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# Energy density functional from the two-body Skyrme force

 Force C0s[0] C1s[0] C0T C1T SkI1 695.860 239.200 120.190 99.573 0.0 0.0 192.660 62.766 1/4 SkI3 84.486 220.360 253.180 113.940 0.0 0.0 92.235 22.777 1/4 SkI4 44.038 231.980 209.030 104.120 0.0 0.0 124.590 37.943 1/4 SkO 373.770 262.960 41.421 84.253 0.0 0.0 70.365 26.590 1/4 SkO' 277.910 262.430 47.082 84.154 -104.090 -9.172 42.791 16.553 1/4 SkX 57.812 180.660 -35.639 81.246 -7.861 -23.669 -4.434 9.514 1/2 SGII 271.110 330.620 61.048 91.676 0.0 0.0 15.291 15.283 1/6 SkP 152.340 366.460 -31.328 78.562 7.713 -41.127 -4.211 9.757 1/6 SkM* 271.110 330.620 31.674 91.187 0.0 0.0 17.109 17.109 1/6 SLy4 -207.820 311.110 153.210 99.737 0.0 0.0 47.048 14.282 1/6 SLy5 -171.360 310.430 151.080 99.133 -14.659 -65.058 45.787 14.000 1/6 SLy6 -201.460 309.940 157.050 100.280 0.0 0.0 48.822 14.655 1/6 SLy7 -215.830 310.100 158.260 100.640 -30.079 -55.951 49.680 14.843 1/6

The standard two-body Skyrme force is given by [2,33]
 = (32)

where ) is the spin-exchange operator, acts to the right, and acts to the left. Calculating the Hartree-Fock expectation value from this force yields the energy functional given in Eq. (12) with the coupling constants:
 = = C0s = C1s = = = C0T = C1T = = = = = = = = 0 = 0 , (33)

nine of which are independent. Although in this approach , many parameterizations of the Skyrme interaction set . That violates the interpretation of the Skyrme functional as an expectation value of a real two-body interaction and removes the rationale for calculating the time-odd coupling constants from (34). For Skyrme interactions with a generalized spin-orbit interaction [42], e.g. for SkI3, SkI4, SkO, or SkO', the spin-orbit coupling constants are given by
 (34)

The resulting terms in the energy functional again cannot be represented as the HF expectation value of a two-body spin-orbit potential (see, e.g., [41]), again violating the assumptions behind the calculation of the time-odd coupling constants in (34).

As Eqs. (34) represent the standard approach to the time-odd coupling constants, it is worthwhile to take a look at the actual values. Table 4 compares them for several Skyrme forces. None of these parameterizations was obtained from observables sensitive to the time-odd terms in the energy functional. Differences among the forces merely reflect various strategies for adjusting the time-even coupling constants. Values of the density-dependent isoscalar coupling constants C0s, either at or at , are scattered in a wide range. This is probably one of the main sources of differences in the predictions of the forces for time-odd corrections to rotational bands. For the SLyx forces, C0s [0] is negative, which is unusual; most often this part of the isoscalar spin-spin interaction is repulsive at all densities. The difference will probably cause visible differences in rotational properties whenever the spin density is large at the surface. All the forces agree on the isovector coupling constant C1s, especially at the saturation density, i.e., MeV fm3. This simply follows from the fact that, assuming Eq. (34), C1s is proportional to the time-even that is fixed from binding energies and radii.

Next: Infinite Nuclear Matter Up: Gamow-Teller strength and the Previous: Local Densities and Currents
Jacek Dobaczewski
2002-03-15