The p-p channel

In the p-p energy density, each operator of the relative momentum, $\hat{\mbox{{\boldmath {$k$}}}}'$ and $\hat{\mbox{{\boldmath {$k$}}}}$, acts on variables of the same density matrix, and thus no recoupling is necessary. Terms of the interaction that are linear in momenta then lead to current densities (47) and (51), while terms that are quadratic in momenta lead to derivatives of local densities and to kinetic densities (41), (45), and (49), because

$$\hat{\mbox{{\boldmath {$k$}}}}^2 = -{\textstyle{\frac{1}{4}}}\big(\mbox{{\boldmath {$\nabla$}}}_1 + ... ...2\big)^2 +\mbox{{\boldmath {$\nabla$}}}_1\cdot\mbox{{\boldmath {$\nabla$}}}_2 ,$$ (142)

$$\hat{\mbox{{\boldmath {$k$}}}}\otimes\hat{\mbox{{\boldmath {$k$}}}} = -{\textstyle{\frac{1}{4}}}\big(\mbox{{\boldmath {$\nabla$}}}_1 + ... ...imes\big(\mbox{{\boldmath {$\nabla$}}}_1 + \mbox{{\boldmath {$\nabla$}}}_2\big)$$

$$+{\textstyle{\frac{1}{2}}}\big(\mbox{{\boldmath {$\nabla$}}}_1\ot... ..._2+\mbox{{\boldmath {$\nabla$}}}_2\otimes\mbox{{\boldmath {$\nabla$}}}_1\big) .$$ (143)

However, in the p-p energy density, indices of Pauli matrices couple together the two density matrices, and hence do require recoupling to the p-p channel. These recoupling formulae can be obtained by means of the standard algebra of angular momentum. A sum of the three Clebsch-Gordan coefficients appropriate to the present case reads [179]

$$4s'_2s_2 \sum_{\mu_1\mu_2}\langle {\textstyle{\frac{1}{2}}}s_1\la... ...! s'_2\rangle \langle\lambda_1-\! \mu_1\lambda_2-\! \mu_2\vert\lambda\mu\rangle$$

$$~~~~~~=\sum_{l'm',lm}(-1)^{\lambda_1-\lambda_2 +l'}(2l'+1)(2l+1) ... ...textstyle{\frac{1}{2}}}s_2\rangle \langle l'-\! m'l-\! m\vert\lambda\mu \rangle$$ (144)

Taking relevant combinations of $\lambda_1,\, \lambda_2 =0,\, 1$ and $\lambda =0,\, 1,\, 2$, one obtains:

$$4s'_2s_2\hat{\delta}^{\sigma}_{s'_1,-s'_2-s_1,-s_2} = \phantom{-}{\textstyle{\frac{1}{2}}}\delta_{s'_2s'_1}\delta_{s_2s... ...boldmath {$\sigma$}}}^*_{s'_2s'_1}\cdot\mbox{{\boldmath {$\sigma$}}}_{s_2s_1} ,$$ (145)

$$4s'_2s_2\hat{P}^\sigma_{s'_1,-s'_2s_1,-s_2} = - {\textstyle{\frac{1}{2}}}\delta_{s'_2s'_1}\delta_{s_2s_1} + {\t... ...boldmath {$\sigma$}}}^*_{s'_2s'_1}\cdot\mbox{{\boldmath {$\sigma$}}}_{s_2s_1} ,$$ (146)

$$4s'_2s_2\hat{\mbox{{\boldmath {$S$}}}}_{s'_1,-s'_2s_1,-s_2} = -i\,\mbox{{\boldmath {$\sigma$}}}^*_{s'_2s'_1}\times\mbox{{\boldmath {$\sigma$}}}_{s_2s_1} ,$$ (147)

$$4s'_2s_2\hat{{\mathsf S}}^{ab}_{s'_1,-s'_2s_1,-s_2} = -{\textstyle{\frac{3}{2}}}\big(\mbox{{\boldmath {$\sigma$}}}^{a*}... ...{\boldmath {$\sigma$}}}^*_{s'_2s'_1}\cdot\mbox{{\boldmath {$\sigma$}}}_{s_2s_1}$$ (148)

and the formulae similar to Eqs. (145) and (146) are obtained for $\hat{\delta}^{\tau}$ and $\hat{P}^{\tau}$, respectively.

