The effective Skyrme force

For a general overview of the foundations and properties of the Skyrme force we refer the reader to the review article [20] and references therein. The Skyrme force is an effective interaction depending on a limited number of parameters,

$$\begin{split}V_{12}=&t_0(1+x_0P_\sigma)\delta +\frac{1}{2}t_1... ...ft( \mathbf k'\times\delta\mathbf k\right)\,,\hfill \end{split}$$ (43)

where $\delta$ is a short notation for $\delta({\mathbf r}_1-{\mathbf r}_2)$, $\mathbf k=\mathbf k_1-\mathbf k_2$ acting on the right and $\mathbf k'=\mathbf k_1-\mathbf k_2$ acting on the left.

The parameters of the Skyrme forces were fitted in the literature to reproduce various bulk nuclear properties as well as selected properties of some nuclei (usually doubly magic nuclei). Simplifications have often been made in the expression of the functional (19), like treatment of the Coulomb exchange term with the Slater approximation and/or omission of the two-body center of mass contribution or of the ``$\,J^2\,$'' terms. The latter corresponds to the fourth line in Eq. (19) and to the two first terms in the spin-orbit mean field, Eq. (29). It is important to keep in mind that one should use each parametrization of the functional within the same simplifications with which it has been adjusted to data.

For all forces implemented in HFBRAD, the force in the particle-particle channel is chosen to be

$$V_{12}'=\left(t_0' +\frac{t_3'}{6}\rho^{\gamma'}\right)\delta\,.$$ (44)

Table 1 gives the different sets of parameters for the force in the particle-hole channel, while table 2 gives the parameters in the particle-particle channel. It is important to keep in mind that these latter parameters have been adjusted along with a given cut-off and cut-off diffuseness, which are two additional parameters of the force.

Table 1: Parameters in the particle-hole channel for the different versions of the Skyrme forces implemented in the code HFBRAD. The last line indicates if the $J^2$ terms are included in the Skyrme functional when the force is used.

	SIII [4]		SkM* [21]		.	SLy4 [22]		.	SLy5 [22]		.	SkP [5]		.
$t_0$	$-$1128	.	75	$-$2645	.	0	$-$2488	.	913	$-$2488	.	345	$-$2931	.	6960
$t_1$	395	.	0	410	.	0	486	.	818	484	.	230	320	.	6182
$t_2$	$-$95	.	0	$-$135	.	0	$-$546	.	395	$-$556	.	690	$-$337	.	4091
$t_3$	14000	.	0	15595	.	0	13777	.	0	13757	.	0	18708	.	96
$x_0$	0	.	45	0	.	09	0	.	834	0	.	776	0	.	2921515
$x_1$	0	.	0	0	.	0	$-$0	.	344	$-$0	.	317	0	.	6531765
$x_2$	0	.	0	0	.	0	$-$1	.	000	$-$1	.	000	$-$0	.	5373230
$x_3$	1	.	0	0	.	0	1	.	354	1	.	263	0	.	1810269
$\gamma$	1	.	0	1	.	6	1	.	6	1	.	6	1	.	6
$W_0$	130	.	0	120	.	0	123	.	0	125	.	00	100	.	000
$J^2$	No		No		.	No		.	Yes		.	Yes		.

Three kinds of pairing forces have been adjusted: (i) the volume pairing (the pairing field follows the shape of the density), below denoted by ``$\rho$'', (ii) the surface pairing (the pairing field is peaked at the surface and follows roughly the variations of the density), below denoted by `' $\delta\rho$'', and (iii) the mixed pairing (a compromise between the two first two), below denoted by ``$\rho$+ $\delta\rho$''. All pairing parameters have been adjusted in order to give a mean neutron gap of 1.245 MeV in $^{120}$Sn.

Table 2: Parameters in the particle-particle channel for the different versions of the Skyrme forces implemented in the code HFBRAD. The three first columns corresponds to the use of a cut-off at 60 MeV with a Fermi shape and a diffuseness of 1 MeV, the three columns on the right corresponds to the regularization procedure of the pairing field described in Sec. 2.7.

	cut-off			regularization
	$\rho$	$\delta\rho$	$\rho+\delta\rho$	$\rho\,^r$	$\delta\rho\,^r$	$(\rho+\delta\rho)^r$
$t_0'$(SIII)	$-$159.6	$-$483.2	$-$248.5	$-$197.6	$-$807.0	$-$316.9
$t_0'$(SkM*)	$-$148.6	$-$452.6	$-$233.9	$-$184.7	$-$798.4	$-$300.7
$t_0'$(SLy4)	$-$186.5	$-$509.6	$-$283.3	$-$233.0	$-$914.2	$-$370.2
$t_0'$(SLy5)	$-$179.9	$-$504.9	$-$275.8	$-$222.7	$-$901.9	$-$356.6
$t_0'$(SkP)	$-$131.6	$-$429.5	$-$213.1	$-$196.6	$-$1023.0	$-$326.5
$t_3'$	0	$-37.5\,t_0'$	$-18.75\,t_0'$	0	$-37.5\,t_0'$	$-18.75\,t_0'$
$\gamma'$	1	1	1	1	1	1