In the p-p energy density, each operator of the relative
momentum,
and
, 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
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]
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:
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||
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1 | 1 | 0 | 0 | 2 | 0 | 0 | |
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12 | 1 | 1 | 0 | 0 | 2 | 0 | 0 | |
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12 | 0 | 0 | 1 | ![]() |
0 | 0 | 0 | |
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0 | 0 | 0 | 0 | 1 | 0 | 0 | |
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0 | 0 | 0 | 0 | 1 | 0 | 0 | |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
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1 | ![]() |
0 | 0 | 0 | 0 | 0 | |
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12 | 1 | ![]() |
0 | 0 | 0 | 0 | 0 | |
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4 | 0 | 0 | 1 | 1 | 0 | ![]() |
4 | |
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6 | 0 | 0 | 1 | 1 | 0 | 5 | 2 | |
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12 | 0 | 0 | 1 | 1 | 0 | ![]() |
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|
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Equivalently, the density-dependent zero-range pairing force
can be used in the p-p channel
[180,181,182,183],
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(152) |
For the Gogny interaction [167], the zero-range
density-dependent term with
=1/3 was used in order to
enforce proper saturation properties. The corresponding exchange
parameter
=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 coming from the spin-orbit interaction
contains the combination of components of the p-p spin-current
density
,
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(155) |
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(156) |
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(157) |