of the measurement revealed significant fine structure of the ISGMR.

Objective: To extract scales by means of wavelet analysis characterizing the observed fine structure of the ISGMR in order to investigate the role of different mechanisms contributing to its decay width.

Methods: Characteristic energy scales are extracted from the fine structure using continuous wavelet transforms. The experimental energy scales are compared to different theoretical approaches performed in the framework of quasiparticle random phase approximation (QRPA) and beyond-QRPA including complex configurations using both non-relativistic and relativistic density functional theory.

Results: All models highlight the role of Landau fragmentation for the damping of the ISGMR especially in the medium-mass region. Models which include the coupling between one-particle–one-hole (1p-1h) and two-particle–two-hole (2p-2h) configurations modify the strength distributions and wavelet scales indicating the importance of the spreading width. The effect becomes more pronounced with increasing mass number.

Conclusions: Wavelet scales remain a sensitive measure of the interplay between Landau fragmentation and the spreading width in the description of the fine structure of giant resonances. The case of the ISGMR is intermediate between the IVGDR, where Landau damping dominates, and the ISGQR, where fine structure originates from coupling to low-lying surface vibrations.

Figure 1

Isoscalar monopole strength distributions obtained with the ($\alpha ,{\alpha }^{\prime }$) reaction at $E_{\alpha }=196\mathrm{MeV}$ on ${}^{208}\mathrm{Pb},{}^{120}\mathrm{Sn},{}^{90}\mathrm{Zr}$, and ${}^{58}\mathrm{Ni}$. See text for details.

Figure 2

Double-differential cross sections measured for the ${}^{24}\mathrm{Mg}$($\alpha ,{\alpha }^{\prime }$) reaction at $E_{\alpha }=196\mathrm{MeV}$ for the angular range ${\theta }_{\text{Lab}}=0^{\circ }–1.{91}^{\circ }$ (blue) and ${\theta }_{\text{Lab}}=2^{\circ }–6^{\circ }$ (red).

Figure 3

The PPC prediction of IS0 strength of ${}^{208}\mathrm{Pb}$. The dotted curve shows the RPA result, and the solid curve corresponds to the result of the calculation taking into account the phonon-phonon coupling. The dashed curve corresponds to the PPC calculation included only the ${[3^{-}\otimes 3^{-}]}_{\text{RPA}}$ configurations. The experimental data are taken from Ref. [16].

Figure 4

Top set (right column): IS0 strength of ${}^{90}\mathrm{Zr}$ obtained using the RCNP-based energy-dependent correction factor as determined in Ref. [16]. Top set (lower right): Density plot of the real part of the CWT coefficients of the data. Top set (left column): Corresponding power spectrum for the excitation-energy region indicated by the vertical dashed lines ($11\text{MeV}\le E_{\text{x}}\le 24\text{MeV}$) in the top right plot. Bottom set: Same as the top set but for the difference spectrum obtained using TAMU-based energy-dependent correction factors. Bottom set (left column): The corresponding power spectrum shown in black, contrasted with the power spectrum from the top set (blue line).

Figure 5

Left column: Experimental IS0 strength in ${}^{58}\mathrm{Ni}$ (top row) in comparison with model predictions (rows 2–5) folded with the experimental energy resolution. The vertical dashed lines indicate the summation region of the wavelet coefficients ($11–24\mathrm{MeV}$) to determine the power spectra. Right column: Corresponding power spectra. Scales are indicated by filled circles with the associated errors, and for the experimental results additionally by vertical grey bars.

Figure 6

Same as Fig. 5, but for ${}^{90}\mathrm{Zr}$.

Figure 7

Same as Fig. 5, but for ${}^{120}\mathrm{Sn}$ and the excitation-energy region from 11 to 20 MeV (experimental data).

Figure 8

Same as Fig. 5, but for ${}^{208}\mathrm{Pb}$ and the excitation-energy region from 11 to 16 MeV over which the wavelet coefficients were summed in order to determine the corresponding power spectra.