Electric and magnetic dipole strength in ${}^{112,114,116,118,120,124}\mathrm{Sn}$

Background: There is renewed interest in electric dipole strength distributions for a variety of reasons including the extraction of the dipole polarizability related to properties of the symmetry energy and a measure for the neutron skin thickness, understanding the structure of low-energy $E1$ strength in nuclei with neutron excess, and establishing the systematics of the isovector giant dipole resonance (IVGDR). Inelastic proton scattering at energies of a few hundred MeV and very forward angles including $0^{\circ }$ has been established as a tool for the study of electric and magnetic dipole strength distributions in nuclei.

Purpose: The present work aims at a systematic investigation of the electric and magnetic dipole strength distributions in the chain of stable even-mass tin isotopes.

Methods: Inelastic proton scattering experiments were performed at the Research Center for Nuclear Physics, Osaka, with a 295-MeV beam covering laboratory angles $0^{\circ }\text{–}{6}^{\circ }$ and excitation energies $6\text{–}22$ MeV. Cross sections due to $E1$ and $M1$ excitations were extracted with a multipole decomposition analysis (MDA) and then converted to reduced transition probabilities with the “virtual photon method” for $E1$ and the “unit cross section method” for $M1$ excitations, respectively. Including a theory-aided correction for the high-excitation-energy region not covered experimentally, the electric dipole polarizability was determined from the $E1$ strength distributions.

Results: Total photoabsorption cross sections derived from the $E1$ and $M1$ strength distributions show significant differences compared to those from previous $(\gamma ,xn)$ experiments in the energy region of the IVGDR. The widths of the IVGDR deduced from the present data with a Lorentz parametrization show an approximately constant value of about 4.5 MeV in contrast to the large variations between isotopes observed in previous work. The IVGDR centroid energies are in good correspondence to expectations from empirical systematics of their mass dependence. Furthermore, a study of the dependence of the IVGDR energies on bulk matter properties is presented. The $E1$ strengths below neutron threshold show fair agreement with results from $(\gamma ,{\gamma }^{\prime })$ experiments on ${}^{112,116,120,124}\mathrm{Sn}$ in the energy region between 6 and 7 MeV, where also isoscalar $E1$ strength was found for ${}^{124}\mathrm{Sn}$. At higher excitation energies, large differences are observed, pointing to a different nature of the excited states with small ground-state branching ratios. The isovector spin-$M1$ strengths exhibit a broad distribution between 6 and 12 MeV in all studied nuclei.

Conclusions: The present results contribute to the solution of a variety of nuclear structure problems including the systematics of the energy and width of the IVGDR, the structure of low-energy $E1$ strength in nuclei, new constraints to energy density functionals (EDFs) aiming at a systematic description of the dipole polarizability across the nuclear chart, from which properties of the symmetry energy can be derived, and the systematics of the isovector spin-$M1$ strength in heavy nuclei.

Figure 1

Particle identification via the correlation of energy loss and corrected time of flight (${\mathrm{ToF}}_{\mathrm{c}}$). Two beam bunches are shown. The time difference between the two bunches corresponds to a beam pulse period of about 60 ns.

Figure 2

Focal plane spectra of the ${}^{12}\mathrm{C}(p,p^{\prime })$ reaction before (left) and after (right) the aberration correction described in the text.

Figure 3

Correlation of the position $y_{fp}$ in the nondispersive direction and the vertical angle ${\varphi }_{fp}$ before (left) and after (right) the restoration of the focusing condition at the focal plane. The vertical angle was corrected in such a way that the background events are distributed symmetrically around $y_c=0$.

Figure 4

Background subtraction procedure using the correlation of the position $y_c$ in nondispersive direction and the vertical angle ${\varphi }_c$ after the correction shown in Fig. 3. The two-dimensional gate including true and background events is indicated by the black rectangle. To determine the background, the data were shifted by a constant value to the left and right in the $y_c$ direction.

Figure 5

Top: Excitation energy spectra of ${}^{124}\mathrm{Sn}$ measured at $0^{\circ }$ corresponding to the three data sets from Fig. 4: True-plus-background (blue), background from shift to the left (orange), and background shift to the right (green). Bottom: Background-subtracted spectrum.

Figure 6

Double differential cross sections of the ${}^{112,114,116,118,120,124}\mathrm{Sn}(p,p^{\prime })$ reactions at $E_0=295$ MeV for spectrometer angles $\theta =0^{\circ }$ (blue), $2.5^{\circ }$ (orange), and $4.5^{\circ }$ (green).

Figure 7

Angular distributions of different multipolarities calculated with the code DWBA07 and QPM transition densities for the ${}^{120}\mathrm{Sn}(p,p^{\prime })$ reaction in the angular range 0–$6^{\circ }$. The maxima of the curves are normalized to unity.