The two zero-order (density-dependent) p-p coupling constants of the energy density (80) are related to the Skyrme parameters in the following way:

$$\breve{C}_{0}^{s} = {\textstyle{\frac{1}{8}}} t_0(1+x_0) + {\textstyle{\frac{1}{48}}}t_3(1+x_3)\rho_0^{\alpha}(\mbox{{\boldmath {$r$}}}),$$ (149)

$$\breve{C}_{1}^{\rho} = {\textstyle{\frac{1}{8}}} t_0(1-x_0) + {\textstyle{\frac{1}{48}}}t_3(1-x_3)\rho_0^{\alpha}(\mbox{{\boldmath {$r$}}}),$$ (150)

and the second-order p-p coupling constants are given in Table 2. Similar to the p-h case, the gauge-invariance conditions (112) are met.

Table 2: Second-order coupling constants of the p-p energy density (80) as functions of parameters of the Skyrme interaction (117), expressed by the formula: $\breve{C}=\frac{A}{96}(at_1+bt_1x_1+ ct_2+dt_2x_2+et_{\mbox{\rm\scriptsize{e}}}+ft_{\mbox{\rm\scriptsize{o}}}+gW_0)$.

	$A$		$a$	$b$	$c$	$d$	$e$	$f$	$g$
$\breve{C}_0^{\Delta s}$	$-$3		1	1	0	0	2	0	0
$\breve{C}_0^{T}$	12		1	1	0	0	2	0	0
$\breve{C}_0^{j}$	12		0	0	1	$-$1	0	0	0
$\breve{C}_0^{\nabla{s}}$	$-$18		0	0	0	0	1	0	0
$\breve{C}_0^{F}$	$-$72		0	0	0	0	1	0	0
$\breve{C}_0^{\nabla{j}}$	0		0	0	0	0	0	0	0
$\breve{C}_1^{\Delta\rho}$	$-$3		1	$-$1	0	0	0	0	0
$\breve{C}_1^{\tau}$	12		1	$-$1	0	0	0	0	0
$\breve{C}_1^{J0}$	4		0	0	1	1	0	$-$10	4
$\breve{C}_1^{J1}$	6		0	0	1	1	0	5	2
$\breve{C}_1^{J2}$	12		0	0	1	1	0	$-$1	$-$2
$\breve{C}_1^{\nabla{J}}$	0		0	0	0	0	0	0	0

Equivalently, the density-dependent zero-range pairing force $V_{\mbox{\rm\scriptsize {pair}}}$ can be used in the p-p channel [180,181,182,183],

$$V_{\mbox{\rm\scriptsize {pair}}}(\mbox{{\boldmath {$r$}}},\m... ...}})\delta(\mbox{{\boldmath {$r$}}}-\mbox{{\boldmath {$r$}}}'),$$ (151)

for

$$f_{\mbox{\rm\scriptsize {pair}}}(\mbox{{\boldmath {$r$}}}) =... ...$r$}}})}{\rho_c}\right]^\alpha (1+x_3\hat{P}^\sigma)\right\} ,$$ (152)

where $\hat{P}^\sigma$ is the spin-exchange operator (120). In such a case, coupling constants (149) read

$$\breve{C}_{0}^{s} = {\textstyle{\frac{1}{8}}} V_0(1+x_0) - {\textstyle{\frac{1}{8}}} V_0(1+x_3) \left[\frac{\rho_0(\mbox{{\boldmath {$r$}}})}{\rho_c}\right]^\alpha ,$$ (153)

$$\breve{C}_{1}^{\rho} = {\textstyle{\frac{1}{8}}} V_0(1-x_0) - {\textstyle{\frac{1}{8}}} V_0(1-x_3) \left[\frac{\rho_0(\mbox{{\boldmath {$r$}}})}{\rho_c}\right]^\alpha .$$ (154)

