The tensor part of the Skyrme energy density functional. I. Spherical
  nuclei
The tensor part of the Skyrme energy density functional. I. Spherical nuclei
T. Lesinski,1, ∗ M. Bender,2, 3, † K. Bennaceur,1, 2 T. Duguet,4 and J. Meyer1
1Université de Lyon, F-69003 Lyon, France; Institut de Physique Nucléaire de Lyon,
CNRS/IN2P3, Université Lyon 1, F-69622 Villeurbanne, France
2DSM/DAPNIA/SPhN, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France
3Université Bordeaux 1; CNRS/IN2P3; Centre d’Études Nucléaires de Bordeaux Gradignan,
UMR5797, Chemin du Solarium, BP120, F-33175 Gradignan, France
4National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA
(Dated: April 4, 2007)
We perform a systematic study of the impact of the J2 tensor term in the Skyrme energy functional
on properties of spherical nuclei. In the Skyrme energy functional, the tensor terms originate both
from zero-range central and tensor forces. We build a set of 36 parameterizations which cover a wide
range of the parameter space of the isoscalar and isovector tensor term coupling constants with a
fit protocol very similar to that of the successful SLy parameterizations. We analyze the impact of
the tensor terms on a large variety of observables in spherical mean-field calculations, such as the
spin-orbit splittings and single-particle spectra of doubly-magic nuclei, the evolution of spin-orbit
splittings along chains of semi-magic nuclei, mass residuals of spherical nuclei, and known anomalies
of radii. The major findings of our study are (i) tensor terms should not be added perturbatively to
existing parameterizations, a complete refit of the entire parameter set is imperative. (ii) The free
variation of the tensor terms does not lower the χ2 within a standard Skyrme energy functional.
(iii) For certain regions of the parameter space of their coupling constants, the tensor terms lead
to instabilities of the spherical shell structure, or even the coexistence of two configurations with
different spherical shell structure. (iv) The standard spin-orbit interaction does not scale properly
with the principal quantum number, such that single-particle states with one or several nodes have
too large spin-orbit splittings, while those of nodeless intruder levels are tentatively too small. Tensor
terms with realistic coupling constants cannot cure this problem. (v) Positive values of the coupling
constants of proton-neutron and like-particle tensor terms allow for a qualitative description of the
evolution of spin-orbit splittings in chains of Ca, Ni and Sn isotopes. (vi) For the same values of
the tensor term coupling constants, however, the overall agreement of the single-particle spectra
in doubly-magic nuclei is deteriorated, which can be traced back to features of the single-particle
spectra that are not related to the tensor terms. We conclude that the currently used central and
spin-orbit parts of the Skyrme energy density functional are not flexible enough to allow for the
presence of large tensor terms.
PACS numbers: 21.10.Dr, 21.10.Pc, 21.30.Fe, 21.60.Jz
I. INTRODUCTION
The strong nuclear spin-orbit interaction in nuclei is
responsible for the observed magic numbers in heavy nu-
clei [1, 2, 3, 4]. While a simple spin-orbit interaction al-
lows for the qualitative description of the global features
of shell structure, the available data suggest that single-
particle energies evolve with neutron and proton number
in a manner that cannot be related to the geometrical
growth of the single-particle potential with N and Z.
Many anomalies of shell structure have been identified
that do not fit into simple experimental systematics, and
that challenge any global model of nuclear structure.
The evolution of shell structure with N and Z as a fea-
ture of self-consistent mean-field models has been known
for long. To quote the pioneering study of shell structure
∗Electronic address: lesinski@ipnl.in2p3.fr
†Electronic address: bender@cenbg.in2p3.fr
in a self-consistent model performed by Beiner et al. [5],
the “most striking effect is the appearance of N = 16,
34 and 56 as neutron magic numbers for unstable nuclei,
together with a weakening of the shell closure at N = 20
and 28”. Various mechanisms that modify the appear-
ance of gaps in the single-particle spectra have been dis-
cussed in detail in the literature. The two most promi-
nent ones that were worked out by Dobaczewski et al. in
Ref. [6], however, play mainly a role for weakly-bound
exotic nuclei far from stability, as they are directly or
indirectly related to the physics of loosely bound single-
particle states, namely that the enhancement of the dif-
fuseness of neutron density distribution reduces the spin-
orbit coupling in neutron-rich nuclei on the one hand,
and the interaction between bound orbitals and the con-
tinuum results in a quenching of shell effects in light and
medium systems on the other hand. The former effect
was also extensively discussed in the framework of rela-
tivistic models by Lalazissis et al. [7, 8], while the latter
triggered a number of studies that discussed the poten-
tial relevance of this so-called “Boguliubov enhanced shell
http://arxiv.org/abs/0704.0731v3
mailto:lesinski@ipnl.in2p3.fr
mailto:bender@cenbg.in2p3.fr
quenching” to explain the abundance pattern from the
astrophysical r-process of nucleosynthesis [9, 10, 11, 12].
These two effects take place in neutron-rich nuclei. In
proton-rich nuclei, the Coulomb barrier suppresses both
the diffuseness of the proton density and the coupling of
bound proton states to the continuum. But the Coulomb
interaction itself can also modify the shell structure: for
super-heavy nuclei, it begins to destabilize the nucleus
as a whole. Mean-field models predict that it ampli-
fies the shell oscillations of the densities for incomplete
filled oscillator shells, which leads to strong variations of
the density profile that feed back onto the single-particle
spectra [13, 14].
Interestingly, most theoretical papers about the evolu-
tion of shell structure from the last decade have specu-
lated about new effects that mainly affect neutron shells
in nuclei far from stability in the anticipation of the rare-
isotope physics that might become accessible with the
next generation of experimental facilities. The known
anomalies, some of which have been known for a long
time, and many more have been identified recently, con-
cern also proton shells and already appear sufficiently
close to stability that “exotic phenomena can be ruled
out for their explanation” in most cases, to paraphrase
the authors of Ref. [15]. By contrast, this suggests that
there exists a mechanism that induces a strong evolution
of single-particle spectra already in stable nuclei that has
been overlooked for long.
There is a prominent ingredient of the nucleon-nucleon
interaction that has been ignored for decades in virtu-
ally all global nuclear structure models for medium and
heavy nuclei, be it macroscopic-microscopic approaches
or self-consistent mean-field methods. It is only very re-
cently, that the systematic discrepancies between model
predictions and experiment have triggered a renaissance
of the tensor force in the description of finite medium-
and heavy-mass nuclei.
The tensor force is a crucial and necessary ingredient
of the bare nucleon-nucleon interaction [16, 17], and con-
sequently is contained in all ab-initio approaches that are
available for light, mainly p-shell nuclei [18, 19]. One of
the first experimental signatures of the tensor force was
the small, but finite quadrupole moment of the deuteron.
In a boson-exchange picture of the bare nucleon-nucleon
interaction, the tensor force originates from the exchange
of pseudoscalar pions, which have both central and tensor
couplings, see for example section 2.3 in Ref. [20] or ap-
pendix 13A of Ref. [21]. In a nuclear many-body system,
the bare tensor force induces a strong correlation between
the spatial and spin orientations in the two-body density
matrix. For two nucleons with parallel spins, the ten-
sor force energetically favors the configuration where the
distance vector is aligned with the spins, while for anti-
parallel spins the tensor force prefers when the distance
vector is perpendicular to the spins, see the discussion
of Fig. 13 in Ref. [22] and of Fig. 3 in Ref. [23]. The
authors of these papers also demonstrate very nicely the
well-known fact [24, 25] that in an approach that starts
from the bare nucleon-nucleon interaction, nuclei are not
bound without taking into account the two-body corre-
lations induced by the tensor force.
The role of the tensor force, however, manifests itself
differently in self-consistent mean-field models, otherwise
called energy density functional (EDF) methods, the tool
of choice for medium and heavy nuclei. The latter meth-
ods use an independent-particle state as a reference state
to express the energy of the correlated nuclear ground
state. Thus, correlations are not explicitly present in the
higher-order density matrices of the reference state, but
rather included under the form of a more elaborate func-
tional of the (local and nearly local parts of the) one-body
density matrix of that reference state. In such a scheme,
most of the effect of the bare tensor force on the binding
energy is integrated out through the renormalization of
the coupling constants associated with a central effective
vertex, in a similar fashion as the tensor part of the bare
interaction is renormalized into the central one when go-
ing from the bare nucleon-nucleon force to a Brueckner G
matrix. The tensor terms of the EDF relate to a residual
tensor vertex, that gives nothing but a correction to the
spin-orbit splittings, which for light p-shell nuclei might
be of the same order as the contribution from the genuine
spin-orbit force. The interplay of spin-orbit and tensor
forces in the mean field of medium and heavy nuclei was
explored in Refs. [26, 27, 28], where the particular role
of spin-unsaturated shells was pointed out.
There are two widely used effective interactions for
non-relativistic self-consistent mean-field models [29], the
zero-range non-local Skyrme interaction [30, 31, 32, 33]
on the one hand and the finite-range Gogny force [34, 35]
on the other hand.
In fact, the effective zero-range non-local interaction
proposed by Skyrme in 1956 [30, 31, 32, 33] already con-
tained a zero-range tensor force. The first applications of
Skyrme’s interaction in self-consistent mean-field models
that became available around 1970, however, neglected
the tensor force, and the simplified effective Skyrme in-
teraction used in the seminal paper by Vautherin and
Brink [36] soon became the standard Skyrme interaction
that was used in most applications ever since. Until very
recently, there was only very little exploratory work on
Skyrme’s tensor force. In their early study, Stancu, Brink
and Flocard [37], who added the tensor force perturba-
tively to the SIII parameterization, pointed out that some
spin-orbit splittings in magic nuclei can be improved with
a tensor force. A complete fit including the terms from
the tensor force that contribute in spherical nuclei was
attempted by Tondeur [38], with the relevant coupling
constants of the spin-orbit and tensor terms adjusted to
selected spin-orbit splittings in 16O, 48Ca and 208Pb. An-
other complete fit of a generalized Skyrme interaction in-
cluding a tensor force was performed by Liu et al. [39],
but the authors did not investigate the effect of the ten-
sor force in detail, nor was the resulting parameterization
ever used in the literature thereafter.
Similarly, the seminal paper by Gogny [34] on the eval-
uation of matrix elements of a finite-range force of Gaus-
sian shape in an harmonic oscillator basis contains the ex-
pressions for a finite-range tensor force, which, however,
was omitted in the parameterizations of Gogny’s force
adjusted by the Bruyères-le-Châtel group [35]. It were
Onishi and Negele [40] who first published an effective
interaction that combined a Gaussian two-body central
force, a finite-range tensor force with a zero-range spin-
orbit force and a zero-range non-local three-body force,
which, however, also fell into oblivion.
The role of the tensor force is slightly different in
Skyrme and Gogny interactions. In the Gogny force,
the contributions from the central and tensor parts re-
main explicitly distinct, although, of course, this does
not prevent a certain entanglement of their physical ef-
fects. In the context of Skyrme’s functional, however, the
contribution of a zero-range tensor force to the spherical
mean-field state of an even-even nucleus has exactly the
same form as a particular exchange term from the non-
local part of the central Skyrme force. When looking at
spherical nuclei only, adding Skyrme’s tensor force simply
allows one to decouple a term that is already provided
by the central force. This indeed makes the effective-
interaction-restricted functional more flexible, as the ad-
ditional degrees of freedom from the tensor force remove
an interdependence between the effective mass, the sur-
face terms and the “tensor terms”. However, one must
always keep in mind that both the central and tensor
part of the effective vertex contribute to the so-called J2t
“tensor” terms of the functional.1
In the context of relativistic mean-field models, the
equivalent of the non-relativistic tensor force appears
as the exchange term of effective fields with the quan-
tum numbers of the pion, which by construction do
not appear in the standard relativistic Hartree models.
Only relativistic Hartree-Fock models contain this tensor
force, with the first predictive parameterizations becom-
ing available just recently [42].
We also mention that there is a large body of work
on the tensor force in the interacting shell model, see
Ref. [43] for a review, that concentrates on a completely
different aspect of the tensor force, namely its unique
contribution to excitations with unnatural parity.
The recent interest in the effect of the tensor force in
the context of self-consistent mean field models was trig-
gered by the observed evolution of single-particle levels
of one nucleon species in dependence of the number of
the other nucleon species. Otsuka et al. [44] proposed
1 As we will outline below, and as was already pointed out in
Ref. [5], this argument does not hold for deformed even-even nu-
clei or any situation where intrinsic time-reversal is broken, for
example odd nuclei or dynamics. There, the tensor and non-local
central parts of the effective Skyrme interaction give contribu-
tions to the mean-fields and the binding energy with different
analytical expressions. This will be discussed in a companion
article [41].
that at least part of the effect is caused by the proton-
neutron tensor force from pion exchange. Many groups
attempt now to explain known, but so far unresolved,
anomalies of shell structure in terms of a tensor force.
A particularly popular playground is the relative shift of
the proton 1g7/2 and 1h11/2 levels in tin isotopes, which is
interpreted as the reduction of the spin-orbit splittings of
both levels with their respective partners with increasing
neutron number [45].
Otsuka et al. [46] added a Gaussian tensor force, ad-
justed on the long-range part of a one-pion+ρ exchange
potential, to a standard Gogny force. After a consis-
tent readjustment of the parameters of its central and
spin-orbit parts, they were able to explain coherently the
anomalous relative evolution of some single-particle levels
without, however, being able to describe their absolute
distance in energy. Dobaczewski [47] has pointed out that
a perturbatively added tensor interaction with suitably
chosen coupling constants in the Skyrme energy density
functional does not only modify the evolution of shell
structure, but does also improve the description of nu-
clear masses around magic nuclei. Brown et al. [48] have
fitted a Skyrme interaction with added zero-range tensor
force with emphasis on the reproduction of single-particle
spectra. While the authors appreciate the qualitatively
correctly described evolution of relative level distances,
they point out that the combination of zero-range spin-
orbit and tensor forces does not and can not correctly
describe the ℓ-dependence of spin-orbit splittings. Colò
et al. [49], and Brink et al. [50] have added Skyrme’s
tensor force perturbatively to the existing standard pa-
rameterization SLy5 [51, 52], and to the SIII [5] one,
respectively. They have investigated some single-particle
energy differences: the 1h11/2 and 1g7/2 proton states in
tin isotopes as well as 1i13/2 and 1h9/2 neutron states in
N = 82 isotones and propose similar parameters as in
Ref. [48]. The effect of the tensor force on the centroid of
the GT giant resonance is also estimated by Colò et al.
using a sum-rule approach and found to be substantial.