Figure 8

Strength distribution of the ISGMR (top) and ISGQR (bottom) in ${}^{120}\mathrm{Sn}$ from $\alpha$ scattering experiments [78]. Lorentzian fits in the resonance region are shown as orange curves.

Figure 9

Theoretical $(p,p^{\prime })$ cross sections of the ISGMR and ISGQR in ${}^{120}\mathrm{Sn}$ calculated with the DWBA07 code.

Figure 10

Spectra of the ${}^{120}\mathrm{Sn}(p,p^{\prime })$ reaction before (blue) and after (orange) subtraction of the ISGMR (green) and ISGQR (red) contributions for two different angles.

Figure 11

Top: Excitation energy spectra of ${}^{120}\mathrm{Sn}$ and excitation energy bins (vertical dashed lines) used to determine a parametrization of the angular dependence of the continuum background. Bottom: Corresponding angular distributions for different energy bins together with the fit of Eq. (2). For better visibility, they are shifted relative to each other by 2 mb/sr.

Figure 12

Typical results of the MDA for the example of ${}^{120}\mathrm{Sn}$ and three different excitation energy bins at 8, 15, and 23 MeV. Top: Spectra and energy bins indicated by the vertical dashed lines. Bottom: Experimental angular distributions and results of Eq. (5) for different multipoles and their sum.

Figure 13

Results of the multipole decomposition analysis for the spectra measured at a spectrometer angle $\theta =0^{\circ }$. Orange: Experimental cross sections after subtraction of the ISGMR and ISGQR contributions. Blue: $E1$ contributions. Green: $M1$ contributions. Red: Contributions from multipoles $\lambda >1$. Purple: Continuum background. The error bars include statistical, systematic, and MDA uncertainties.

Figure 14

Photoabsorption cross sections obtained in this work (blue circles) in comparison to $(\gamma ,xn)$ experiments by Fultz et al. [32] at Livermore (green left triangles), Leprêtre et al. [33] at Saclay (red right triangles), and Sorokin et al. [34, 35] (orange downward triangles). $(\gamma ,n)$ data from Utsunomiya et al. [36, 37] are shown as black upward triangles. The neutron thresholds are indicated by vertical dashed lines.

Figure 15

Same as Fig. 14, but restricted to the energy region $8\text{–}13$ MeV.

Figure 16

Centroid energies $E_{\mathrm{GDR}}$ (top) and widths ${\mathrm{\Gamma }}_{\mathrm{GDR}}$ (bottom) of Lorentzian fits to the IVGDR in tin isotopes determined from the data shown in Fig. 14. The purple line shows the phenomenological mass dependence of the centroid energy, Eq. (9).

Figure 17

Average IVGDR peak positions, Eq. (10), along the chain of tin isotopes. Compared are experimental data (blue) with results from various Skyrme parametrizations as indicated (for details, see text). Each panel collects variation of one NMP. The results of the original SV-bas interaction [89] are always shown in green.

Figure 18

$B\left(E1\right)$ strength distributions for ${}^{112}\mathrm{Sn}$ and ${}^{116}\mathrm{Sn}$ below the neutron threshold in 200-keV bins from the present work (blue) in comparison with results from NRF experiments [29, 98] (orange).

Figure 19

Electric dipole strength distributions for ${}^{124}\mathrm{Sn}$ in 200-keV bins from different experiments. Top: $B\left(E1\right)$ strength distributions for ${}^{124}\mathrm{Sn}$ from the present work (blue) in comparison with NRF results [98] (orange). Bottom: Isoscalar $E1$ strength distributions deduced from (${}^{17}\mathrm{O},{}^{17}\mathrm{O}^{\prime }\gamma$) experiment [103] (red) and differential cross sections from an ($\alpha ,{\alpha }^{\prime }\gamma$) experiment [104] (green).

Figure 20

Running sums of the dipole polarizability deduced from the present ($p,p^{\prime }$) data. Red: Contribution from 6 MeV to $S_{\mathrm{n}}$. Blue: Contribution from $S_{\mathrm{n}}$ to 20 MeV. Orange: Contribution above 20 MeV from QPM calculations; see text for details.

Figure 21

Contribution to the dipole polarizability in the energy region from $S_n$ to 20 MeV deduced from the present data (blue circles), Ref. [32] (green diamonds), and Ref. [33] (orange squares).

Figure 22

Dipole polarizability of stable even-mass tin isotopes in the energy region from 20 to 29.6 MeV deduced from the data of Ref. [32] (green diamonds) compared with the theory-based estimate used for the present results (red pentagons).

Figure 23

$B\left(M1\right)$ strength distributions extracted with the method described in the text. Additionally, for ${}^{120}\mathrm{Sn}$ results based on the measurement of spin-transfer observables from Ref. [8] are shown. The vertical lines indicate the neutron threshold energies.