Mass correction

Last modification: 1996/3/25
To evaluate the size of the error, one needs the HF+BCS solution for vanishingly small mesh size. It can be obtained without difficulty concerning spherical solutions, because even personal computers can execute spherical HF+BCS codes with very small radial-grid spacing. Therefore, we have chosen to compare the total energies between the solutions of a spherical HF+BCS code (footnote) and the Cartesian-mesh code EV8 with constraints of vanishing quadrupole moments. (footnote) In the following, we designate the total energy obtained with the spherical code as E0 while that from the Cartesian-mesh code simply as E. Our aim is to construct a formula for the energy correction so that the corrected energy Ec = E +(correction) has a much smaller r.m.s. deviation from E0 than E has.

In the least-square fitting to determine the parameters of the formula, we have used 1005 nuclei (among the 1029 nuclei) whose paring gaps (both for proton and neutron) coincide within 0.1 MeV between the results of the two codes. With the simplest fitting function of Ec = E + c1 A, the r.m.s. difference between Ec and E0 can be decreased from 6.7 MeV to 0.35 MeV. By adding temrs up to the second order in N and Z,

Ec= E + c1 A + c2 (N-Z) + c3 A2 + c4 A(N-Z) + c5 (N-Z)2,
the r.m.s. value of Ec - E0 is decreased to 114 keV with c1-40.2, c2=-20.6, c3= 0.033, c4=-0.081, and c5=-0.080 (keV). Since this size of error is much smaller than the typical precision of the mass formulae in the market place (about 0.5 MeV) [MNM94,APD92,TUY88], we have decided to adopt the above formula (footnote) .

We have tested the accuracy of the above correction formula for deformed solutions by comparing Ec with an energy extrapolated to a - > 0. In the extrapolation, first, we calculate the total energy for seven values of a ranging from 1 fm to 0.56 fm (six values ranging from 1 fm to 0.6 fm for 240Pu), while keeping the box size constant. Second, we fit a polynomial

E(a) = Eext + b1 a2 + b2 a6
to the seven or six sets of values (a, E) using Eext, b1, and b2 as the fitting parameters. This form of the fitting function has been chosen on the following grounds: At a ~ 1 fm, the error is dominated by a term of order a6, which originates in the seven-point approximation to the first derivatives. At a ~ 0.5 fm, the contribution from lower-order terms becomes comparable to that of the a6 term. These terms seem to arise principally from the error in the Coulomb energy, whose leading order term is a2. The ambiguity in the extrapolated energy Eext is roughly estimated to be ~ 0.2 MeV. 
nucleus  oblate   spherical prolate  correction
                                     formula
 72Se   2.9 MeV   3.1 MeV   2.8 MeV   2.83 MeV
100Zr   4.4       4.7       4.2       4.30
130Nd   4.9       5.1       4.7       4.99
170Er   7.1       7.3       6.7       7.14
240Pu  10.1      10.0       9.9      10.05
In the above table the difference between the extrapolated energy to a - > 0 and the energy calculated with a=1 fm is shown for oblate, spherical (obtained with constraints of vanishing mass quadrupole moments), and prolate solutions of five nuclei. The fifth column shows the energy correction given by the correction formula. For these 15 solutions, the mean and the r.m.s. values of the difference between the correction formula and the extrapolations are 0.0 MeV and 0.21 MeV, respectively, which are smaller than or of the same size as the accuracy of the extrapolation. This agreement shows the applicability of the correction formula to deformed solutions as well as to spherical ones. It also confirms that the spherical code and the Cartesian code with the spherical constraint are indeed equivalent to each other.