Long et al. [53], demonstrate that the tensor force that
emerges naturally in relativistic Hartree-Fock also im-
proves the relative shifts of the proton 1g7/2 and 1h11/2
levels in tin isotopes.
The work on the tensor force published so far aims
at an optimal single parameterization, that establishes a
best fit to either the underlying bare tensor force [46, 48]
or empirical data [38, 47, 49]. The published results,
as well as our first exploratory studies, however, suggest
that adding a tensor force to the existing mean-field mod-
els gives only a local improvement of the relative change
of certain single-particle energies, but not necessarily a
global improvement of single-particle spectra or other ob-
servables. In the framework of the Skyrme interaction,
that we will employ throughout this work, there is also
the already mentioned ambiguity that the contribution
from the tensor force to spherical nuclei has the same
structure as a term from the central force. In view of
this situation, we will pursue a different strategy and in-
vestigate the effect of the tensor terms on a multitude
of observables in nuclei though a set of Skyrme interac-
tions with systematically varied coupling constants of the
tensor terms.
The present study was motivated by the finding that
the performance of the existing Skyrme-type effective in-
teractions for masses and spectroscopic properties is lim-
ited by systematic deficiencies of the single-particle spec-
tra [54, 55, 56, 57] that seem to be impossible to remove
within the standard Skyrme interaction. The details of
single-particle spectra were so far somewhat outside the
focus of self-consistent mean-field methods, on the one
hand as they do not correspond directly to empirical
single-particle energies (we will come back to that be-
low), and on the other hand because many of the ob-
servables that are usually calculated with self-consistent
mean-field methods are not very sensitive to the exact
placement of single-particle levels. By contrast, there
is an enormous body of work that examines the infi-
nite and semi-infinite nuclear matter properties of the
effective interactions that are the analog of liquid-drop
and droplet parameters in great detail. The reason is,
of course, that the global trends over the whole chart of
nuclei have to be understood before one can look into
details. The last few years have seen an increasing de-
mand on predictive power. Moreover, beyond-mean-field
approaches of the projected generator coordinate method
(GCM), or Bohr-Hamiltonian type, have become widely
used tools to analyze and predict spectroscopic properties
in medium and heavy nuclei, employing either Gogny or
Skyrme interactions. The underlying single-particle spec-
tra thus now deserve more attention, as many of the spec-
troscopic properties of interest turn out to be extremely
sensitive to even subtle details of the single-particle spec-
tra. As the tensor force is the most obvious missing piece
in all standard mean-field interactions, it is the natural
starting point for the systematic investigation of possi-
ble generalizations with the ultimate goal to improve the
predictive power of the interactions for spectroscopy.
In the present paper, we will outline the formalism of
a Skyrme interaction with added tensor force, describe
the fit of the parameterizations, analyze the role of the
tensor terms for single-particle spectra, masses and radii
of spherical even-even nuclei. A second paper [41] studies
the surface and deformation properties of these Skyrme
interactions for even-even nuclei, and future work will ex-
amine the stability of nuclear matter and the role of the
time-odd terms from the tensor force in odd and rotating
nuclei. Only deformed nuclei and, in particular, observ-
ables sensitive to the time-odd contributions, will pos-
sibly allow to distinguish clearly between the non-local
central and tensor parts of the Skyrme force.
II. THE SKYRME INTERACTION WITH
TENSOR TERMS
A. The energy density functional
The usual ansatz for the Skyrme effective interac-
tion [51, 52] leads to an energy density functional which
can be written as the sum of a kinetic term, the Skyrme
potential energy functional that models the effective
strong interaction in the particle-hole channel, a pairing
energy functional corresponding to a density-dependent
contact pairing interaction, the Coulomb energy func-
tional (calculated using the Slater approximation [58])
and correction terms to approximately remove the ex-
citation energy from spurious motion caused by broken
symmetries
E = Ekin + ESkyrme + Epairing + ECoulomb + Ecorr . (1)
B. The Skyrme energy density functional
Throughout this work, we will use an effective Skyrme
energy functional that corresponds to an antisymme-
trized density-dependent two-body vertex in the particle-
hole channel of the strong interaction, that can be decom-
posed into a central, spin-orbit and tensor contribution
vSkyrme = vc + vt + vLS . (2)
Other choices for the writing of the Skyrme energy func-
tional are possible and have been made in the literature,
which might affect the form of the effective interaction,
its interpretation and the results obtained from it. We
will come back to that in section IID below.
The Skyrme energy density functional is a functional
of local densities and currents
ESkyrme =
d3r HSkyrme(r) , (3)
which has many technical advantages compared to finite-
range forces such as the Gogny force. All exchange terms
have the same structure as the direct terms, which greatly
reduces the number of necessary integrations during a
calculation.
1. Local densities and currents
Throughout this paper we will assume that we have
pure proton and neutron states. The formal framework
of the general case including proton-neutron mixing is
discussed in Ref. [59]. Without making reference to any
single-particle basis, we start from the density matrices
of protons and neutrons in coordinate space [60]
ρq(rσ, r
′σ′) = 〈â
r′σ′q ârσq〉
ρq(r, r
′)δσσ′ +
sq(r, r
′) · 〈σ′|σ̂|σ〉
where
ρq(r, r
ρq(rσ, r
sq(r, r
ρq(rσ, r
′σ′) 〈σ′|σ̂|σ〉 . (5)
The Skyrme energy functional up to second order in
derivatives that we will introduce below can be expressed
in terms of seven local densities and currents [59] that are
defined as
ρq(r) = ρq(r, r
sq(r) = sq(r, r
τq(r) = ∇ ·∇
′ ρq(r, r
Tq,µ(r) = ∇ ·∇
′ sq,µ(r, r
jq(r) = −
(∇−∇′) ρq(r, r
Jq,µν(r) = −
(∇µ −∇
µ) sq,ν(r, r
Fq,µ(r) =
sq,ν(r, r
which are the density ρq(r), the kinetic density τq(r),
the current (vector) density jq(r), the spin (pseudovec-
tor) density sq(r), the spin kinetic (pseudovector) density
Tq(r), the spin-current (pseudotensor) density Jq,µν(r),
and the tensor-kinetic (pseudovector) density Fq(r).
ρq(r), τq(r) and Jq,µν(r) are time-even, while sq(r),
Tq(r), jq(r) and Fq(r) are time-odd. For a detailed dis-
cussion of their symmetries see Ref. [60]. There are other
local densities up to second order in derivatives that can
be constructed, but when constructing an energy func-
tional they either cannot be combined with others to
terms with proper symmetries or they lead to terms that
are not independent from the others [61].
The cartesian spin-current pseudotensor density Jµν
can be decomposed into pseudoscalar, (anti-symmetric)
vector and (symmetric) traceless pseudotensor parts, all
of which have well-defined transformation properties un-
der rotations
Jµν(r) =
δµν J
(0)(r)+ 1
ǫµνκ J
κ (r)+J
µν (r) , (7)
where δµν is the Kronecker symbol and ǫµνκ the Levi-
Civita tensor. The pseudoscalar, vector and pseudoten-
sor parts expressed in terms of the cartesian tensor are
given by
J (0)(r) =
Jµµ(r) , (8)
J (1)κ (r) =
µ,ν=x
ǫκµν Jµν(r) ,
J (2)µν (r) =
[Jµν(r) + Jνµ(r)] −
Jκκ(r) .
The vector spin current density J(1)(r) ≡ J(r) is often
called spin-orbit current, as it enters the spin-orbit energy
density. 2
For the formal discussion of the physical content of the
Skyrme energy functional it is of advantage to recouple
the proton and neutron densities to isoscalar and isovec-
tor densities, for example
ρ0(r) = ρn(r) + ρp(r) ,
ρ1(r) = ρn(r)− ρp(r) (9)
and similar for all others. As we assume pure proton
and neutron states, only the Tz = 0 component of the
isovector density is non-zero, which we exploit to drop
the index Tz from the isovector densities ρ1Tz (r) etc.
2. Skyrmes’s central force
We will use the standard density-dependent central
Skyrme force
vc(R, r) = t0 (1 + x0P̂σ) δ(r)
t3 (1 + x3P̂σ) ρ
α(R) δ(r)
t1 (1 + x1P̂σ)
′2 δ(r) + δ(r) k̂2
+ t2 (1 + x2P̂σ) k̂
′ · δ(r) k̂ (10)
where we use the shorthand notation
r = r1 − r2 ,
R = 1
(r1 + r2) , (11)
while k̂ is the usual operator for relative momenta
k̂ = − i
(∇1 −∇2) (12)
and k̂′ its complex conjugated acting on the left. Finally,
P̂σ is the spin exchange operator that controls the relative
strength of the S = 0 and S = 1 channels for a given term
in the two-body interaction
P̂σ =
(1 + σ̂1 · σ̂2) . (13)
As said above, we restrict ourselves to a parameterization
of the Skyrme energy functional as obtained from the
average value of an effective two-body vertex in the ref-
erence Slater determinant. We decompose the isoscalar
and isovector parts of the resulting energy density func-
tional Hc into a part H
c,even
t that is composed entirely of
time-even densities and currents, and a part H
c,odd
t that
contains terms which are bilinear in time-odd densities
2 Some authors call J(r) spin density, which is ambiguous and
confusing when discussing the complete energy density functional
including terms that contain the time-odd s(r).
and currents and vanishes in intrinsically time-reversal
invariant systems
Hc(r) =
t=0,1
c,even
t (r) +H
c,odd
t (r)
. (14)
Both H
c,even
t and H
c,odd
t are of course constructed such
that they are time-even; they are given by [59, 62]
c,even
t = A
t [ρ0] ρ
t ρt∆ρt +A
t ρtτt
µ,ν=x
Jt,µνJt,µν ,
c,odd
t = A
t [ρ0] s
+A∆st st ·∆st +A
t st ·Tt , (15)
where A
t [ρ0] and A
t [ρ0] are density dependent coupling
constants that depend on the total (isoscalar) density.
The detailed relations between the coupling constants of
the functional and the central Skyrme force are given
in appendix A. The notation reflects that two pairs of
terms in H
c,even
t and H
c,odd
t are connected by the require-
ment of local gauge invariance of the Skyrme energy func-
tional [63].
3. A zero-range spin-orbit force
The spin-orbit force used with most standard Skyrme
interactions
vLS(r) = iW0 (σ̂1 + σ̂2) · k̂
′ × δ(r) k̂ (16)
is a special case of the one proposed by Bell and
Skyrme [32, 33]. Again, the corresponding energy func-
tional [59, 62] can be separated into a time-even and a
time-odd term
HLS(r) =
t=0,1
LS,even
t (r) +H
LS,odd
t (r)
where
LS,even
t = A
t ρt∇ · Jt
LS,odd
t = A
t st · ∇ × jt (18)
which share the same coupling constant as again both
terms are linked by the local gauge invariance of the en-
ergy functional. The relation between the A∇·Jt and the
one coupling constant of the two-body spin-orbit force
W0 is given in appendix A.
4. Skyrme’s tensor force
By convention, the tensor operator in the tensor force
is constructed using the unit vectors in the direction of
the relative coordinate er = r/|r| and subtracting σ̂1 · σ̂2
Ŝ12 = 3(σ̂1 · er)(σ̂2 · er)− σ̂1 · σ̂2 , (19)
such that its mean value vanishes for a relative S state,
which decouples the central and tensor channels of the
interaction. The operator Ŝ12 commutes with the total
spin [Ŝ12, Ŝ
2] = 0, therefore it does not mix partial waves
with different spin, i.e. spin singlet and spin triplet states.
In particular, it does not act in spin singlet states at
all, as Ŝ12P̂S=0 = 0 (see section 13.6 of Ref. [21]). As
a consequence, there is no point in multiplying a tensor
force with an exchange operator (1+xtP̂σ) as done for the
central force, as this will only lead to an overall rescaling
of its strength.
The derivation of the general energy functional from a
zero-range two-body tensor force is discussed in detail in
Refs. [59, 64]. We repeat here the details relevant for our
discussion, starting from the two zero-range tensor forces
proposed by Skyrme [30, 31]
vt(r) = 1
3 (σ1 · k
′) (σ2 · k
′)− (σ1 · σ2)k
δ(r) + δ(r)
3 (σ1 · k) (σ2 · k)− (σ1 · σ2)k
3 (σ1 · k
′) δ(r) (σ2 · k)− (σ1 · σ2)k
′ · δ(r)k
where r, k̂ and k̂′ are defined as above, Eqs. (11) and (12).
The corresponding energy density functional can again be
decomposed in a time-even and a time-odd part
Ht(r) =
t=0,1
t,even
t (r) +H
t,odd
t (r)
with [59]
t,even
t = −B
µ,ν=x
Jt,µνJt,µν −
Jt,µµ
µ,ν=x
Jt,µνJt,νµ
t,odd
t = B
t st ·Tt +B
t st · Ft
+B∆st st ·∆st +B
t (∇ · st)
2 , (22)
where we already used the local gauge invariance of the
energy functional [59] for the expressions of the coupling
constants. The actual expressions for the coupling con-
stants expressed in terms of the two coupling constants
te and to of the tensor forces are given in appendix A.
The “even” term proportional to te in the two-body
tensor force (20) mixes relative S and D waves, while
the “odd” term proportional to to mixes relative P and
F waves. Thus, due to the fact that both act in spin-
triplet states only, antisymmetrization implies that the
former acts in isospin-singlet states (and hence con-
tributes to the neutron-proton interaction only) and the
latter in isospin-triplet states (contributing both to the
like-particle and neutron-proton interactions). The cen-
tral and spin-orbit interactions as we use them, however,
do not containD or F wave interactions. From this point
of view, one might suspect a mismatch when combining
the various interaction terms. From the point of view
of the energy functional (22), however, all contributions
from the zero-range tensor force are of the same second
order in derivatives as the contributions from the non-
local part of the central Skyrme force (15) and from the
spin-orbit force (18).
