Incompressibility in finite nuclei and nuclear matter

The incompressibility (compression modulus) $K_0$ of infinite symmetric nuclear matter at saturation density has become one of the major constraints on mean-field models of nuclear many-body systems as well as of models of high density matter in astrophysical objects and heavy-ion collisions. It is usually extracted from data on the giant monopole resonance (GMR) or calculated using theoretical models. We present a comprehensive reanalysis of recent data on GMR energies in even-even ${}^{112–124}$Sn and ${}^{106,100–116}$Cd and earlier data on $58\le A\le 208$ nuclei. The incompressibility of finite nuclei $K_A$ is calculated from experimental GMR energies and expressed in terms of $A^{-1/3}$ and the asymmetry parameter $\beta =(N-Z)/A$ as a leptodermous expansion with volume, surface, isospin, and Coulomb coefficients $K_{\mathrm{vol}}$, $K_{\mathrm{surf}}$, $K_{\tau }$, and $K_{\mathrm{Coul}}$. Only data consistent with the scaling approximation, leading to a fast converging leptodermous expansion, with negligible higher-order-term contributions to $K_A$, were used in the present analysis. Assuming that the volume coefficient $K_{\mathrm{vol}}$ is identified with $K_0$, the $K_{\mathrm{Coul}}=-(5.2\pm 0.7)$ MeV and the contribution from the curvature term $K_{\mathrm{curv}}{A}^{-2/3}$ in the expansion is neglected, compelling evidence is found for $K_0$ to be in the range 250 $<K_0<$ 315 MeV, the ratio of the surface and volume coefficients $c=K_{\mathrm{surf}}/K_{\mathrm{vol}}$ to be between $-2.4$ and $-1.6$ and $K_{\tau }$ between $-840$ and $-350$ MeV. In addition, estimation of the volume and surface parts of the isospin coefficient $K_{\tau }$, $K_{\tau ,v}$, and $K_{\tau ,s}$, is presented. We show that the generally accepted value of $K_0$ = (240 $\pm$ 20) MeV can be obtained from the fits provided $c\sim -1$, as predicted by the majority of mean-field models. However, the fits are significantly improved if $c$ is allowed to vary, leading to a range of $K_0$, extended to higher values. The results demonstrate the importance of nuclear surface properties in determination of $K_0$ from fits to the leptodermous expansion of $K_A$. A self-consistent simple (toy) model has been developed, which shows that the density dependence of the surface diffuseness of a vibrating nucleus plays a major role in determination of the ratio $K_{\mathrm{surf}}/K_{\mathrm{vol}}$ and yields predictions consistent with our findings.

Figure 1

$E_{\mathrm{GRM}}$ = $\widetilde{E_3}=\sqrt{\frac{m_3}{m_1}}$ as a function of (N-Z)/A as reported in ${}^{58,60}$Ni, ${}^{56}$Fe [46] (TAMU), ${}^{90}$Zr [62] (TAMU), ${}^{106,110-116}$Cd [49] (RCNP), ${}^{110,116}$Cd [60] (TAMU), ${}^{112-124}$Sn [43, 44] (RCNP) and ${}^{112,124}$Sn [61] (TAMU). We also display $E_{\mathrm{GMR}}$(GF) obtained from (8) for ${}^{92}$Mo [82] (Duhamel), ${}^{110,116}$Cd [60] (TAMU), ${}^{112,124}$Sn [61] (TAMU), ${}^{112-116,120,124}$Sn, ${}^{144}$Sm and ${}^{148}$Sm [29] (Sharma). For more details see text.

Figure 2

The same as Fig. 1 but for $E_{\mathrm{GRM}}$ = $\widetilde{E_1}=\sqrt{\frac{m_1}{m_{-1}}}$ as reported in ${}^{58,60}$Ni, ${}^{56}$Fe [46] (TAMU), ${}^{90}$Zr [62, 83] (TAMU), ${}^{106,110-116}$Cd [49] (RCNP), ${}^{110,116}$Cd [60] (TAMU), ${}^{112-124}$Sn [43, 44] (RCNP), ${}^{112,124}$Sn [61] (TAMU), ${}^{116}$Sn, ${}^{144}$Sm and ${}^{208}$Pb [83] (TAMU). Note the y-scale is the same as in Figs. 1 and 3.

Figure 3

The same as Fig. 1 but for $E_{\mathrm{GRM}}$ = $\widetilde{E_0}=\frac{m_1}{m_0}$ as reported in ${}^{58,60}$Ni, ${}^{56}$Fe [46] (TAMU), ${}^{90}$Zr [83, 84] (TAMU), ${}^{106,110-116}$Cd [49] (RCNP), ${}^{110,116}$Cd [60] (TAMU), ${}^{112-124}$Sn [43, 44] (RCNP), ${}^{112,124}$Sn [61] (TAMU), ${}^{116}$Sn, ${}^{144}$Sm [62, 83] (TAMU) and ${}^{208}$Pb [62, 83] (TAMU) and [85] (Uchida 2003). Note the y-scale is the same as in Figs. 1 and 2.