Note that when only the isovector pairing is used, as in most LDA applications to date, the exchange parameters $x_0$ and $x_3$ are redundant in the definition of the isovector coupling constant $\breve{C}_{1}^{\rho}$, and hence are usually set to 0. However, if one wants to independently model the isoscalar and isovector pairing intensity, one has to use non-zero values of $x_0$ and $x_3$.

For the Gogny interaction [167], the zero-range density-dependent term $t_3$ with $\alpha$=1/3 was used in order to enforce proper saturation properties. The corresponding exchange parameter $x_3$=1 was used to prevent this zero-range force from contributing to the isovector pairing channel. However, such a choice, when applied literally to the proton-neutron mixing case, might lead to a very strong repulsive isoscalar pairing interaction.

The term of $\breve{\chi}$ coming from the spin-orbit interaction contains the combination of components of the p-p spin-current density $\vec{\breve{\mathsf J}}$,

$$\sum_{ab}\left( \vec{\breve{\mathsf J}}^{\ast}_{aa}\circ\ve... ...}\vert^2 - \vert\underline{ \vec{\breve{\mathsf J}}} \vert^2 ,$$ (155)

that is different from that coming from the tensor $t_{\mbox{\rm\scriptsize {o}}}$ term,

$$\sum_{ab}\left(\vec{\breve{\mathsf J}}^{\ast}_{ab}\circ\vec{\brev... ...thsf J}}^{\ast}_{ab}\circ\vec{\breve{\mathsf J}}_{ba}\right) ~~~~~~~~~~~~~~~~~~$$

$$~~~~~~~~~~~~~~~~~~~~~~~~ = - {\textstyle{\frac{5}{3}}} \vert\vec{... ... - {\textstyle{\frac{1}{2}}}\vert\underline{ \vec{\breve{\mathsf J}}} \vert^2 ,$$ (156)

and from that coming from the central $t_2$ term,

$$\vert\vec{\breve{\mathsf J}}\vert^2 = \sum_{ab}\left(\vec{\b... ...}\vert^2 + \vert\underline{ \vec{\breve{\mathsf J}}} \vert^2 .$$ (157)

Therefore, by setting appropriate values of the $t_2(1+x_2)$, $W_0$, and $t_{\mbox{\rm\scriptsize {o}}}$ parameters, one can obtain arbitrary values of the spin-current coupling constants $\breve{C}_1^{J0}$, $\breve{C}_1^{J1}$, and $\breve{C}_1^{J2}$. Similarly, parameter $t_2(1-x_2)$ allows for fixing an arbitrary value of the current coupling constant $\breve{C}_0^{j}$. On the other hand, parameter $t_1(1+x_1)$ defines two isoscalar coupling constants, $\breve{C}_0^{\Delta s}$ and $\breve{C}_0^{T}$, parameter $t_{\mbox{\rm\scriptsize {e}}}$ defines another two isoscalar coupling constants, $\breve{C}_0^{\nabla{s}}$ and $\breve{C}_0^{F}$, and parameter $t_1(1-x_1)$ defines two isovector coupling constants, $\breve{C}_1^{\Delta\rho}$ and $\breve{C}_1^{\tau}$; hence, these pairs of coupling constants are not independent from one another. These three pairs of dependencies reflect, in fact, the three gauge invariance conditions (104). In this way, seven Skyrme force parameters determine ten coupling constants in the p-p channel. Finally, the Skyrme interaction does not give any non-zero values for the spin-orbit coupling constants $\breve{C}_0^{\nabla{j}}$ and $\breve{C}_1^{\nabla{J}}$. Therefore, up to the gauge invariance conditions, the Skyrme interaction fully determines the energy density in the p-p channel.