In the time-even part of the energy functional H
t,even
there appear three different combinations of the carte-
sian components of the spin current tensor. The term
proportional to BTt contains the symmetric combination
JµνJµν as it already appeared in the energy functional
from the central Skyrme interaction (15), while the term
proportional to BFt contains two different terms, namely
the antisymmetric combination JµνJνµ and the square of
the trace of Jνµ.
5. Combining central and tensor interactions
The Skyrme energy functional representing central,
tensor, and spin-orbit interactions is given by
ESkyrme = Ec + ELS + Et
t=0,1
t [ρ0] ρ
t + C
t [ρ0] s
t + C
t ρt∆ρt + C
t (∇ · st)
2 + C∆st st ·∆st + C
t (ρtτt − j
st ·Tt −
µ,ν=x
Jt,µνJt,µν
+ CFt
st · Ft −
Jt,µµ
µ,ν=x
Jt,µνJt,νµ
+C∇·Jt (ρt∇ · Jt + st · ∇ × jt)
. (23)
This functional contains all possible bilinear terms up to
second order in the derivatives that can be constructed
from local densities and that are invariant under spatial
and time inversion, rotations, and local gauge transfor-
mations [59].
Some of the coupling constants are completely defined
by the standard central Skyrme force, i.e. C
t = A
Cst = A
t , C
t = A
t , and C
t = A
t , two by the
spin-orbit force, C∇Jt = A
t , others by the tensor force,
CFt = B
t and C
t = B
t , while some are the sum of
coupling constants from both central and tensor forces,
CTt = A
t , and C
t = A
The three terms bilinear in Jµν can be recoupled into
terms bilinear in its pseudoscalar, vector, and pseudoten-
sor components J (0), J (1), and J (2), Eq. (8), which is
prefered by some authors [59]
µ,ν=x
Jt,µνJt,µν =
µ,ν=x
t,µνJ
t,µν (24)
Jt,µµ
µ,ν=x
Jt,µνJt,νµ
µ,ν=x
t,µνJ
t,µν . (25)
After combining (23) with the kinetic, Coulomb, pairing
and other contributions from (1), the mean-field equa-
tions are obtained by standard functional derivative tech-
niques from the total energy functional [29, 59].
The complete Skyrme energy functional (23) has quite
complicated a structure, and in the most general case
leads to seven distinct mean fields in the single-particle
Hamiltonian [59]. As already mentioned, we want to di-
vide the examination of those terms that contain two
derivatives and two Pauli matrices in the complete func-
tional, i.e. those terms from the central Skyrme force
that are often neglected and all the terms from the ten-
sor Skyrme force, into three distinct steps: First, in the
present paper, we enforce spherical symmetry which re-
moves all time-odd densities and all but one out of the
nine components of the spin current tensor Jµν as will
be outlined in the following section. A subsequent pa-
per [41] will discuss deformed even-even nuclei where the
complete spin current tensor Jµν is present, and future
work will address the time-odd part of the energy func-
tional (23).
C. The Skyrme energy functional in spherical
symmetry
For the rest of this paper, we will concentrate on spher-
ical nuclei, enforcing spherical symmetry of the N -body
wave functions. As a consequence, the canonical single-
particle wave functions Ψi [65] can be labeled by ji, ℓi
and mi. The index ni labels the different states with
same ji and ℓi. The functions Ψi separates into a radial
part ψ and an angular and spin part, represented by a
tensor spherical harmonic Ωjℓm
Ψnjℓm(r) =
ψnjℓ(r) Ωjℓm(θ, φ) . (26)
Spherical symmetry also enforces that all magnetic sub-
states of Ψnjℓm have the same occupation probability
v2njℓm ≡ v
njℓ for all −j ≤ m ≤ j. For a static spherical
state, all time-odd densities are zero sq(r) = Tq(r) =
jq(r) = Fq(r) = 0, as are the corresponding mean fields
in the single-particle Hamiltonian.
Enforcing spherical symmetry also greatly simplifies
the spin-current tensor, both the pseudoscalar and pseu-
dotensor parts of Jµν vanish. From the vector spin-orbit
current, only the radial component is non-zero, which is
given by [36]
Jq(r) =
n,j,ℓ
(2j + 1) v2njℓ
j(j + 1)− ℓ(ℓ+ 1)− 3
ψ2njℓ(r) (27)
so that there is only one out of the nine components of the
spin-current tensor density that contributes in spherical
nuclei. Unlike the total density ρ and the kinetic den-
sity τ , that are bulk properties of the nucleus and grow
with the size of the nucleus, the spin-orbit current is a
shell effect that shows strong fluctuations. Assume the
two shells with same n and ℓ which are split by the spin-
orbit interaction, one coupled with the spin to j = ℓ+ 1
the other to j = ℓ − 1
. It is easy to verify that their
contributions to Jq(r) are equal but of opposite signs
such that they cancel when (i) both shells are completely
filled and (ii) their radial wave functions are identical
ψn,ℓ+1/2,ℓ = ψn,ℓ−1/2,ℓ. Although the latter condition is
never exactly fulfilled, this demonstrates that the spin-
orbit current is not a bulk property, but a shell effect
that strongly fluctuates with N and Z. It nearly van-
ishes in so-called spin-saturated nuclei, where all spin-
orbit partners are either completely occupied or empty,
and it might be quite large when only the j = ℓ+1/2 level
out of one or even several pairs of spin-orbit partners is
filled.
Altogether, the Skyrme part of the energy density func-
tional in spherical nuclei is reduced to
HSkyrme =
t=0,1
t [ρ0] ρ
t + C
t ρt∆ρt + C
t ρtτt
CJt J
t + C
t ρt∇ · Jt
, (28)
where we have introduced an effective coupling constant
CJt of the J
t tensor terms at sphericity, such that the
corresponding contribution to the energy functional is
given by
t=0,1
CJt J
t=0,1
CTt +
t . (29)
The effective coupling constants can be separated back
into contributions from the non-local central and tensor
forces
CJt = A
t (30)
which are given by
AJ0 =
AJ1 =
BJ0 =
(te + 3to) =
(T + 3U)
BJ1 =
(to − te) =
(U − T ) , (31)
where we also give the expressions using the notation
T = 3te and U = 3to employed in [37, 49, 64].
For the following discussion it will be also illuminating
to recouple this expression to a representation that uses
proton and neutron densities, where we use the notation
introduced in Ref. [37]
Ht = 1
α (J2n + J
p) + β Jn · Jp , (32)
α = CJ0 + C
1 , β = C
0 − C
CJ0 =
(α+ β) , CJ1 =
(α− β) . (33)
The proton-neutron coupling constants α = αC+αT and
β = βC + βT can again be separated into contributions
from central and tensor forces
(t1 − t2)−
(t1x1 + t2x2) ,
βC = −
(t1x1 + t2x2) ,
(te + to) =
(T + U) . (34)
As could be expected, the isospin-singlet tensor force
contributes only to the proton-neutron term, while the
isospin-triplet tensor force contributes to both.
The spin-orbit potential of the neutrons is given by
Wn(r) =
δJn(r)
2∇ρn +∇ρp) + αJn + β Jp . (35)
The expression for the protons is obtained exchanging
the indices for protons and neutrons. In spherical sym-
metry, the tensor force gives a contribution to the spin-
orbit potential, but does not alter the structure of the
spin-orbit terms in the single-particle Hamiltonian as
such. This will be different in the case of deformed mean
fields [41, 59].
The dependence of the spin-orbit potential Wq(r) on
the spin-orbit current Jq(r) through the tensor terms is
the source of a potential instability. When the spin-orbit
splitting becomes larger than the splitting of the cen-
troids of single-particle states with different orbital angu-
lar momentum ℓ, the reordering of levels might increase
the number of spin-unsaturated levels, which increases
the spin-orbit current Jn and feeds back on the spin-orbit
potential by increasing it even further, which ultimately
leads to an unphysical shell structure. An example will
be given in appendix B.
D. A brief history of tensor terms in the central
Skyrme energy functional
For the interpretation of the parameterizations we will
describe below it is important to point out that within
our choice of the effective Skyrme interaction as an an-
tisymmetrized vertex the two coupling constants of the
contribution from the central force toHT , Eq. (29), either
represented through AJ0 , A
1 or through αC , βC , are not
independent from the coupling constants Aτ0 , A
1 , A
and A
1 , that appear in Eq. (28). Through the expres-
sions given in appendix A, all six of them are determined
by the four coupling constants t1, x1, t2, and x2 from the
central Skyrme force, Eq. (10). As a consequence, a ten-
sor force is absolutely necessary to decouple the values of
the CJt from those of the C
t and C
t , which determine
the isoscalar and isovector effective masses and give the
dominant contribution to the surface and surface asym-
metry coefficients, respectively.
This interpretation of the Skyrme interaction is, how-
ever, far from being common practice and a source of
confusion and potential inconsistencies in the literature.
Many authors have used parameterizations of the central
and spin-orbit Skyrme energy functional with coupling
constants that in one way or the other do not exactly
correspond to the functional obtained from Eqns. (10)
and (16), which, depending on the point of view, can be
seen as an approximation to or a generalization of the
original Skyrme interaction. As the most popular mod-
ification concerns the tensor terms, a few comments on
the subject are in order. Again, the practice goes back
to the seminal paper by Vautherin and Brink [36], who
state that “the contribution of this term to [the spin-
orbit potential] is quite small. Since it is difficult to in-
clude such a term in the case of deformed nuclei, it has
been neglected”. This choice was further motivated by
the interpretation of the effective Skyrme interaction as
a density-matrix expansion (DME) [25, 66, 67, 68]. All
early parameterizations as SI and SII [36], SIII-SVI [5],
SkM [69] and SkM∗ [70] followed this example and did
not contain the J2 terms. Beiner et al. [5] weakened
the case for J2 terms further by pointing out that they
might lead to unphysical single-particle spectra. During
the 1980s and later, however, it became more popular
to include them, for example in SkP [65], the parame-
terizations T1-T9 by Tondeur et al. [71], Eσ and Zσ by
Friedrich and Reinhard [72]. Some of the recent param-
eterizations come in pairs, where variants without and
with J2 terms are fitted within the same fit protocol, for
example (SLy4, SLy5) and (SLy6, SLy7) in Ref. [52], or
(SkO, SkO’) in Ref. [73].
Interestingly, all but one parameterization of the cen-
tral Skyrme interaction found in the literature set the
coupling constants of the J2 terms either to their Skyrme
force value (A1) or strictly to zero. The exception is
Ref. [38] by Tondeur, where an independent fit of the cou-
pling constants of the J2 terms was attempted, making
explicit reference to a DME interpretation of the energy
functional.
Setting the coupling constants of a term to zero when
one does not know how to adjust its parameters is of
course an acceptable practise when permitted by the cho-
sen framework. For Skyrme interactions fitted without
the J2 terms, the situation becomes confusing when one
looks at deformed nuclei and any situation that breaks
time-reversal invariance. First of all, Galilean invariance
of the energy functional dictates that the coupling con-
stant of the s · T terms is also set to zero, as already
indicated by the presentation of the energy functional
in Eq. (23). Second, using a DME interpretation of the
Skyrme energy functional in one place, but the interre-
lations from the two-body Skyrme force in all others is
not entirely satisfactory. Many authors who drop the J2
terms rarely show scruples to keep most of the time-odd
terms in the Skyrme energy functional (23) with coupling
constants Ast and A
t from (A1), although they are not
at all constrained in the common fit protocols employ-
ing properties of even-even nuclei and spin-saturated nu-
clear matter. For a list of exceptions see Sect. II.A.2.d of
Ref. [29]. An alternative is to set up a hierarchy of terms,
as it was attempted by Bonche, Flocard and Heenen in
their mean-field and beyond codes, which set A∆st = 0 in
addition to the coupling constant of the J2 terms, as all
three terms have in common that they couple two Pauli
matrices with two derivatives in different manners, see
the footnote on page 129 of [74].
There are also inconsistent applications of parameter-
izations without J2 − s · T terms to be found in the lit-
erature. For example, almost all applications of Skyrme
interactions to the Landau parameters gℓ and g
ℓ and the
properties of polarized nuclear matter, include the con-
tribution from the s · T terms, although it should be
dropped for parameterizations fitted without J2 terms.
Similarly, most RPA and QRPA codes include them for
simplicity, see the discussion in Refs. [75, 76, 77].
As it is relevant for the subject of the present paper,
we also mention another generalization of the Skyrme in-
teraction that invokes the interpretation of the Skyrme
energy functional in a DME framework. The spin-orbit
force (16) fixes the isospin mix of the corresponding
terms in the Skyrme energy functional (23) such that
A∇J0 = 3A
1 (A2). There are a few parameterizations
as MSkA [78], SkI3 and SkI4 [79], SkO and SkO’ [73] and
SLy10 [52] that liberate the isospin degree of freedom in
the spin-orbit functional. A DME interpretation of the
energy functional is mandatory for this generalization. It
is motivated by the better performance of standard rela-
tivistic mean-field models for the kink of the charge radii
in Pb isotopes. Note that the standard RMF models are
effective Hartree theories without exchange terms, and
that the standard Lagrangians have very limited isovec-
tor degrees of freedom [29], both of which supress a strong
isospin dependence of the spin-orbit interaction. It is in-
teresting to note that the existing fits of Skyrme energy
functionals with generalized spin-orbit interaction do not
improve spin-orbit splittings [14].
III. THE FITS
A. General remarks
In order to study the effect of the J2 terms, we have
built a set of 36 effective interactions that systematically
cover the region of coupling constants CJ0 and C
1 that
give a reasonable description of finite nuclei in connec-
tion with the standard central and spin-orbit Skyrme
forces. At variance with the perturbative approach used
in Refs. [37, 49], each of these parameterizations has been
fitted separately, following a procedure nearly identical to
that used for the construction of the SLy parameteriza-
tions [51, 52], so that we can keep the connection between
the new fits with parameterizations that have been ap-
plied to a large variety of observables and phenomena.
