Results

In our study we performed a set of calculations for semi-magic nuclei starting from $Z=8$ or $N=8$ and ending with $Z=82$ or $N=126$. The ground states properties were calculated within the HFB method by using the code HOSPHE [23], whereas the monopole strength functions were obtained by implementing in the same code the QRPA method within the Arnoldi iterative method.

We decided to use two different Skyrme functionals - SLy4 [29] and UNEDF0 [30]. Both of them were tuned (among other observables) to reproduce the main properties of the infinite nuclear matter. In particular, they correspond to the same value of nuclear incompressibility (29) of $K_{\infty}=230$MeV and differ in their values of the effective mass of $m^*/m=0.70$ and 1.11 for SLy4 and UNEDF0, respectively.

The present study is focused on comparing incompressibilities obtained with two different pairing interactions, namely, the standard zero-range force, $V_\tau(\bm{r},\bm{r}')=-V_{0\tau} \delta(\bm{r}-\bm{r}')$, and separable force presented in Sec. 3. To make the comparison meaningful, we adjusted the strength parameters, $G_\tau$ and $V_\tau$, so as to obtain for both forces very similar neutron (proton) pairing gaps in $Z=50$ isotopes ($N=50$ isotones). The resulting gaps roughly correspond to the experimental odd-even mass staggering along the $Z=50$ and $N=50$ chains of nuclei. Theoretical pairing gaps, $\Delta_n$ and $\Delta_p$, were determined as in Ref. [31], namely,

$$\Delta_\tau = \frac{\mbox{Tr}' (\rho_{\tau} \Delta_{\tau})}{\mbox{Tr} \rho_{\tau}}$$ (33)

where Tr$A= \sum_k A_{kk}$ and Tr$' A = \sum_{k>0} A_{k\bar{k}}$. For the separable pairing, in Eq. (26) we used only one Gaussian term with $a_1=0.66$fm.

In this way, in the calculations we used the separable-force strength parameters of $G_n=631$ and 473MeVfm$^3$ ($G_p=647$ and 521MeVfm$^3$) for the SLy4 and UNEDF0 functionals, respectively, and similarly, for the zero-range force: $V_n=195$ and 126MeVfm$^3$ ($V_p=221$ and 157MeVfm$^3$). All calculated neutron and proton pairing gaps are shown in Figs. 1 and 2, respectively. One can see that the results obtained for both pairing forces are fairly similar. The HFB iterations were carried out using a linear mixing of densities from the current and previous iteration defined by a constant mixing parameter [23]. With this recipe, for some of the nuclei, the HFB iterations did not end in converged solutions. Such cases were excluded from the analysis of pairing properties and the subsequent QRPA calculations.

We note here that no energy cut-off is needed for calculations using the separable force, and thus in our calculations the entire harmonic-oscillator basis up to $N_0=20$ shells was used. On the other hand, for the zero-range force we used the cut-off energy of 60MeV applied within the two-basis method [32,33].

Figure 1: (Color online) Neutron pairing gaps in the $Z=8$, 20, 28, 50, and 82 isotopes (see the legend shown in Fig. 5). Upper and lower panels show results obtained for the SLy4 and UNEDF0 functionals, respectively. Left and right panels show results obtained for the zero-range and separable pairing, respectively.

Figure 2: (Color online) Same as in Fig. 1 but for the proton gaps in the $N=8$, 20, 28, 50, 82, and 126 isotones (see the legend shown in Fig. 6).

First, in Fig. 3 we illustrate the high reliability of the Arnoldi method in determining the key factors of our analysis, namely, the ratios of moments of the monopole strength functions. To obtain a perfectly stable result, only about 70 Arnoldi iterations suffice. In this way, the QRPA result is achieved within the CPU time that is of the same order as that needed to obtain a converged HFB ground state. Note that the Arnoldi iteration conserves all odd moments, so during the iteration, the moment ${m_1}$ does not change; thus the convergence of ${m_1}/{m_0}$ simply illustrates the convergence of ${m_0}$ alone.

