Fitting theories of nuclear binding energies

In developing theories of nuclear binding energy such as density-functional theory, the effort required to make a fit can be daunting because of the large number of parameters that may be in the theory and the large number of nuclei in the mass table. For theories based on the Skyrme interaction, the effort can be reduced considerably by using the singular value decomposition to reduce the size of the parameter space. We find that the sensitive parameters define a space of dimension four or so, and within this space a linear refit is adequate for a number of Skyrme parameters sets from the literature. We find no marked differences in the quality of the fit among the SLy4, the BSk4, and SkP parameter sets. The root-mean-square residual error in even-even nuclei is about 1.5 MeV, half the value of the liquid drop model. We also discuss an alternative norm for evaluating mass fits, the Chebyshev norm. It focuses attention on the cases with the largest discrepancies between theory and experiment. We show how it works with the liquid drop model and make some applications to models based on Skyrme energy functionals. The Chebyshev norm seems to be more sensitive to new experimental data than the root-mean-square norm. The method also has the advantage that candidate improvements to the theories can be assessed with computations on smaller sets of nuclei.

Figure 1

Eigenvalues of the least-squares fitting matrix, Eq. (2).

Figure 2

Least-squares refit as a function of the dimension of the parameter space. The r.m.s. residuals of the linear refit are shown as circles, connected by lines to guide the eye. The triangles show the actual r.m.s., residuals for the parameters determined by the linear refit. However, not all nuclei converged for the $N=6$ and $N=8$ parameter sets, and they were not included in the r.m.s. residual.

Figure 3

Residuals between theory and experiment as a function of N, with nuclei of the same atomic number joined by lines. The top plot shows the original SLy4 and the $N=4$ linear refit is shown on the bottom.

Figure 4

Chart of nuclei showing the critical nuclei of the liquid drop model and of the linearized refit of the SCMF based on the SLy4 interaction. Critical nuclei are indicated by triangles, with the orientation of the triangle distinguishing overbound (Δ) and underbound (∇) nuclei. The cross area shows the nuclei whose masses have been measured.