The Saclay-Lyon fit protocol focuses on the simultaneous
reproduction of nuclear bulk properties such as binding
energies and radii of finite nuclei and the empirical char-
acteristics of infinite nuclear matter (i.e. symmetric and
pure neutron matter). The latter establishes an impor-
tant, though highly idealized, limiting case as it permits
to confront the energy functional with calculations from
first principles using the bare nucleon-nucleon force [80].
The region of effective coupling constants (CJ0 , C
1 ) of
the J2 terms acting in spherical nuclei, as defined in
Eq. (28), that we will explore, is shown in Fig. 1. The
parameterizations are labeled TIJ , where indices I and
J refer to the proton-neutron (β) and like-particle (α)
coupling constants in Eq. (32) such that
α = 60 (J − 2) MeV fm5,
β = 60 (I − 2) MeV fm5. (36)
The corresponding values of CJt can be obtained through
Eq. (33) or from Fig. 1. On the one hand, we cover
the positions of the most popular existing Skyrme in-
teractions that take the J2 terms from the central force
into account, which are SLy5 [52], SkP [65], Zσ [72],
T6 [71], SkO’ [73] and BSk9 [81]. On the other hand,
among recent parameterizations including a tensor term,
i.e. Skxta [48], Skxtb [48, 82] as well as those published
by Colò et al. [49] and Brink and Stancu [50], most fall
in a region of negative CJ1 and vanishing C
0 , that is to
the lower left of Fig. 1. Parameterizations of this region,
which also includes a part of the triangle advocated in
the perturbative study of Stancu et al. [37], gave unsat-
isfactory results for many observables. Moreover, when
attempting to fit parameterizations with large negative
coupling constants, we sometimes obtained unrealistic
single-particle spectra or even ran into the instabilities al-
ready mentioned and outlined in appendix B. Parameter-
izations further to the lower and upper right also have un-
realistic deformations properties. The contribution from
the J2 terms vanishes for T22, which will serve as the
reference point. For the parameterizations T2J , only the
proton-proton and neutron-neutron terms in Ht are non-
zero (β = 0), while for the parameterizations TI2, only
the proton-neutron term in Ht contributes (α = 0). Note
that the earlier parameterizations T6 and Zσ have a pure
like-particle J2 terms as a consequence of the constraint
x1 = x2 = 0 employed for both (and most other early
parameterizations of Skyrme’s interaction).
B. The fit protocol and procedure
The list of observables used to construct the cost
function χ2 minimized during the fit (see Eq. (4.1) in
Ref. [51]) reads as follows: binding energies and charge
radii of 40Ca, 48Ca, 56Ni, 90Zr, 132Sn and 208Pb; the bind-
ing energy of 100Sn; the spin-orbit splitting of the neutron
3p state in 208Pb; the empirical energy per particle and
density at the saturation point of symmetric nuclear mat-
ter; and finally, the equation of state of neutron matter
as predicted by Wiringa et al. [16].
Furthermore, some properties of infinite nuclear mat-
ter are constrained through analytic relations between
-60 -30  0  30  60  90  120  150  180  210  240  270
CJ0 [MeV fm
BSk9T6
Skxta
Skxtb
Brink
FIG. 1: Values of CJ0 and C
1 for our set of parameteriza-
tions (circles). Diagonal lines indicate α = CJ0 +C
1 = 0 (pure
neutron-proton coupling) and β = CJ0 − C
1 = 0 (pure like-
particle coupling). Values for classical parameter sets are also
indicated (dots), with SLy4 representing all parameterizations
for which J2 terms have been omitted in the fit. Recent pa-
rameterizations with tensor terms are indicated by squares.
coupling constants in the same manner as they were in
Refs. [51, 52]: the incompressibility modulus K∞ is kept
at 230 MeV, while the volume symmetry energy coeffi-
cient aτ is set to 32 MeV. The isovector effective mass, ex-
pressed through the Thomas-Reiche-Kuhn sum rule en-
hancement factor κv, is taken such that κv = 0.25.
When using a single density-dependent term in the
central Skyrme force (10), the isoscalar effective mass
m∗0 cannot be chosen independently from the incompress-
ibility modulus for a given exponent α of ρ0. We fol-
low here the prescription used for the SLy parameteriza-
tions [51, 52] and use α = 1/6, which leads to an isoscalar
effective mass close to 0.7 in units of the bare nucleon
mass for all TIJ parameterizations. This value allows for
a correct description of dynamical properties, as for ex-
ample the energy of the giant quadrupole resonance [83].
Using such a protocol we cannot reproduce the isovec-
tor effective mass consistent with recent ab-initio predic-
tions [84]. Regarding the present exploratory study of the
tensor terms this is not a critical limitation, in particular
as the influence of this quantity on static properties of
finite nuclei turns out to be small.
There are three modifications of the fit protocol com-
pared to [51, 52]. The obvious one is that the values
for CJ0 and C
1 are fixed beforehand as the parameters
that will later on label and classify the fits. The second
is that we have added the binding energies of 90Zr and
100Sn to the set of data. Indeed, we observed that the
latter nucleus is usually significantly overbound when not
   -60   0   60   120   180   240
(T11)
β [MeV fm5] α [MeV fm5]
FIG. 2: Values of the cost function χ2 as defined in the fit
procedure, for the set of parameterizations TIJ . The label
“T11” indicates the position of this parameterization in the
(α,β)-plane as obtained from Eqs. (36). Contour lines are
drawn at χ2 = 11, 12, 15, 20, 25, and 30. The minimum
value is found for T21 (χ2 = 10.05), the maximum for T61
(χ2 = 37.11).
included in the fit. The third is that we have dropped
the constraint x2 = −1 that was imposed on the SLy pa-
rameterizations [51, 52] to ensure the stability of infinite
homogeneous neutron matter against a transition into
a ferromagnetic state. On the one hand, this stability
criterion is completely determined by the coupling con-
stants of the time-odd terms in the energy functional [76],
that we do not want to constrain here, accepting that
the parameterizations might be of limited use beyond
the present study. On the other hand, the tensor force
brings many new contributions to the energy per parti-
cle of polarized nuclear matter that lead to a much more
complex stability criterion. We postpone the entire dis-
cussion concerning the stability in polarized systems in
the presence of a tensor force to future work that will
also address finite-size instabilities [84]. It also has to be
stressed that the actual stability criterion, as all proper-
ties of the time-odd part of the Skyrme energy functional,
depends on the choices made for the interpretation of its
coupling constants, i.e. antisymmetrized vertex or den-
sity functional [76].
The properties of the finite nuclei entering the fit are
computed using a Slater determinant without taking
pairing into account. The cost function χ2 was mini-
mized using a simulated annealing algorithm. The an-
nealing schedule was an exponential one, with a charac-
teristic time of 200 iterations (also referred to as “simu-
lated quenching”) Thus, assuming a reasonably smooth
cost function, we strive to obtain satisfactory convergence
to its absolute minimum in a single run, allowing a sys-
tematic and straightforward production of a large series
of forces. The coupling constants for all 36 parameteri-
zations can be found in the Physical Review archive [85].
Figure 2 displays the value of χ2 after minimization as
-150 -120 -90 -60 -30  0  30  60  90  120  150  180
BJ0 [MeV fm
FIG. 3: The contributions from the tensor force BJ0 and B
to the effective coupling constants of the J2 term at sphericity.
Diagonal lines as in Fig. 1. The diagonal where BJ0 + B
αT = 0 (pure proton-neutron contribution) additionally cor-
responds to an isospin-singlet force with to ≡ U = 0.
a function of the recoupled coupling constants α and β.
The first striking feature is the existence of a “valley” at
β = 0, i.e. a pure like-particle tensor term ∼ (J2n + J
The abrupt rise of χ2 around this value can be attributed
to the term depending on nuclear binding energies, as
sharp variations of energy residuals can be seen between
neighboring magic nuclei with functionals of the T6J se-
ries (β = 240). For example, 48Ca and 90Zr tend to be
significantly overbound in this case. We will come back
later to discussing the implications for the quality of the
functionals.
C. General properties of the fits
The coupling constants of the energy functional for
spherical nuclei (28) obtained for T22 are very similar to
those of SLy4, except for a slight readjustment coming
from the inclusion of the binding energies of 90Zr and
100Sn in the fit as well as the abandoned constraint on
x2. With its value of −0.945, the x2 obtained for T22
still stays close to the value −1 enforced for SLy4, which
confirms that this is not too severe a constraint for pa-
rameterizations without effective J2 terms at sphericity.
Increasing the effective tensor term coupling constants
CJt , however, the values for x2 start to deviate strongly
from the region around −1, which is to a large extent due
to the feedback from the contribution of the J2 terms to
the surface and surface symmetry energy coefficients in
the presence of constraints on isoscalar and isovector ef-
    -60    0
    60    120
    180    240
-60  
120  
180  
240  
W0 [MeV fm
(T11)
β [MeV fm5]α [MeV fm5]
W0 [MeV fm
FIG. 4: Value of spin-orbit coupling constant W0 for each
of the parameterizations TIJ , vs. indices I and J (The
“(T11)” label indicates the position of this parameterization
in the (α, β)-plane). The contour lines differ by 20 MeV fm5.
The values plotted here range from 103.7 MeV fm5 (T11) to
195.3 MeV fm5 (T66).
fective masses, all of which also depend on x2. A more de-
tailed discussion of the contribution of the J2 terms to the
surface energy coefficients will be given elsewhere [41].
From the constrained coupling constants CJ0 and C
the respective contributions BJ0 and B
1 from the tensor
force can be deduced afterwards using the expressions
given in Sect. II C. Their values, shown in Fig. 3, are
less regularly distributed, which is a consequence of the
the non-linear interdependence of all coupling constants.
Still, a general trend can be observed, such that all
parameterizations are shifted towards the “south-west”
compared to Fig. 1. In turn, this indicates that the con-
tribution from the central Skyrme force always stays in
the small region outlined by SkP, SLy5, Zσ, etc in Fig. 1,
with values that range between 28 and 104 MeV fm5 for
AJ0 and between 38 to 62 MeV fm
5 for AJ1 , respectively.
This justifies a posteriori to use the tensor force as a
motivation to decouple the J2t terms from the central
part of the effective Skyrme vertex. We note in passing
that all our parameterizations TI4 correspond to an al-
most pure proton-neutron or isospin-singlet tensor force,
i.e. the term ∝ te in Eq. (20), as they are all located close
to the αT = 0 line.
We also find a particularly strong and systematic vari-
ation of the coupling constant W0 of the spin-orbit force,
which varies from W0 = 103.7 MeV fm
5 for T11 to
W0 = 195.3 MeV fm
5 for T66, see Fig. 4. This variation
is of course correlated to the strength of the tensor force.
As already shown, the tensor force has the tendency to
reduce the spin-orbit splittings in spin-unsaturated nu-
clei. To maintain a given spin-orbit splitting in such a
nucleus, the spin-orbit coupling constant W0 has to be
increased.
Ni, T440.015
0.010
0.005
0.000
ρsat/2
0 1 2 3 4 5 6 7
r [fm]
-0.010
-0.005
0.000
0.005
0.010
0.015
0.020
Jn [fm
FIG. 5: (Color online) Radial component of the neutron
spin-orbit current for the chain of Ni isotopes, plotted against
radius and neutron number N . The solid line on the base
plot indicates the radius where the total density has half its
saturation value.
IV. RESULTS AND DISCUSSION
The calculations presented below include open-shell
nuclei treated in the Hartree-Fock-Bogoliubov (HFB)
framework. In the particle-particle channel, we use a
zero-range interaction with a mixed surface/volume form
factor (called DFTM pairing in Ref. [86]). The HFB
equations were regularized with a cutoff at 60 MeV in
the quasiparticle equivalent spectrum [87]. The pair-
ing strength was adjusted in 120Sn with the particle-hole
mean field calculated using the parameter set T33. The
resulting strength was kept at the same value for all pa-
rameterizations, which is justified by the fact that the
effective mass parameters are the same. Moreover, we
thus avoid including, in the adjustment of the pairing
strength, local effects linked with changes in details of
the single-particle spectrum.
A. Spin-orbit currents and potentials
As a first step in the analysis of the role of the tensor
terms and their interplay with the spin-orbit interaction
in spherical nuclei, we analyze the spin-orbit current den-
sity and its relative contribution to the spin-orbit poten-
tial. We choose the chain of nickel isotopes, Z = 28, as it
covers the largest number of spherical neutron shells and
subshells (N = 20, 28, 40 and 50) of any isotopic chain,
two of which are spin-saturated (N = 20 and 40), while
the other two are not. Figure 5 displays the radial com-
ponent of the neutron spin-orbit current Jn for isotopes
from the proton to the neutron drip-lines. The calcula-
tions are performed with T44, but the spin-orbit current
is fairly independent from the parameterization. Starting
from N = 20, which corresponds to a completely filled
and spin-saturated sd-shell, the next magic number at
N = 28 is reached by filling the 1f7/2 shell, which leads
to the steeply rising bump in the plot of Jn in the fore-
ground, peaked around r ≃ 3.5 fm. Then, from N = 28
to N = 40 the rest of the fp shell is filled, which first
produces the small bump at small radii that corresponds
to the filling of the 2p3/2 shell, but ultimately leads to
a vanishing spin-orbit current when the 1f and 2p lev-
els are completely filled for the N = 40 isostope, visible
as the deep valley in Fig. 5. Adding more neutrons, the
filling of the 1g9/2 shell leads again to a strong neutron
spin-orbit current at N = 50. For the remaining isotopes
up to the neutron drip line, the evolution of Jn is slower
with the filling of the 2d and 3s orbitals.
A few further comments are in order. First, the spin-
orbit current clearly reflects the spatial probability dis-
tribution of the single-particle wave function in pairs of
unsaturated spin-orbit partners. Within a given shell,
the high-ℓ states contribute at the surface, represented
by the solid line on the base of Fig. 5, while low-ℓ states
contribute at the interior. The peak from the high-ℓ or-
bitals, however, is always located on the inside of the nu-
clear surface, as defined by the radius of half saturation
density. Second, within a given shell, the largest contri-
butions to the spin-orbit current density obviously come
from the levels with largest ℓ, as they have the largest
degeneracy factors in (27), and because they do not have
nodes, which leads to a single, sharply peaked contribu-
tion. Third, the spin-orbit current is not exactly zero
for nominally “spin-saturated” nuclei, exemplified by the
N = 20 and N = 40 isotopes in Fig. 5, as the radial
single-particle wave functions are not exactly identical
for all pairs of spin-orbit partners, which is a necessary
requirement to obtain Jn = 0 at all radii (Cf. the example
of the ν 2d states in 132Sn in Fig. 16 below). Fourth, pair-
ing and other correlations will always smooth the fluctu-
ations of the spin-orbit current with nucleon numbers, as
levels in the vicinity of the Fermi energy will never be
completely filled or empty.