Figure 3: Convergence of the ratio of first and zero moments ${m_1}/{m_0}$ calculated in $^{112}$Sn as a function of the number of Arnoldi iterations.

In Fig. 4 we compare our QRPA results with raw experimental data obtained in Ref. [3]. In this work, a Lorentzian fit to data was performed in the region of energies of 10.5-20.5MeV, and the experimental values of ${m_1}/{m_0}$ were determined from the corresponding fitted curve (its moments were calculated for energies from zero to infinity). In determining our theoretical values of ${m_1}/{m_0}$, we also perform the integration in the entire energy domain. We have checked that the integration of theoretical curves in the fixed region of 10.5-20.5MeV does not bring meaningful results, because, in the wide region of masses studied here, the GMR peaks move too much, and extend beyond the above narrow range of energies. Our QRPA strength functions were obtained from the discrete Arnoldi strength distributions by using the smoothing methods explained in Ref. [22]. We also note that in our QRPA calculations, the high-energy shoulder of the strength function is not obtained, cf. discussion in Ref. [3].

Figure 4: (Color online) The QRPA monopole strength function in $^{112}$Sn (solid line) compared to raw experimental data [3] and Lorentzian fit to data (dashed line) performed in the region of energies of 10.5-20.5MeV [3].

Figs. 5 and 6 present the overview of all obtained finite-nucleus incompressibilities $K_A$, Eqs. (30) and (31), calculated along the isotopic and isotonic chains, respectively. One can see that for both Skyrme functionals, SLy4 and UNEDF0, values corresponding to the zero-range (full symbols) and separable (open symbols) pairing forces are very similar.

Figure 5: (Color online) Incompressibility $K_A$ calculated for the isotopic chains of semimagic nuclei with $Z=8$, 20, 28, 50, and 82. Left and right panels show results obtained for the SLy4 and UNEDF0 functionals, respectively. Full (empty) symbols correspond to the zero-range (separable) pairing force.

Figure 6: (Color online) Same as in Fig. 5, but for the isotonic chains with $N=8$, 20, 28, 50, 82, and 126.

To see effects of the pairing interaction in more detail, we focus on the results obtained for chains of tin and lead isotopes. In Figs. 7 and 8 we compare theoretical results with the experimental data for $^{208}$Pb and $^{112-124}$Sn, taken from Refs. [2,4,3]. A comparison of the two types of pairing interactions, and two different Skyrme functionals, leads to the conclusion that the calculated incompressibilities $K_A$ depend on the interactions in the particle-particle channel as well as the particle-hole channel of the two Skyrme functionals used in our study - SLy4 and UNEDF0 - only weakly. Of course, we can expect that using Skyrme parametrizations tuned to higher (lower) values of $K_{\infty}$ may lead to uniformly higher (lower) values of $K_A$.

Figure 7: (Color online) Incompressibility $K_A$ calculated for chains of the $Z=50$ and 82 isotopes. Results obtained by using the separable (squares) and zero-range (circles) pairing with the UNEDF0 functional are compared to the available experimental data [2,4,3].

Figure 8: (Color online) Same as in Fig. 7, but for the UNEDF0 (squares) and SLy4 (circles) functionals and separable pairing force.

To check a weak dependence of $K_A$ on the intensity of pairing correlations, we have repeated the calculations by using values of neutron pairing strengths varied in a wide range, $G_n=631\pm150$MeVfm$^3$ and $V_n=195\pm30$MeVfm$^3$. Such variations induce very large changes of neutron pairing gaps, shown in Fig. 9; the ones that are certainly beyond any reasonable range of uncertainties related to adjustments of pairing strengths to data. In Figs. 10 and 11, we show the influence of the varied pairing strengths on the calculated incompressibilities $K_A$. We see clearly that even such large variations cannot induce changes compatible with discrepancies with experimental data.

