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Within the localdensity approximation, the energy functional is given
by the spatial integral of the local energy density

(4) 
The energy density is composed of the kinetic term
,
the Skyrme energy density
that describes the
effective strong interaction between the nucleons, and a term arising
from the
electromagnetic interaction
:

(5) 
For the electromagnetic interaction, we take the standard Coulomb
expression,
including the Slater approximation for the exchange
term. The energy functional discussed here contains all possible
terms bilinear in local densities and currents and up to second order
in the derivatives that are invariant under reflection,
timereversal, rotation, translation, and isospin rotation
[20].
Timereversal invariance requires the energy density
to be bilinear in either timeeven densities or timeodd densities,
so the Skyrme energy density can be separated into a ``timeeven"
part
and a ``timeodd" part
:

= 

(6) 
The sum runs over the isospin t and its third component t_{3}.
Only the t_{3} = 0 component of the isovector t = 1
terms contribute to nuclear ground states and the rotational bands
discussed later, while the
components contribute
only to chargeexchange (e.g. GT) excitations. In the notation of
Refs. [3,20],
the timeeven and timeodd Skyrme energy densities read
Isospin invariance of the Skyrme interaction makes the coupling
constants independent of the isospin zprojection.
All coupling constants might be density dependent. Following the
standard ansatz for the Skyrme interaction, we neglect such a
possibility
except in
and C_{t}^{s}, for which we restrict the density
dependence to the following form
Here
is the isoscalar scalar density, and
is its
value in saturated infinite nuclear matter. The exponent
that
specifies the density dependence of
must be
about 0.25 for the incompressibility coefficient
to be
correct[21,22,23,24]. Although this fact does
not restrict the analogous power in
,
Eq. (10), we keep
equal to
for simplicity here.
Usually we will consider energy functionals that
are invariant under local gauge transformations [3], which
generalize the Galilean invariance of the Skyrme interaction discussed
in
[2]. Gauge invariance links three pairs of timeeven and
timeodd terms in the energy functional:
C_{t}^{j} 
= 

(11) 
These relations fix all orbital timeodd terms, leaving only
timeodd terms corresponding to the spinspin interaction with free
coupling constants. Relations (11) lead to a simplified
form of Eqs. (6)(8),
The timeeven terms of the energy functional can be directly
related to nuclear bulk properties such as E/A, the saturation density
,
incompressibility, symmetry energy, surface and
surface symmetry energy, and spinorbit splittings. The remaining
timeodd terms cannot.
We will set the coupling constant
to 0. The term it multiplies comes
from a local twobody tensor force considered in Skyrme's original
papers [25] and discussed by Stancu et al. [26],
but omitted in all modern Skyrme parameterizations except the force SL1
introduced by Liu et al. [27], which has not been used since.
Next: Existing parameterizations
Up: A generalized Skyrme energy
Previous: Basics of energy density
Jacek Dobaczewski
20020315