Next, we compare the contributions from the tensor
terms and from the spin-orbit force to the spin-orbit po-
tentials of protons and neutrons, Eq. (35). The contri-
butions from the tensor force to the spin-orbit poten-
tial are proportional to the spin-orbit currents of pro-
tons and neutrons. For the Ni isotopes, the proton spin-
orbit current is very similar to that of the neutrons at
N = 28 displayed in Fig. 5. For the parameterization
T44 we use here as an example, we have contributions
from both proton and neutron spin-orbit currents, which
come with equal weights. Their combined contribution
to the spin-orbit potential of the neutronWn might be as
large as 4 MeV, see Fig. 6. This is more than a third of
the maximum contribution from the spin-orbit force to
Wn, see Fig. 7. The latter is proportional to a combina-
tion of the gradients of the proton and neutron densities,
2∇ρn(r) + ∇ρp(r), see Eq. (35). As a consequence, it
has a smooth behavior as a function of particle number,
Ni, T44
ρsat/2
0 1 2 3 4 5 6 7
r [fm]
  20 N
Wn,t [MeV fm]
FIG. 6: (Color online) Contribution from the tensor terms to
the neutron spin-orbit potential for the chain of Ni isotopes
as obtained with the parameterization T44. The solid line on
the base plot indicates the radius where the isoscalar density
ρ0 crosses half its saturation value.
Ni, T44
ρsat/2
0 1 2 3 4 5 6 7
r [fm]
  20 N
Wn,so [MeV fm]
FIG. 7: (Color online) Contribution from the spin-orbit force
to the neutron spin-orbit potential for the chain of Ni isotopes
as obtained with the parameterization T44. The solid line on
the base plot indicates the radius where the isoscalar density
ρ0 crosses half its saturation value.
with slowly and monotonically varying width, depth and
position. Only limited local variations can be seen on
the interior due to small variations of the density profile
originating from the successive filling of different orbits.
Furthermore, one can easily verify that the contribution
from the spin-orbit force is peaked at the surface of the
nucleus (the solid line on the base plot). The strongest
variation of the depth of this potential occurs just be-
fore the neutron drip line at N = 62, where is becomes
wider and shallower due to the development of a diffuse
Ni, T44
ρsat/2
0 1 2 3 4 5 6 7
r [fm]
Wn [MeV fm]
FIG. 8: (Color online) Total neutron spin-orbit potential for
the chain of Ni isotopes as obtained with the parameterization
T44. The solid line on the base plot indicates the radius where
the isoscalar density ρ0 crosses half its saturation value.
Ni, T44
ρsat/2
0 1 2 3 4 5 6 7
r [fm]
Wp [MeV fm]
FIG. 9: (Color online) Total proton spin-orbit potential for
the chain of Ni isotopes as obtained with the parameterization
T44. The solid line on the base plot indicates the radius where
the isoscalar density ρ0 crosses half its saturation value.
neutron skin, which reduces the gradient of the neutron
density [6, 7, 8].
Adding the contributions from the proton and neutron
tensor terms to that from the spin-orbit force, the total
neutron spin-orbit potential for neutrons in Ni isotopes
is shown in Fig. 8. For the parameterization T44 used
here (and most others in the sample of parameterizations
used in this study) the dominating contributions from
the spin-orbit and tensor forces to the spin-orbit poten-
tial are of opposite sign. For Ni isotopes, Jp is always
quite large, while Jn varies as shown in Fig. 5. Notably,
both are peaked inside of the surface. When examining
 20  30  40  50  60
1g9/2
2p1/2
1f5/2
2p3/2
1f7/2
1d3/2
2s1/2
1d5/2
2d3/2
3s1/2
2d5/2
1g9/2
2p1/2
1f5/2
2p3/2
1f7/2
2s1/2
1d3/2
FIG. 10: (Color online) Single-particle spectra of neutrons
(upper panel) and protons (lower panel) for the chain of Ni
isotopes, as obtained with the parameterization T22 with van-
ishing combined J2 terms. The thick solid line in the upper
panel denotes the Fermi energy for neutrons.
the combined contribution from the spin-orbit and tensor
forces to the spin-orbit potential (35), one must keep in
mind that they are peaked at different radii. Moreover,
the variation of tensor-term coupling constants among a
set of parameterizations implies a rearrangement of the
spin-orbit term strength, as will be discussed later. As a
consequence, taking into account the tensor force modi-
fies the width and localization of the spin-orbit potential
Wq(r) much more than it modifies its depth through the
variation of the spin-orbit currents.
Our observations also confirm the finding of Otsuka
et al. [46] that the spin-orbit splittings might be more
strongly modified by the tensor force than they are by
neutron skins in neutron-rich nuclei through the reduc-
tion of the gradient of the density.
Figure 9 shows the spin-orbit potential of the protons
for the chain of Ni isotopes. Here, the contribution from
the spin-orbit force has a larger contribution coming from
the gradient of the proton density that just grows with
the mass number, without being subject to varying shell
fluctuations. The same holds for the proton contribution
from the tensor terms. Only the neutron contribution
from the tensor terms varies rapidly, proportional to Jn
displayed in Fig. 5, which has a very limited effect on the
total spin-orbit potential, though.
With that, we can examine how the tensor terms af-
fect the evolution of single-particle spectra. To that end,
Fig. 10 shows the single-particle energies of protons and
neutrons along the chain of Ni isotopes for the parameter-
ization T22 with vanishing combined tensor terms, which
 20  30  40  50  60
1g9/2
2p1/2
1f5/2
2p3/2
1f7/2
1d3/2
2s1/2
1d5/2
2d3/2
3s1/2
2d5/2
1g9/2
2p1/2
1f5/2
2p3/2
1f7/2
2s1/2
1d3/2
FIG. 11: (Color online) The same as Fig. 10, obtained with
T44 with proton-neutron and like-particle tensor terms of
equal strength.
will serve as a reference, while Fig. 11 shows the same for
the parameterization T44 with proton-neutron and like-
particle tensor terms of equal strength. For the latter,
the variation of the neutron spin-orbit current with N in-
fluences both neutron and proton single-particle spectra.
The effect of the tensor terms is subtle, but clearly visi-
ble: for T22, the major change of the single-particle en-
ergies is their compression with increasing mass number,
while for T44 the level distances oscillate on top of this
background correlated to the neutron shell and sub-shell
closures at N = 20, 28, 40 and 50. As shown above, the
neutron spin-orbit current vanishes for N = 20, where
it consequently has no effect on the spin-orbit potentials
and splittings. By contrast, the neutron spin-orbit cur-
rent is large for N = 28 and 50, where its contribution
to the spin-orbit potential reduces the splittings from the
spin-orbit force.
The strong variation of the spin-orbit current with
nucleon numbers is typical for light nuclei up to about
mass 100. For heavier nuclei, its variation becomes much
smaller. This is exemplified in Fig. 12 for the neutron
spin-orbit current in the chain of Pb isotopes. There
remain the fast fluctuations at small radii which we al-
ready saw for the Ni isotopes and that reflect the subse-
quent filling of low-ℓ levels with many nodes, but which
have a very limited impact on the spin-orbit splittings
when fed into the spin-orbit potential. The dominating
peak of the spin-orbit current, just beneath the surface
shows only small fluctuations, as the overlapping spin-
orbit splittings of levels with different ℓ never give rise to
a spin-saturated configuration in heavy nuclei.
Pb, T44
0.010
0.005
0.000
ρsat/2
0 2 4 6 8 10r [fm]
-0.005
0.000
0.005
0.010
0.015
Jn [fm
FIG. 12: (Color online) Radial component of the Neutron
spin-orbit current for the chain of Pb isotopes plotted in the
same manner as in Fig. 5.
Note that both the spin-orbit current J and the spin-
orbit potential are exactly zero at r = 0 as they are
vectors with negative parity.
B. Single-particle energies
As a next step, we analyze the modifications that the
presence of J2 terms brings to single-particle energies in
detail. Before we do so, a few general comments on the
definition and interpretation of single-particle energies
are in order. From an experimental point of view, empir-
ical single-particle energies in a doubly-magic nucleus are
determined as the separation energies between the even-
even doubly magic nucleus and low-lying states in the
adjacent odd-A nuclei, i.e. they are differences of bind-
ing energies. In nuclear models, however, it is customary
to discuss shell structure and single-particle energies in
terms of the spectrum of eigenvalues ǫi of the Hartree-
Fock mean-field Hamiltonian (in even-even nuclei), as we
have done already in Figs. 10 and 11:
ĥΦi = ǫiΦi . (37)
In the nuclear EDF approach without pairing, the ref-
erence state is directly constructed as a Slater determi-
nant of eigenstates of ĥ; hence, the corresponding eigen-
values are directly connected to the fundamental build-
ing blocks of the theory and reflect the mean field in
the nucleus. The density of single-particle levels around
the Fermi surface drives the magnitude of pairing cor-
relations, the relative distance of single-particle levels at
sphericity and their quantum numbers determine to a
large extent the detailed structure of the deformation en-
ergy landscape which in turn, determines the collective
spectroscopy. The spectroscopic properties of even-even
nuclei, in particular when they exhibit shape coexistence,
provide valuable benchmarks for the underlying single-
particle spectrum [56]. The link between the spectrum
of single-particle energies on the one hand and the col-
lective excitation spectrum on the other hand, however,
always remains indirect.
On the other hand, “single-particle” states near the
Fermi level of a magic nucleus can be observed by adding
or removing a particle in one of these states, and thus cor-
respond to the ground and excited states of the neighbor-
ing odd-mass nuclei. Assuming an infinitely stiff magic
core, which is neither subject to any rearrangement or po-
larization, nor to any collective excitations following the
addition (or removal) of a nucleon, the separation ener-
gies with the states in the odd-mass neighbors are equal
to the single-particle energies as defined through (37).
This highly idealized situation is modified by static [88]
and dynamic [89, 90] correlations, often called “core po-
larization” (see chapter 7 of Ref. [91]) and “particle-
vibration coupling” (see section 9.3.3 of Ref. [92]) in the
literature, that alter the separation energies. The main
effect of the correlations is that they compress the spec-
trum, pulling down the levels from above the Fermi en-
ergy and pushing up those from below. The gross fea-
tures, i.e. the ordering and relative placement of single-
particle states, however, are more weakly affected by
correlations. The particle-vibration coupling, however,
is also responsible for the fractionization of the single-
particle strength. When the latter is too large, the naive
comparison between the calculated ǫi given by Eq. (37)
and the energy of the lowest experimental state with the
same quantum numbers is not even qualitatively mean-
ingful anymore [48].
We mention that a part of the static correlations orig-
inate from the non-vanishing time-odd densities in the
mean-field ground-state of an odd-A nucleus, that also
cannot be truly spherical, so that the complete energy
functional from Eq. (23) should be considered in a fully
self-consistent calculation of the separation energies.
The effective single-particle energies that are used to
characterize the underlying shell structure in the inter-
acting shell model [93] have a slightly different mean-
ing. Their definition usually renormalizes polarization
and particle-vibration coupling effects around a doubly-
magic nucleus whereas their evolution is discussed in
terms of monopole shifts [94]. A collection of effective
single-particle energies and their evolution was collected
by Grawe [95, 96]. Note that the SkX parameterization
of the Skyrme energy functional by Brown and its vari-
ants [48, 97] were constructed aiming at a description of
effective single-particle energies along these lines.
It should be kept in mind that the obvious, coarse dis-
crepancies between the calculated spectra of ǫµ and the
empirical single-particle energies are often larger than the
uncertainties coming from the missing correlations, as
long as one observes some elementary precautions. We
took care to ensure that the states used in the analy-
sis below were one-quasiparticle states weakly coupled
to core phonons. First, we checked that the even-even
nucleus of interest could be described as spherical, indi-
cated by a sufficiently high-lying 2+ state. Second, we
avoided all levels which were obviously correlated with
the energies of 2+ states in the adjacent semi-magic se-
ries, as this indicates strong coupling with core excita-
tions. Finally, we carefully examined states, lying above
the 2+ energy and/or twice the pairing gap of adjacent
semi-magic nuclei, in order to eliminate those more accu-
rately described as an elementary core excitation coupled
to one or more quasiparticles, which generally appear as
a multiplet of states. We did not attempt to use energy
centroids calculated with use of spectroscopic factors, as
these are not systematically available. Indeed, our re-
quirement is that if some collectivity is present, it should
be similar among all nuclei considered, in order to be eas-
ily subtracted out. Empirical single-particle levels shown
below are determined from the lowest states having given
quantum numbers in an odd-mass nucleus.
1. Spin-orbit splittings
The primary effect one expects from a tensor term
is that it affects spin-orbit splittings by altering the
strength of the spin-orbit field in spin-unsaturated nuclei,
according to Eq. (35). One should remember, though,
that the spin-orbit coupling itself is readjusted for each
pair of coupling constants CJ0 , and C
1 . The effect of this
readjustment is generally opposite to that of the variation
of the isoscalar tensor term coupling constant. It should
thus be stressed that the effects described result from the
balance between the variation of tensor and spin-orbit
terms, which for most of our parameterizations pull into
opposite directions.
Common wisdom states that the energy spacing be-
tween levels that are both above or both below the magic
gap are not much affected by correlations, even when
their absolute energy changes; hence it is common prac-
tice to confront only the spin-orbit splittings between
pairs of particle or hole states with calculated single-
particle energies from the spherical mean field. Figure 13
shows the relative error of single-particle splitting of such
levels for doubly-magic nuclei throughout the chart of nu-
clei. The calculated values are typically 20 to 60% larger
than the experimental ones, with the exception of 16O,
where the splittings of the neutron and proton 1p states
are acceptably reproduced at least for the parameteri-
zations T22, T24 and T42, i.e. those with the weakest
tensor terms in the sample.
