Constraining the surface properties of effective Skyrme interactions

Abstract

Background: Deformation energy surfaces map how the total binding energy of a nuclear system depends on the geometrical properties of intrinsic configurations, thereby providing a powerful tool to interpret nuclear spectroscopy and large-amplitude collective-motion phenomena such as fission. The global behavior of the deformation energy is known to be directly connected to the surface properties of the effective interaction used for its calculation. Purpose: The precise control of surface properties during the parameter adjustment of an effective interaction is key to obtain a reliable and predictive description of nuclear properties. The most relevant indicator is the surface-energy coefficient asurf. There are several possibilities for its definition and estimation, which are not fully equivalent and require a computational effort that can differ by orders of magnitude. The purpose of this study is threefold: first, to identify a scheme for the determination of asurf that offers the best compromise between robustness, precision, and numerical efficiency; second, to analyze the correlation between values for asurf and the characteristic energies of the fission barrier of 240Pu; and third, to lay out an efficient and robust procedure for how the deformation properties of the Skyrme energy density functional (EDF) can be constrained during the parameter fit. Methods: There are several frequently used possibilities to define and calculate the surface energy coefficient asurf of effective interactions built for the purpose of self-consistent mean-field calculations. The most direct access is provided by the model system of semi-infinite nuclear matter, but asurf can also be extracted from the systematics of binding energies of finite nuclei. Calculations can be carried out either self-consistently [Hartree-Fock (HF)], which incorporates quantal shell effects, or in one of the semiclassical extended Thomas-Fermi (ETF) or modified Thomas-Fermi (MTF) approximations. The latter is of particular interest because it provides asurf as a numerical integral without the need to solve self-consistent equations. Results for semi-infinite nuclear matter obtained with the HF, ETF, and MTF methods will be compared with one another and with asurf, as deduced from ETF calculations of very heavy fictitious nuclei. Results: The surface energy coefficient of 76 parametrizations of the Skyrme EDF have been calculated. Values obtained with the HF, ETF, and MTF methods are not identical, but differ by fairly constant systematic offsets. By contrast, extracting asurf from the binding energy of semi-infinite matter or of very large nuclei within the same method gives the same result within the numerical uncertainties. Conclusions: Despite having some drawbacks compared to the other methods studied here, the MTF approach provides sufficiently precise values for asurf such that it can be used as a very robust constraint on surface properties during a parameter fit at negligible additional cost. While the excitation energy of superdeformed states and the height of fission barriers is obviously strongly correlated to asurf, the presence of shell effects prevents a one-to-one correspondence between them. As in addition the value of asurf providing realistic fission barriers depends on the choices made for corrections for spurious motion, its “best value” (within a given scheme to calculate it) depends on the fit protocol. Through the construction of a series of eight parametrizations SLy5s1–SLy5s8 of the standard Skyrme EDF with systematically varied asurf value, it is shown how to arrive at a fit with realistic deformation properties.