Figure 9: (Color online) Neutron pairing gaps calculated in tin isotopes for low (triangles), central (squares), and high (circles) values of pairing strength parameters given in captions of Figs. 10 and 11.

To illustrate the effect of isospin asymmetry, in Figs. 10 and 11 we plotted the results as functions of $N/Z$, whereby $^{124}$Sn and $^{208}$Pb are located at almost the same point of the abscissa. These figures clearly show that the discrepancies with data are probably not related to the isospin dependence of $K_A$. Indeed, for both types of pairing, in the region of $1.0 < N/Z < 1.6$, the results obtained for tin and lead isotopes roughly follow each other.

Figure 10: (Color online) Incompressibility $K_A$ calculated for the SLy4 functional and separable pairing force in tin (squares) and lead (circles) isotopes compared to the available experimental data. Theoretical results are plotted together with uncertainties pertaining to variations of the neutron strength parameter in the range of $G_n=631\pm150$MeVfm$^3$.

Figure 11: (Color online) Same as in Fig. 10, but for the zero-range pairing force and uncertainties pertaining to variations of the neutron strength parameter in the range of $V_n=195\pm30$MeVfm$^3$.

Finally, to illustrate the fact that nuclear radii are fairly robust and cannot significantly influence the values of $K_A$, determined from Eqs. (30) and (31), we show values of $m_1/m_0$ alone in Figs. 12 and 13. We see that for both types of pairing, in tin and lead the calculated values of $m_1/m_0$ overestimate and underestimate the measured ones by 0.6-0.8 and 0.4MeV, respectively. Exactly the same pattern was obtained within the relativistic nuclear energy density functionals studied in Ref. [13], where the corresponding discrepancies were equal to 0.8-1.0 and 0.2MeV. We also note that this comparison directly relates calculations to data, without using the intermediate and model-dependent definition of $K_A$.

Figure 12: (Color online) Same as in Fig. 10, but for the centroids $m_1/m_0$.

Figure 13: (Color online) Same as in Fig. 12, but for the zero-range pairing force.

To conclude our analysis, we have also performed adjustments of the liquid-drop formula (32) to our microscopically calculated values of $K_A$. The obtained parameters are collected in Table 1. We see that the liquid-drop formula is able to provide an excellent description of the QRPA results, with average deviations of the order of 5MeV, that is, about 3% of the typical value of $K_A$. Similarly the values of the volume incompressibility $K_V$ are determined to about 2% of precision. The least precisely determined liquid-drop parameter is the surface-symmetry incompressibility $K_{S,\tau}$, estimated up to 25% of precision. We also note that, within the fit precision, the volume parameter $K_V$ averaged over both functionals and both pairing forces equals to 254$\pm$5MeV, which is significantly higher than the corresponding infinite-matter incompressibility of $K_{\infty}$=230MeV.

Table 1: Parameters (in MeV) of the liquid-drop formula (32) with standard errors, obtained by a fit to the values of $K_A$ calculated in $M$ semi-magic nuclei across the mass chart. The parameter $\chi$ was determined as the square root of the sum of fit residuals squared divided by the number of fit degrees of freedom ($M-5$ in our case).

		SLy4						UNEDF0
		separable				zero-range				separable				zero-range
$K_{V}$		252		$\pm$		5		258		$\pm$		5		249		$\pm$		5		257		$\pm$		4
$K_{S}$		$-$391		$\pm$		14		$-$406		$\pm$		13		$-$397		$\pm$		14		$-$412		$\pm$		13
$K_{\tau}$		$-$460		$\pm$		30		$-$500		$\pm$		30		$-$510		$\pm$		30		$-$550		$\pm$		30
$K_{S,\tau}$		410		$\pm$		110		560		$\pm$		100		570		$\pm$		120		740		$\pm$		100
$K_{C}$		$-$5.2		$\pm$		0.4		$-$5.4		$\pm$		0.4		$-$4.5		$\pm$		0.4		$-$5.1		$\pm$		0.4
$M$		210				211				204				195
$\chi$		5.0				4.7				5.3				4.4