It is noteworthy that the calculated splittings depend
much more sensitively on the tensor terms for light nuclei
with spin-saturated shells (protons and neutrons in 16O,
protons in 90Zr) than for the heavy doubly-magic 132Sn
and 208Pb, which are quite robust against a variation of
the tensor terms. The reason will become clear below.
ν1p π1p
132Sn
ν2d π2d
208Pb
ν3p π2d
FIG. 13: (Color online) Relative error of the spin-orbit split-
tings in doubly-magic nuclei for ℓ ≤ 2 levels.
2. Connection between tensor and spin-orbit terms
The finding that our parameterizations systematically
overestimate the spin-orbit splittings deserves an expla-
nation. It was earlier already noted that all standard
Skyrme interactions, including the SLy parameteriza-
tions that share our fit protocol, have an unresolved trend
that overestimates the spin-orbit splittings in heavy nu-
clei [14, 29, 98]. Adding the tensor terms, however,
further deteriorates the overall description of spin-orbit
splittings, instead of improving it. It is particularly dis-
turbing that the spin-orbit splitting of the 3p level in
208Pb that was used to constrain W0 in the fit is overes-
timated by 30 to 40%, which is larger than the relative
tolerance of 20% included in the fit protocol. In fact,
it turns out that the coupling constant W0 of the spin-
orbit force is more tightly constrained by the binding
energies of light nuclei than by this or any other spin-
orbit splitting. In the HF approach used during the fit,
the structure of 40Ca, 48Ca, and 56Ni differs by the occu-
pation of the neutron and proton 1f7/2 levels. First, we
have to note that the terms in the energy functional that
contain the spin-orbit current play an important role for
the energy difference between 40Ca and 56Ni. The com-
bined contribution from the tensor and spin-orbit terms
varies from a near-zero value in the spin-saturated 40Ca
to about −60 MeV in 56Ni for all our parameterizations,
which is a large fraction of the −142 MeV difference in
total binding energy between both nuclei. The Z = 40
subshell and Z = 50 shell are another example of abrupt
variation of the spin-orbit current with the filling of the
1g9/2 level, which strongly affects the relative binding
energy of N = 50 isotones 90Zr and 100Sn. Second, the
fit to phenomenological data can take advantage of the
large relative variation of these terms to mock up missing
physics in the energy functional that should contribute
to the energy difference, but that is absent in it. The
consequence will be a spurious increase of the spin-orbit
and tensor term coupling constants. The resulting energy
functional will correctly describe the mass difference, but
not the physics of the spin-orbit and tensor terms.
In order to test the above interpretation, we performed
a refit of selected TIJ parameterizations without taking
into account the masses of 40Ca, 48Ca, 56Ni and 90Zr in
the fit procedure. In the resulting parameterizations, the
spin-orbit coefficient W0 is typically 20% lower than in
the original ones. As a consequence, the empirical value
for the spin-orbit splitting of the neutron 3p level in 208Pb
is met well within tolerance, at the price of binding en-
ergy residuals in light nuclei being unacceptably large,
i.e. 56Ni being underbound by 5 MeV while 40Ca and
90Zr are overbound by up to 10 MeV. While the global
trend of the spin-orbit splittings shown in Fig. 13 is enor-
mously improved with these fits, in particular for heavy
nuclei, the overall agreement of the single-particle spectra
with experiment is not, so that we had to discard these
parameterizations. This finding hints at a deeply rooted
deficiency of the Skyrme energy functional. The spin-
orbit and, when present, tensor terms indeed do simu-
late missing physics of the energy functional at the price
of unrealistic spin-orbit splittings. This also hints why
perturbative studies, as those performed in [37, 49] give
much more promising results than what we will find be-
low with our complete refits. We will discuss mass resid-
uals in more detail in Sect. IVC1 below.
During the fit, the masses of light nuclei do not only
compromise the spin-orbit splittings, they also establish
a correlation betweenW0 and C
0 in all our parameteriza-
tions. The combined spin-orbit and spin-current energy
of a given spherical nucleus (N,Z) is given by (keeping
only the isoscalar part since we shall focus on the N = Z
nuclei 40Ca and 56Ni)
0 (N,Z) = C
0 (N,Z) + C
0 (N,Z) (38)
I∇J0 (N,Z) =
d3r ρ0∇ · J0 (39)
IJ0 (N,Z) =
d3r J20 . (40)
-30 -15  0  15  30  45  60  75  90 105 120
J [MeV fm5]
FIG. 14: Correlation between the values of spin-orbit cou-
pling constant C∇J0 and the isoscalar spherical effective spin-
current coupling constant CJ0 . Dots: values for the actual
parameterizations TIJ , solid line: trend estimated through
Eq. (42) (see text).
The difference of E
0 between
56Ni and 40Ca
= ∆Espin (41)
turns out to be fairly independent from the parameteri-
zation. Averaged over all 36 parameterizations TIJ used
here, ∆Espin has a value of −58.991MeV with a standard
deviation as small as 3.202 MeV, or 5.4%.
The integrals in Eqs. (39,40) are fairly independent
from the actual parameterization. For a rough estimate,
we can replace them in Eq. (38) by their average values.
Plugged into Eq. (41) this yields
C∇J0 =
∆Espin − CJ0 〈I
− IJ0
〈I∇J0 (
56Ni)− I∇J0 (
40Ca)〉
. (42)
Figure 14 compares the values of C∇J0 as obtained
through (42) with the values for the actual parameter-
izations. The estimate works very well, which demon-
strates that C∇J0 = −
W0 and C
0 are indeed correlated
and cannot be varied independently within a high qual-
ity fit of the energy functional (28). As the combined
strength of the spin-orbit and tensor terms in the energy
functional is mainly determined by the mass difference
of the two N = Z nuclei 40Ca and 56Ni, the spin-orbit
coupling constant W0 depends more or less linearly on
the isoscalar tensor coupling constant CJ0 , while for all
practical purposes it is independent from the isovector
one, see also Fig. 4 above.
3. Splitting of high-ℓ states and the role of the radial form
factor
As stated above, it is common practice to confront only
the spin-orbit splittings between pairs of particle or hole
states with calculated single-particle energies from the
spherical mean field. The spin-orbit splitting of intruder
states is rarely examined. Figure 15 displays the relative
ν1f π1f
132Sn
ν1h π1g
208Pb
ν1i π1h
FIG. 15: (Color online) Spin-orbit splittings of high-ℓ levels
in magic nuclei across the Fermi energy. The calculated values
are less robust against correlation effects than those shown in
Fig. 13 and have to be interpreted with caution (see text).
deviation of the spin-orbit splittings of the intruder states
with ℓ ≥ 3 that span across major shell closures and
are thus given by the energy difference of a particle and
a hole state. These splittings are not “safe”, i.e. they
can be expected to be strongly decreased by polarization
and correlation effects [88, 89, 90]. To leave room for
this effect, a mean-field calculation should overestimate
the empirical spin-orbit splittings. We observe, however,
that mean-field calculations done here give values that
are quite close to the experimental ones, or even smaller
for parameterizations with large positive isoscalar tensor
coupling (cf. the evolution from T22 to T66).
This means that the spin-orbit splittings are not too
large in general, as might be concluded from Fig. 13,
but that there is a wrong trend of the splittings with ℓ
with the strength of the spin-orbit potential establishing
a compromise between the in-shell splittings of small ℓ
orbits that are too large and the across-shell splittings
of the intruders that are tentatively too small. In fact,
the levels in Fig. 15 obviously have in common that their
radial wave functions do not have nodes, while the levels
in Fig. 13 have one or two nodes, with the notable excep-
tion of the 1p levels in 16O, for which we also find smaller
deviations of the spin-orbit splittings than for the other
levels in Fig. 13.
Underestimating the spin-orbit splittings of intruder
levels has immediate and obvious consequences for the
performance of an effective interaction, as this closes the
magic gaps in the single-particle spectra and compro-
mises the predictions for doubly-magic nuclei, as we will
demonstrate in detail below. By contrast, the spin-orbit
splittings of the low-ℓ states within the major shells have
no obvious direct impact on bulk properties. Their devi-
ation from empirical data is less dramatic, as the typical
bulk observables discussed with mean-field approaches
are not very sensitive to them. It is only in applica-
tions to spectroscopy that their deficiencies become ev-
ident. It is noteworthy that the parameterization T22
without effective tensor terms at sphericity provides a
reasonable compromise between the tentatively underes-
timated splittings of the intruder levels shown in Fig. 15
and the tentatively overestimated splittings of the lev-
els within major shells shown in Fig. 13 above, while for
parameterizations with tensor terms this balance is lost.
There clearly is a proton-neutron staggering in Figs. 13
and 15, such that calculated proton splittings are rela-
tively smaller than the neutron ones. The effect appears
both when comparing proton and neutron levels with dif-
ferent ℓ in the same nucleus, and when comparing proton
and neutron levels with the same ℓ in the same or dif-
ferent nuclei (see the 1h levels in 132Sn and 208Pb). The
staggering for the intruder levels is even amplified for pa-
rameterizations with large proton-neutron tensor term,
as T62, T64 or T66. The effect is particularly promi-
nent for the heavy 132Sn and 208Pb with a large proton-
to-neutron ratio N/Z, which might hint at unresolved
isospin dependence of the spin-orbit interaction, although
alternative explanations that involve how single-particle
states in different shells should interact through tensor
and spin-orbit forces are possible as well, see also the
next paragraph.
Note that also the spin-orbit splittings of the low-ℓ
levels shown in Fig. 13 exhibit a staggering, which is
of smaller amplitude, though. It has been pointed out
by Skalski [99], that an exact treatment of the Coulomb
exchange term (compared to the Slater approximation
used here and nearly all existing literature) does indeed
slightly increase the spin-orbit splittings of protons across
major shells. This effect might give a clue to the stagger-
ing observed for the N = Z nucleus 56Ni, but the magni-
tude of the effect reported in [99] is too small to explain
the large staggering we find for the heavier N 6= Z nuclei.
Next, we use the example of 132Sn to demonstrate why
the spin-orbit splittings of nodeless high-ℓ states are more
sensitive to the tensor terms than low-ℓ states with one
or several nodes, see Fig. 16. The lower panel shows the
neutron spin-orbit potential in 132Sn for four different
parameterizations, while the upper panel shows selected
radial single-particle wave functions. The ν 1h11/2 and
 0  1  2  3  4  5  6  7  8  9  10
r [fm]
ν2d3/2
ν2d5/2
ν1h11/2
π1g9/2
FIG. 16: (Color online) Neutron spin-orbit potential (top)
and the radial wave function of selected orbitals (bottom) in
132Sn.
π 1g9/2 levels give the main contribution to the neutron
and proton spin-orbit currents in this nucleus, and con-
sequently to the tensor contribution to the spin-orbit po-
tential. Indeed, the largest differences between the spin-
orbit potentials from the chosen parameterizations are
caused by the varying contribution from the tensor terms
and appear for the region between 3 and 6 fm, where the
wave functions of the 1g and 1h states are peaked. This
region corresponds to the inner flank of the spin-orbit po-
tential well, while the outer flank is much less affected.
While the 1g and 1h wave functions are peaked at the in-
ner flank, the 2d orbitals have their node in this region.
Consequently, the splittings of the 1g and 1h levels are
strongly modified by the tensor terms, while those of the
2d orbitals are quite insensitive.
As a rule of thumb, the tensor contribution to the
spin-orbit potential in doubly-magic nuclei comes mainly
from the nodeless intruder states, which, when present, in
turn mainly affect their own spin-orbit splittings, leaving
the splittings of the low-ℓ states with one or more nodes
nearly unchanged for reasons of geometrical overlap.
We note in passing that the slightly different radial
wave functions of the 2d orbitals demonstrate nicely that
their contribution to the spin-orbit current, Eq. (27), can-
not completely cancel.
In fact, when regarding more specifically the evolution
of the spin-orbit potential between the parameterizations
T22 and T66, it is striking that for T66 it is essentially
narrowed and its minimum slightly pushed towards larger
radii, while its depth remains unaltered. Recalling that
] 132Sn, ν  (a)
1h centroid
Exp. T22
T42 T62 T24 T44 T64 T26 T46 T66
(7/2+)
(5/2+)
(1/2+)
(11/2-)
(3/2+)
(7/2-)
(3/2-)
(9/2-)
(1/2-)
1g7/2
2d5/2
3s1/2
2d3/2
1h11/2
2f7/2
3p3/2
1h9/2
3p1/2
2f5/2
132Sn, π  (b)
1g centroid
Exp. T22
T42 T62 T24 T44 T64 T26 T46 T66
(1/2-)
(9/2+)
(7/2+)
(5/2+)
(3/2+)
(11/2-)
2p1/2
1g9/2
1g7/2
2d5/2
2d3/2
3s1/2
1h11/2
FIG. 17: Single-particle energies in 132Sn for a subset of our
parameterizations. We also show the centroid of the intruder
levels, defined through Eq. (43) Top panel: neutron levels,
bottom panel: proton levels. A thick mark indicates the Fermi
level.
T66 shows a pathological behavior of too weak spin-orbit
splitting of the intruder states, it appears that a cor-
rect ℓ-dependence of spin-orbit splittings might require
to modify the radial dependence of the spin-orbit poten-
tial such that it becomes wider towards smaller radii.
This uncalled-for modification of the shape of the spin-
orbit field has previously been put forward by Brown
et al. [48] as an argument for a negative like-particle J2
coupling constant α. However, as will be discussed in
paragraph IVB6 below, the evolution of single-particle
levels along isotopic chains calls for α > 0, see also [48].
Additionally, as we will show in appendix B, large nega-
tive values of α pose the risk of instabilities towards the
transition to states with unphysical shell structure.