Figure 4

(a) Peak energy $E_{\mathrm{peak}}$ and (b) corresponding width $\Gamma$ obtained from various fits to the experimental GMR strength function. Data are taken from ${}^{58,60}$Ni, ${}^{56}$Fe [46] (TAMU Gaussian), ${}^{92}$Mo [82] (Duhamel Gaussian), ${}^{90}$Zr [63] (Uchida 2004 Breit-Wigner), ${}^{106,110-116}$Cd [49] (RCNP Lorentzian), ${}^{110,116}$Cd [60] (TAMU Gaussian), ${}^{112-124}$Sn [43, 44] (RCNP Lorentzian), ${}^{112,124}$Sn [61] (TAMU Gaussian), ${}^{112-116,120,124}$Sn [29] (Sharma Gaussian), ${}^{116}$Sn [63] (Uchida 2004 Breit-Wigner), ${}^{144}$Sm [29] (Sharma Gaussian), ${}^{144}$Sm [86] (Itoh Breit-Wigner), ${}^{148}$Sm [29] (Sharma Gaussian), ${}^{208}$Pb (Breit-Wigner fit) [63] (Uchida 2004 Breit-Wigner). The y-scale in the left panel is the same as in Figs. 1, 2, 3.

Figure 5

$K_A$ calculated for Sn isotopes using $\widetilde{E_3}=\sqrt{\frac{m_3}{m_1}}$ and the mean charge (circle), matter (triangle up), determined from nuclear skin, and matter (triangle down) radius obtained from elastic alpha-particle scattering. See text for more detail.

Figure 6

(a) Minimization of $\sigma$ as a function of $K_0$ for combined data on Sn and Cd isotopes [43, 44, 49]. $K_0$ was varied with the step 0.1 MeV. (b) Fit to data, represented by the left-hand side of (11), as a function of $\frac{{\beta }^2}{1+c{A}^{-1/3}}$ for the $K_0$, corresponding to minimum $\sigma$.

Figure 7

Fit to experimental data in set RCNP-E with $K_{\mathrm{coul}}$ = −(5.2 $\pm$ 0.7) MeV for (a) $c=-1.0$ and (b) $c=-1.6$. Note that both x and y coordinates are $A$ and $c$ dependent. For more explanation see text.

Figure 8

The same as Fig. 7, but for set TAMU-E and (b) $c$ = −2.0.

Figure 9

The same as Fig. 7, but for set GF-M and (b) $c=-2.4$.

Figure 10

Linear (solid) and quadratic (dashed) fits to experimental $K_A$ as a function of $A^{-1/3}$ for data set RCNP-E.

Figure 11

MESH fit to RCNP-E data including the curvature term. Values of $K_0$, $c$, $K_{\tau }$ and $K_{\mathrm{curv}}$ as a function $\sigma$ are displayed, showing a well-defined unique minimum, indicated by the vertical dashed line. For more detail see text.

Figure 12

(a) Comparison of experimental and calculated GMR energies as a function of $K_0$ as presented in (Ref. [17]) and using current experimental and calculated values for (b) ${}^{208}$Pb, (c) ${}^{116}$Sn and (d) ${}^{90}$Zr. Experimental data are taken from [83, 84] (${}^{90}$Zr), [44, 62, 83] (${}^{116}$Sn) and [62, 83, 85] (${}^{208}$Pb). Horizontal lines (black) depict ranges of currently available GMR energies in ${}^{90}$Zr, ${}^{116}$Sn and ${}^{208}$Pb, the dashed lines (blue) illustrate the range of theoretical predictions. Details of the calculation with references are given in Table 12. For more information see text.

Figure 13

Comparison of experimental and theoretical GMR energies in ${}^{116}$Sn for constrained (CON), centroid (CEN) and scaling (SCA) approximations. Experimental data are taken from [83] (TAMU1999), [62] (TAMU2004) and [44] (RCNP2010). Hartree-Fock (HF)+RPA with KDE0v1 [87], HF with KDE0v1, SIII, SIV, SkA Skyrme interactions [88, 89] and the Hartree-Fock-Bogolyubov + QRPA [39] with SkP, SkM* and SLy4 Skyrme interactions and RMF with FSU, NL3 and Hybrid [77] and BSP, IUFSU and IUFSU* Lagrangians. Tick labels on x-axis indicate experimental data (−3, −2, −1) and various calculations (1–11).