Spurious finite-size instabilities in nuclear energy density functionals

Background: It is known that some well established parametrizations of the nuclear energy density functional (EDF) do not always lead to converged results for nuclei. Earlier studies point towards the existence of a qualitative link between this finding and the appearance of finite-size instabilities of symmetric nuclear matter (SNM) near saturation density when computed within the random phase approximation (RPA).

Purpose: We aim to establish a stability criterion based on computationally friendly RPA calculations that can be incorporated into fitting procedures of the coupling constants of the EDF. Therefore, a quantitative and systematic connection between the impossibility to converge self-consistent calculations of nuclei and the occurrence of finite-size instabilities in SNM is investigated for the scalar-isovector ($S=0$, $T=1$) instability of the standard Skyrme EDF.

Results: Tuning the coupling constant $C_1^{\rho \Delta \rho }$ of the gradient term that triggers scalar-isovector instabilities of the standard Skyrme EDF, we find that the occurrence of instabilities in finite nuclei depends strongly on the numerical scheme used to solve the self-consistent mean-field equations. Once the critical value of the coupling constant $C_1^{\rho \Delta \rho }$ is determined in nuclei, one can extract the corresponding lowest density ${\rho }_{\text{crit}}$ at which a pole appears at zero energy in the RPA response function.

Conclusions: Instabilities of finite nuclei can be artificially hidden due to the choice of inappropriate numerical schemes or overly restrictive, e.g., spherical, symmetries. Our analysis suggests a twofold stability criterion to avoid scalar-isovector instabilities.

Figure 1

Binding energy of (a) ${}^{40}$Ca and (b) ${}^{208}$Pb obtained from hosphe and SLy5${}^{\prime }$ [34] as a function of $C_1^{\rho \Delta \rho }$. Calculations are performed for several values of the number of shells $N_{\text{sh}}$. The curves end at the largest value of $C_1^{\rho \Delta \rho }$ for which the convergence criteria are reached within 40 000 iterations.

Figure 2

Maximum value of $C_1^{\rho \Delta \rho }$ for which a solution is found for the ground state of a given nucleus. Results are displayed for SLy5${}^{\prime }$ [(a) and (c)] and LNS${}^{\prime }$ [(b) and (d)]. Results are displayed in the upper (lower) panel as a function of the mesh (basis) size used in lenteur (hosphe).

Figure 3

Contribution of $E_1^{\rho \Delta \rho }$ to the binding energy of ${}^{40}$Ca as a function of the number of iterations. Four modified SLy5${}^{\prime }$ [34] parametrizations with values of $C_1^{\rho \Delta \rho }$ around its critical value $C_{1,\text{crit}}^{\rho \Delta \rho }$ are represented. Calculations are performed with the ev8 code for a value $dx=0.4$ fm of the Cartesian mesh. During the iterations, the Coulomb term in the EDF is switched off, such that the exact value of $E_1^{\rho \Delta \rho }$ should be zero for ${}^{40}$Ca.

Figure 4

$C_{1,\text{crit}}^{\rho \Delta \rho }$ obtained for ${}^{40}$Ca with the ev8 code for the various (modified) parametrizations as a function of the step size $dx$.

Figure 5

The function ${\rho }_p\left(q_{\text{ph}}\right)$ for a SLy5${}^{\prime }$ parametrization corresponding to $C_1^{\rho \Delta \rho }=30.0$ MeV fm${}^5$. The minimum of the ${\rho }_p\left(q_{\text{ph}}\right)$ defines the lowest density ${\rho }_{\text{min}}$ at which a pole occurs for this value of $C_1^{\rho \Delta \rho }$.

Figure 6

${\rho }_{\text{min}}$ as defined in Fig. 5 as a function of $C_1^{\rho \Delta \rho }$. The vertical band $C_{1,\text{crit}}^{\rho \Delta \rho }$ intersects the curve to define the horizontal band ${\rho }_{\text{crit}}$. The dashed line denotes the saturation density ${\rho }_{\text{sat}}$ of SNM corresponding to SLy5${}^{\prime }$.

Figure 7

${\rho }_p\left(q_{\text{ph}}\right)$ for the parametrizations given in Table 2 at the nominal value of the coupling constant $C_1^{\rho \Delta \rho }$.

Figure 8

${\rho }_p\left(q_{\text{ph}}\right)$ for the parametrizations given in Table 2 at the critical value of the coupling constant $C_1^{\rho \Delta \rho }$. Note the smaller scale of the $y$ axis in comparison to Fig. 7.

Figure 9

Critical density ${\rho }_{\text{crit}}/{\rho }_{\text{sat}}$. The uncertainty band comes both from the numerical extraction and the variation with $dx$. The vertical bar indicates the value for the lowest mesh size used $dx=0.4$ fm. The dashed line indicates the interval $[1,{\rho }_{\text{cent}}/{\rho }_{\text{sat}}]$.

Figure 10

Square of the relative momentum distribution $f_n^2\left(q\right)$ at $R=0$ for neutrons in ${}^{40}$Ca with SLy5${}^{\prime }$ taking $C_1^{\rho \Delta \rho }$ slightly above $C_{1,\text{crit}}^{\rho \Delta \rho }$. The four curves (see text) correspond to different numbers of Hartree-Fock iterations and the vertical dashed line indicates $q=2k_F$.

Figure 11

Same as Fig. 10 but for SQMC700${}^{\prime }$ ($C_1^{\rho \Delta \rho }$ slightly above $C_{1,\text{crit}}^{\rho \Delta \rho }$).