4. Single-particle spectra of doubly-magic nuclei
After we have examined the predictions for spin-orbit
splittings, we will now turn to the overall quality of the
single-particle spectra of doubly-magic nuclei. Figure 17
shows the single-particle spectrum of 132Sn. It is evi-
dent that as a consequence of the underestimated spin-
orbit splittings of the intruder levels that we discussed
in the last section, the spectrum is deteriorated for large
positive isoscalar tensor term coupling constants CJ0 (see
T66), as, for example, a decrease of the spin-orbit split-
208Pb, ν  (a)
1i centroid
Exp. T22
T42 T62 T24 T44 T64 T26 T46 T66
13/2+
11/2+
15/2-
3p3/2
2f5/2
1i13/2
3p1/2
2g9/2
1i11/2
3d5/2
4s1/2
1j15/2
2g7/2
3d3/2
208Pb, π  (b)
1h centroid
Exp. T22
T42 T62 T24 T44 T64 T26 T46 T66
11/2-
2d5/2
2d3/2
3s1/2
1h11/2
1h9/2
2f7/2
1i13/2
2f5/2
FIG. 18: Same as Fig. 17 for 208Pb.
ting of the neutron 1h shell pushes the 1h11/2 further up,
closing the N = 82 gap. As a consequence, the presence
of the tensor terms cannot remove the problem shared by
all standard mean-field methods that always wrongly put
the neutron 1h11/2 level above the 2d3/2 and 3s1/2 lev-
els [29], which compromises the description of the entire
mass region. For the same reason, the proton spectrum
of 132Sn also excludes interactions with large positive CJ0 ,
which reduces the Z = 50 gap between the 1g levels to
unacceptable small values.
Figure 17 also shows the energy centroids of the ν 1h
and π 1g levels, defined as
εcentqnℓ =
2ℓ+ 1
εqnℓ,j=ℓ+1/2 +
2ℓ+ 1
εqnℓ,j=ℓ−1/2 . (43)
The position of the centroid is fairly independent from
the parameterization. Assuming that the calculated en-
ergy of the centroid of an intruder state is more robust
against corrections from core polarization and particle-
vibration coupling that its spin-orbit splitting, we see
that the ν 1h centroid is clearly too high in energy by
about 1 MeV. In combination with its tentatively too
small spin-orbit splitting, see Fig. 15, this offers an ex-
planation for the notorious wrong positioning of the
ν 1h11/2, 2d3/2 and 3s1/2 levels in
132Sn [29]. The near-
degeneracy of the ν 2d3/2 and 3s1/2 levels is always well
reproduced, while the 1h11/2 comes out much too high.
As the 1h11/2 is the last occupied neutron level, self-
consistency puts it close to the Fermi energy, which, in
turn, pushes the 2d3/2 and 3s1/2 levels down in the spec-
trum.
The overall situation is similar for 208Pb, see Fig. 18.
Again, the high-ℓ intruder states move too close to the
Z = 82 and N = 126 gaps for large positive CJ0 . The
effect is less obvious than for 132Sn as the intruders and
their spin-orbit partners are further away from the gaps.
Still, the level ordering and the size of the Z = 82 gap
become unacceptable for parameterizations with large
tensor coupling constants. For strong tensor term cou-
pling constants (both like-particle and proton-neutron), a
Z = 92 gap opens in the single-particle spectrum of the
protons that is also frequently predicted by relativistic
mean-field models [14, 88] but absent in experiment [100].
The single-particle spectra for the light doubly magic
nuclei 40Ca (Fig. 19), 48Ca (Fig. 20), 56Ni (Fig. 21), 68Ni
(Fig. 22) and 90Zr (Fig. 23), all have in common that
the relative impact of the J2 terms on the ordering and
relative distance of single-particle levels is even stronger
than for the heavy nuclei discussed above. But not all
of the strong dependence on the coupling constants of
the J2 terms that we see in the figures is due to the ac-
tual contribution of the tensor terms to the spin-orbit
potential. This is most obvious for 40Ca, where protons
and neutrons are spin-saturated so that the J2 terms do
not contribute to the spin-orbit potentials. Still, increas-
ing their coupling constants increases the spin-orbit split-
tings, which manifests the readjustment of the spin-orbit
force to a given set of CJ0 and C
1 (see Fig. 4). The evolu-
tion of the spin-orbit splittings in 40Ca visible in Fig. 19
is the background which we have to keep in mind when
discussing the impact of the tensor terms on nuclei with
non-vanishing spin-orbit currents. Note that the spin-
orbit coupling constant W0 is correlated with isoscalar
tensor coupling constant CJ0 , such that the single-particle
spectra obtained with T24 and T42 are very similar, as
they are for T26, T44 and T62.
For 48Ca, Fig. 20, the protons are still spin-saturated
with vanishing proton spin-orbit current Jp, while for
neutrons we have a large Jn. Depending on the nature
of the tensor terms in the energy functional – i.e. like-
particle or proton-neutron or a mixture of both – the
spin-orbit current will either contribute to the spin-orbit
potential of the neutrons or that of the protons or both,
see Eq. (35). For the parameterizations with dominating
like-particle J2 term, for example T24 and T26, the situ-
ation for the protons is the same as for 40Ca: there is no
contribution from the tensor terms to the proton spin-
orbit splittings, but compared to T22 the proton Z = 20
gap is reduced through the readjustment of the spin-orbit
force, leading to values that are too small. For the same
parameterizations, the large contribution from Jn to Wn
opens up the N = 20 gap to values that are tentatively
too large, as it reduces the neutron spin-orbit splittings
and thereby compensates, even overcompensates, the ef-
fect from the readjustment of the spin-orbit force. At the
same time the N = 28 gap is reduced. The opposite effect
is seen for parameterizations with large proton-neutron
tensor term, for example T42 or T62. For those, the pro-
ton spin-orbit splitting is reduced, opening up the Z = 20
40Ca, ν  (a)
Exp. T22
T24 T26 T42 T44 T46 T62 T64 T66
1d5/2
2s1/2
1d3/2
1f7/2
2p3/2
40Ca, π  (b)
Exp. T22
T24 T26 T42 T44 T46 T62 T64 T66
2s1/2
1d3/2
1f7/2
2p3/2
FIG. 19: Same as Fig. 17 for 40Ca.
48Ca, ν  (a)
Exp. T22
T24 T26 T42 T44 T46 T62 T64 T66
2s1/2
1d3/2
1f7/2
2p3/2
2p1/2
1f5/2
48Ca, π  (b)
Exp. T22
T24 T26 T42 T44 T46 T62 T64 T66
2s1/2
1d3/2
1f7/2
2p3/2
FIG. 20: Same as Fig. 17 for 48Ca.
gap compared to T22, while the neutron spin-orbit split-
tings are increased by the background effect from the
readjusted spin-orbit force.
56Ni, ν  (a)
Exp. T22
T24 T26 T42 T44 T46 T62 T64 T66
1f7/2
2p3/2
1f5/2
2p1/2
56Ni, π  (b)
Exp. T22
T24 T26 T42 T44 T46 T62 T64 T66
1f7/2
2p3/2
1f5/2
2p1/2
FIG. 21: Same as Fig. 17 for 56Ni.
68Ni, ν  (a)
Exp. T22
T24 T26 T42 T44 T46 T62 T64 T66
(5/2-)
(1/2-)
(9/2+)
2p3/2
1f5/2
2p1/2
1g9/2
68Ni, π  (b)
Exp. T22
T24 T26 T42 T44 T46 T62 T64 T66
(7/2-)
1f7/2
2p3/2
1f5/2
2p1/2
FIG. 22: Same as Fig. 17 for 68Ni.
For 56Ni, Fig. 21, we have large Jn and Jp. In
this N = Z nucleus, the like-particle or proton-neutron
parts of the tensor terms cannot be distinguished. The
spectra depend only on the overall coupling constant of
the isoscalar tensor term CJ0 , on the one hand directly
through the contribution of the tensor terms to the spin-
orbit potentials, and on the other hand through the back-
ground readjustment of W0 that is correlated to C
well. As already mentioned, results for T24 and T42 are
very similar, as they are for T26, T44 and T62. All pa-
rameterizations have in common that the proton and neu-
tron gaps at 28 are too small. The variation of the single-
particle spectra among the parameterizations is smaller
than for 40Ca, mainly because the tensor terms compen-
sate the background drift from the readjustment of W0.
The slightly neutron-rich 68Ni combines a spin-
saturated sub-shell closure N = 40 that gives a vanishing
neutron spin-orbit current with the magic Z = 28 that
gives a strong proton spin-orbit current. The variation of
the single-particle spectra in dependence of the coupling
constants of the tensor terms is similar to those of 48Ca,
with the roles of protons and neutrons exchanged.
The nucleus 90Zr combines the spin-saturated proton
sub-shell closure Z = 40 with the major neutron shell
closure N = 50. The high degeneracy of the occupied ν
1g9/2 level leads to a very strong neutron spin-orbit cur-
rent, while the proton spin-orbit current is zero. Even
in the absence of a tensor term contributing to their
spin-orbit potential for parameterizations with pure like-
particle tensor terms, the proton single-particle spectra
are dramatically changed by the feedback effect from the
readjusted spin-orbit force; see the evolution from T22
to T26. The π 1g9/2 comes down, and closes the Z = 40
sub-shell gap. For parameterizations with pure proton-
neutron tensor term, one has the opposite effect, this time
because the contribution from the tensor terms overcom-
pensates the background effect from the spin-orbit force.
The effect of the tensor terms on the neutron spin-orbit
splittings is less dramatic, but still might be sizable.
We have to point out that the calculations displayed
in Fig. 23 were performed without taking pairing into
account, as the HFB scheme breaks down in the weak
pairing regime of doubly magic nuclei. For some ex-
treme (and unrealistic) parameterizations, however, the
gaps disappear which, in turn, would lead to strong pair-
ing correlations if the calculations were performed within
the HFB scheme. This happens, for example, for neu-
trons in 90Zr when using T26 and T46. Interestingly,
the pairing correlations for neutrons break the spin sat-
uration, which leads to a substantial neutron spin-orbit
current Jn. As these parameterizations use values of the
like-particle coupling constant significantly larger than
the neutron-proton one, Jn feeds back onto the neutron
spin-orbit potential only, Eq. (35). As the correspond-
ing coupling constant α is positive for T26 and T46, the
contribution from the tensor terms reduces the spin-orbit
splittings, in particular those of the 1g9/2 and 1f5/2. As
a result, this counteracts the reduction of the N = 40
gap predicted by T26 and T46 in calculations without
pairing.
90Zr, ν  (a)
Exp. T22
T24 T26 T42 T44
T46 T62 T64 T66
(11/2-)
2p3/2
1f5/2
2p1/2
1g9/2
2d5/2
3s1/2
2d3/2
1g7/2
1h11/2
90Zr, π  (b)
Exp. T22 T24
T42 T44 T46 T62 T64 T66
1f5/2
2p3/2
2p1/2
1g9/2
FIG. 23: Same as Fig. 17 for 90Zr.
5. Evolution along isotopic chains: np coupling
In the preceding sections, we have analyzed character-
istics of the single-particle spectra for isolated doubly-
magic nuclei. We found that larger tensor terms do not
lead to an overall improvement of the single-particle spec-
tra. However, we also argued that it might be essentially
due to deficiencies of the central (and possibly spin-orbit)
interactions and that it should not be used to discard the
tensor terms as such. In any case, the results gathered so
far on single-particle spectra of doubly-magic nuclei do
not permit to narrow down a region of meaningful cou-
pling constants of the tensor terms. The analysis must be
complemented by looking at other observables. A better
suited observable is provided by the evolution of spin-
orbit splittings along an isotopic or isotonic chain, which
ideally reflects the nucleon-number-dependent contribu-
tion from the J2 terms to the spin-orbit potentials. Un-
fortunately, safe experimental data for the evolution of
spin-orbit partners are scarce; hence, one has to content
oneself to the evolution of the energy distance of lev-
els with different ℓ, assuming that the effect is primarily
caused by the evolution of the spin-orbit splittings of each
level with its respective partner. A popular playground
for such studies is the chain of Sn isotopes, where two
such pairs of levels have gained attention; the π 2d5/2 and
π 1g7/2 on the one hand, and the π 1g7/2 and π 1h11/2 on
the other hand. Figure 24 shows these two sets of results
for a selection of our parameterizations.
Experimentally, the 2d5/2 and 1g7/2 levels cross be-
 56  60  64  68  72  76  80  84
π1g7/2 - π2d5/2
π1h11/2 - π1g7/2
FIG. 24: (Color online) Distance of the proton 1h11/2 and
1g7/2 levels (top) and of the proton 2d5/2 and 1g7/2 levels
(bottom), for the chain of tin isotopes. The “best” param-
eterization cannot and should not be determined with a χ2
criterion, see text.
tween N = 70 and 72, such that the 2d5/2 provides the
ground state of light odd-A Sb isotopes, and 1g7/2 that of
the heavy ones, see for example Ref. [101]. The crossing
as such is predicted by many mean-field interactions and
most of the parameterizations of the Skyrme interaction
we use here. It has also been studied in detail with the
standard Gogny force (without any tensor term) using
elaborate blocking calculations of the odd-A nuclei [102].
The crossing, however, is never predicted at the right
neutron number, see Fig. 24. As we have learned above,
we should not assume that the absolute distance of the
two levels will be correctly described by any of our param-
eterizations (as the centroids of the ℓ shells will not have
the proper distance and the spin-orbit splittings have a
wrong ℓ dependence within a given shell). Hence, the
neutron number where the crossing takes place cannot
and should not be used as a quality criterion. What does
characterize the tensor terms is the bend of the curves
in Fig. 24, as ideally it reflects how the spin-orbit split-
tings of both levels change in the presence of the ten-
sor terms. Similar caution has to be exercised in the
 28  32  36  40  44
π1f5/2 - π2p3/2
FIG. 25: (Color online) Distance of the proton 1f5/2 and
2p3/2 in the chain of Ni isotopes.
analysis of the unusual relative evolution of the proton
1g7/2 and 1h11/2 levels that was brought to attention
by Schieffer et al. [45]. Their spacing has been investi-
gated in terms of the tensor force before [44, 46, 48, 49].
Again, we pay attention to the qualitative nature of the
bend without focusing too much on the precise value by
which the splitting changes when going from N ≈ 58 to
N = 82. Indeed, the matching of the lowest proton frag-
ment with quantum number 1h11/2 seen experimentally
with the corresponding empirical single-particle energy is
unsafe because of the fractionization of the strength as
discussed in Ref. [48].
