Dipole polarizability of ${}^{120}\mathrm{Sn}$ and nuclear energy density functionals

The electric dipole strength distribution in ${}^{120}\mathrm{Sn}$ between 5 and 22 MeV has been determined at the Research Center for Nuclear Physics, Osaka, from polarization transfer observables measured in proton inelastic scattering at $E_0=295$ MeV and forward angles including $0^{\circ }$. Combined with photoabsorption data, a highly precise electric dipole polarizability ${\alpha }_{\mathrm{D}}\left({}^{120}\mathrm{Sn}\right)=8.93\left(36\right){\mathrm{fm}}^3$ is extracted. The dipole polarizability as isovector observable par excellence carries direct information on nuclear symmetry energy and its density dependence. The correlation of the new value with the well-established ${\alpha }_{\mathrm{D}}\left({}^{208}\mathrm{Pb}\right)$ serves as a test of its prediction by nuclear energy density functionals. Models based on modern Skyrme interactions describe the data fairly well while most calculations based on relativistic Hamiltonians cannot.

Figure 1

(a) Double differential cross sections (black squares) and decomposition into non-spinflip (red diamonds) and spinflip (blue circles) parts of the ${}^{120}\mathrm{Sn}\left(\vec{p},{\vec{p}}^{\prime }\right)$ reaction at $E_0=295$ MeV and $\theta =0^{\circ }–{2.5}^{\circ }$. The green solid line shows the cross sections due to excitation of the ISGQR estimated as described in the text. (b) Total spin transfer from Eq. (2).

Figure 2

Comparison of the $B\left(E1\right)$ strength distribution in ${}^{120}\mathrm{Sn}$ determined by the present work (red circles) and in ($\gamma ,xn$) experiments (blue diamonds [38], green squares [39], and black upward triangles [40]).

Figure 3

Correlation of the experimental ${\alpha }_{\mathrm{D}}$ values for ${}^{120}\mathrm{Sn}$ and ${}^{208}\mathrm{Pb}$ with uncertainties shown as yellow shaded bands. (a) Comparison with models based on Skyrme interactions ${\mathrm{SkM}}^{*}$ [51] (open red diamond), SkP [52] (open green upward triangle), SkT6 [53] (full red diamond), SG-II [54] (open blue circle), SkI3 [55] (full green upward triangle), SLy6 [56] (full red downward triangle), BSk4 [57] (full green star), SV-bas [13] (open blue downward triangle), and UNEDF2 [58] (open blue square). The SV-min [13] (full red circle) and RD-min [59] (full green square) interactions additionally provide theoretical error bars [13, 60]. The dashed black line indicates the correlation between both ${\alpha }_{\mathrm{D}}$ values. (b) Comparison with relativistic mean field models DD-PC-min [61] (blue squares) and DD-ME-min [62] (black circles), FSU [63] (red diamonds), and FSU2 [64] (green upward triangle). Full symbols denote the results of optimum parameter sets. Open symbols show results varying the symmetry energy parameter $J$. The dashed lines serve to guide the eye. The dashed black line from (a) is repeated for direct comparison of SHF and RMF models.

Figure 4

Relationship between ${\alpha }_{\mathrm{D}}$ and $r_{\mathrm{skin}}$ for ${}^{120}\mathrm{Sn}$ predicted by the SV-min [13] (red circle) and RD-min [59] (green square) interactions. Full symbols are the results of the optimum parameter sets and open symbols correspond to a variation of the symmetry energy parameter $J$ as in Fig. 3. Dashed lines are to guide the eye. The horizontal lines denote the range of $r_{\mathrm{skin}}$ values compatible with the experimental polarizability shown as a yellow shaded band.