For both pairs of levels, the evolution of their distance
can be attributed to the tensor coupling between the pro-
ton levels and neutrons filling the 1h11/2 level below the
N = 82 gap. Unfortunately, this introduces an addi-
tional source of uncertainty: as can be seen in Fig. 17,
the ordering of the neutron levels in 132Sn is not prop-
erly reproduced by any of our parameterizations, with the
1h11/2 level being predicted above the 2d3/2 level, while
it is the other way round in experiment. This means that
in the calculations, the contribution from the 1h11/2 level
to the neutron spin-orbit current builds up at larger N
than what can be expected in experiment. As a conse-
quence, the prediction for the relative evolution of the
levels might be shifted by up to four mass units to the
right compared to experiment for both pairs of levels we
examine here.
In the end, the trend of both splittings is best repro-
duced when using a positive value of the neutron-proton
Jn · Jp coupling constant β such that the filling of the
neutron 1h11/2 shell decreases the spin-orbit splittings of
the proton shells. The parameterizations from the T4J
and T6J series indeed do reproduce the bend of empirical
data, with, however, a clear shift in the neutron number
where it occurs, as expected from the previous discussion.
A value of β = 120 MeV fm5, which corresponds to the
series of T4J parameterizations, matches its magnitude
best (see for example T44).
A similar analysis can be performed for the proton
1f5/2 and 2p3/2 levels in the chain of Ni isotopes, see
Fig. 25. This case is interesting as no distinctive feature
can be observed in the empirical spectra, yet the standard
parameterizations without tensor terms like T22 do not
reproduce them. In fact, to keep the 1f5/2 and 2p3/2 at a
constant distance, two competing effects have to cancel.
First, the increasing diffuseness of the neutron density
with increasing neutron number diminishes the proton
spin-orbit splittings through its reduced gradient in the
expression for the proton spin-orbit potential when going
from N = 32 to N = 40. Second, the filling of the neu-
tron 1f5/2 state reduces the neutron spin-orbit current
which in turn increases the proton spin-orbit splittings
for interactions with sizable proton-neutron tensor con-
tribution to the proton spin-orbit potential when going
from N = 32 to N = 40. The former effect can be clearly
seen for parameterizations T2J with vanishing proton-
neutron tensor term, β = 0. Again, parameterizations
of the T4J series seem to be the most appropriate to
describe the evolution of these levels.
The evolution of single-particle levels is the tool of
choice to determine the sign and magnitude of the
proton-neutron tensor coupling constant. The value
which we favor, as a result of our semi-qualitative analy-
sis is β = 120 MeV fm5. This value is only slightly larger
than the value of 94 to 96 MeV fm5 advocated by Brown
et al. in Ref. [48], which was adjusted to theoretical level
shifts in the chain of tin isotopes obtained from a G-
matrix interaction. We can consider this as a reasonable
agreement.
Let us defer the discussion of this value to the end of
this section and study in the next paragraph the like-
particle tensor-term coupling constant α.
6. Evolution along isotopic chains: nn coupling
In order to narrow down an empirical value for the
neutron-neutron tensor coupling constant, the ideal ob-
servable would be the evolution of neutron single-particle
levels along an isotopic chain. Unfortunately, these are
only accessible at the respective shell closures. We shall
therefore compare neutron single-particle spectra of pairs
of doubly-magic nuclei belonging to the same isotopic
chain. Again, the necessity to extract pure single-particle
effects calls for precautions. We choose pairs of parti-
cle or hole levels which are close enough in energy that
their absolute spacing is not much affected by particle-
vibration coupling. Of course, one also has to be careful if
both states appear at relatively high excitation energy in
the neighboring odd isotope because the fractionization
of their strength could again interfere with the analysis.
In the following, we choose pairs of orbitals which are as
safe as possible.
To remove the uncertainties from the deficiencies of the
central and spin-orbit parts of the effective interaction
that we have identified above, we will look at a double
-60  0  60  120  180  240
α [MeV fm5]
β = 0
β = 120
β = 240
FIG. 26: Shift of the distance between the neutron 1d3/2 and
2s1/2 levels when going from
40Ca to 48Ca, Eq. (44) (top) and
of the neutron 1f5/2 and 2p1/2 levels when going from
and 68Ni, Eq. (45) (bottom).
difference, where, first, we construct the energy difference
between the neutron 1d3/2 and 2s1/2 levels separately
for 40Ca and 48Ca, and then compare the value of this
difference in both nuclei
δCa =
1d3/2
2s1/2
1d3/2
2s1/2
. (44)
Assuming that the problems from the central and spin-
orbit forces discussed in Sects. IVB 1 and IVB4 have the
same effect in both nuclei, they will cancel out in δCa.
The interesting feature of this pair of states is that
they are separated by more than 2 MeV in 40Ca, while
they are nearly degenerate in 48Ca, see Figs. 19 and 20.
Such a shift can only be reproduced with a positive (140-
180 MeV fm5) value of α, which decreases the splitting
of the neutron 1d shell when the neutron 1f7/2 level is
filled.
A similar analysis can be performed for the 1f5/2 and
2p1/2 neutron states in the Ni isotopes
56Ni and 68Ni
δNi =
1f5/2
2p1/2
1f5/2
2p1/2
. (45)
Going from 56Ni to 68Ni, the neutron 1f5/2 level comes
further down in energy than the 2p1/2 level for param-
eterizations without tensor terms (T22), see Figs. 21
and 22. The reason for this trend is the geometrical
growth of the nucleus, which on the one hand lowers the
centroid of the 1f levels in the widening potential well,
and on the other hand pushes the spin-orbit field to larger
radii, which has opposite effects on the splittings of 2p
and 1f states. The like-particle tensor terms can com-
pensate this trend through a reduction of the spin-orbit
splitting of the 1f levels. The observed downward shift
by 0.3 MeV can be recovered with a value of α around
120 MeV fm5, see Fig. 26.
It is also gratifying to see that the analysis of Ca and
Ni isotopes suggests nearly the same value for the like-
particle tensor term coupling constant α.
C. Binding energies
Our ultimate goal, although far beyond the scope of
the present paper, is the construction of a universal nu-
clear energy density functional that simultaneously de-
scribes bulk properties like masses and radii, giant res-
onances, and low-energy spectroscopy, such as quasipar-
ticle configurations and collective rotational and vibra-
tional states. To crosscheck how our findings on single-
particle spectra and spin-orbit splittings translate into
bulk properties, we will now analyze the evolution of
mass residuals and charge radii along isotopic and iso-
tonic chains. It has been repeatedly noted in the liter-
ature that the mass residuals from mean-field calcula-
tions show characteristic arches [29, 52, 54, 65, 72, 103,
104, 105], where heavy mid-shell nuclei are usually un-
derbound compared to the doubly magic ones that are
located at the bottom of deep ravines. For light nuclei,
the patterns are often less obvious. Part of this effect can
be explained and removed taking large-amplitude corre-
lations from collective shape degrees of freedom into ac-
count through suitable beyond-mean-field methods. In
turn, this means that the mass residuals should leave
room for the extra binding of mid-shell nuclei from cor-
relations. However, it turns out that for typical effective
interactions the amplitude of the arches is larger than
what is brought by correlations [54]. Furthermore, this
effect seems not to be of the same size for isotopic and
isotonic chains, which altogether hints at deficiencies of
the current effective interactions.
Recently, Dobaczewski pointed out [47] that the
strongly fluctuating contribution brought by the J2 terms
to the total binding energy could remove at least some
of the ravines found in the mass residuals around magic
numbers. The hypothesis was motivated by calculations
that evaluate the tensor terms either perturbatively, or
self-consistently, using in this case an existing standard
parameterization without tensor terms for the rest of the
energy functional. Our set of refitted parameterizations
with varied coupling constants of the tensor terms gives
us a tool to check how much of the argument persists to
a full fit.
1. Semi-magic series
Figure 27 displays binding energy residuals along var-
ious isotopic and isotonic chains of semi-magic nuclei for
a selection of our parameterizations: T22 is the reference
with vanishing J2 terms at sphericity; T24 has a sub-
stantial like-particle coupling constant α and vanishing
proton-neutron coupling constant β, which is similar to
most of the published parameterizations which take the
J2 terms from the central Skyrme force into account; T42
and T62 are parameterizations with substantial proton-
neutron coupling constant β and vanishing like-particle
coupling constant; T44 has a mixture of like-particle and
proton-neutron tensor terms that is close to what we
found preferable for the evolution of spin-orbit splittings
above; and T46 is a parameterization that gives the best
root-mean-square residual of binding energies for spher-
ical nuclei, as we will see below. Finally, T66 is a pa-
rameterization with large and equal proton-neutron and
like-particle tensor-term coupling constants.
The tensor terms have opposite effects in light and
heavy nuclei: The curves obtained with T22, the parame-
terization without J2 term contribution at sphericity, are
relatively flat for the light isotopic and isotonic chains,
but show very pronounced arches with an amplitude of
5 or even more MeV for the heavy Sn and Pb isotopic
chains. By contrast, the most striking effect of the J2
terms is that they induce large fluctuations of the mass
residuals in light nuclei, while they flatten the curves in
the heavy ones.
The strong variation between the parameter sets for
light nuclei are of course the direct consequence of the
strong variation of the spin-orbit current J that enters the
spin-orbit and tensor terms when going back and forth
between nuclei where the configuration of at least one
nucleon species is spin-saturated. The variations seen
are a result of the modifications of tensor-term coupling
constants and the associated readjustment of the spin-
orbit strength W0. For example,
48Ca is overbound with
respect to 40Ca and 56Ni for parameterizations with a
proton-neutron coupling constant β > 0, while the like-
particle coupling constant α has a more limited effect.
Since only the neutron core is spin-unsaturated in this
nucleus, this must be attributed to the increase in the
readjusted spin-orbit strength W0 (correlated with C
(α + β)) which dominates when β is increased and α
kept at zero, and counterbalances the effect of α when
the latter varies. See the parameter sets T62 and T66
in Figures 27 and 28. The large overbinding of nuclei
around 90Zr (Z = 40, N = 50) for parameterizations
with large proton-neutron tensor coupling constant has
the same origin. For a given parameterization and a given
nucleus, the energy gain from the spin-orbit term seems
to be almost always larger than the energy loss from the
J2 one, see Fig. 28 for Ca isotopes and Fig. 29 for Sn
isotopes. Of course other terms in the energy functional
compensate for a part of the gain from the spin-orbit
term, but the overall trends of the mass residuals suggest
that the spin-orbit energy has a much larger contribution
to the differences between the parameterizations visible
in Fig. 27 than the J2 terms.
We have to note that the spin-orbit current does not
completely vanish for the nominally proton and neutron
spin-saturated 40Ca for parameterizations with large cou-
 16  20  24  28
 32  36  40  44  48
 48  52  56  60  64  68
 16  20  24  28  32
 28  32  36  40  44
 50  60  70  80
 100  110  120  130
FIG. 27: (Color online) Mass residuals Eth − Eexp along selected isotopic and isotonic chains of semi-magic nuclei for the
parameterizations as indicated. Positive values of Eth − Eexp denote underbound nuclei, negative values overbound nuclei.
pling constants of the J2 terms. For those, the gap at 20
is strongly (and nonphysically) reduced, see Fig. 19. The
small gap at 20 does not suppress pairing correlations
anymore in our HFB approach. The resulting scattering
of particles from the sd shell to the fp shell breaks the
spin-saturation, such that there is a finite, in some cases
quite sizable, contribution from the spin-orbit term to
the total binding energy. Owing to the compensation be-
tween all contributions, the total energy gain compared
to a HF calculation without pairing is usually small and
rests on the order of 200 keV for the parameterizations
shown in Fig. 27.
It is also important to note that some of the light
chains in Fig. 27 are sufficiently close to or even cross the
N = Z line that they are subject to the Wigner energy,
which still lacks a satisfying explanation, not to men-
tion a description in the framework of mean-field meth-
ods [106]. The Wigner energy is not taken into account
 16  20  24  28  32
FIG. 28: (Color online) Evolution of spin-orbit current (J2t )
energy (bottom panel, zero by construction for T22) and spin-
orbit energy (top panel) with neutron number N in the chain
of Ca isotopes (Z = 20).
in our fits, while it turned out to be a crucial ingredient
of any HFB [107, 108, 109] or other mass formula. In
fact, as shown in Fig. 14 of Ref. [54], the missing Wigner
energy clearly sticks out from the mass residuals for SLy4
(which is very similar to T22) when they are plotted for
isobaric chains. This local trend around N = Z is, how-
ever, overlaced with a global trend with mass number,
such that the missing Wigner energy cannot be spotted
anymore when looking at the mass residuals for the iso-
topic chain of Ca isotopes, similar to what is seen for T22
in Fig. 27. Within our fit protocol, the correlation be-
tween the masses of 40Ca, 48Ca and 56Ni, that is brought
by the spin-orbit force (see Sect. IVB 2) does not tolerate
a correction for the Wigner energy for standard central
and spin-orbit Skyrme forces, as this will lead to an un-
acceptable underbinding of 48Ca. This, however, might
change when the J2 terms are added. Indeed, Fig. 27
suggests that adding a phenomenological Wigner term
around 40Ca and 56Ni to a parameter set like T44, which
is consistent with the evolution of single-particle levels,
would flatten the curves for the mass residuals in the Ca,
Ni and N = 28 chains. The mass residuals for the chain
of oxygen isotopes that are not shown here would be im-
proved in a similar manner. However, extreme caution
should be exercised before jumping to premature conclu-
sions, as the spin-orbit splittings and level distances in
light nuclei are far from realistic for all our parameter-
 50  60  70  80
FIG. 29: (Color online) Same as Fig. 28 for tin isotopes
(Z = 50).
izations; as a consequence it is difficult to judge if the
room we find for the Wigner energy is fortuitous or in-
deed a feature of well-tuned J2 terms. Note that the HFB
mass formulas that do include a correction for the Wigner
energy side-by-side with the J2 terms from the central
Skyrme force give satisfying mass residuals for light nu-
clei [107, 108, 109], but have nuclear matter properties
that are quite different from ours; cf. BSk1 and BSk6
with SLy4 in Table I of Ref. [110]. Our constraints on
the empirical nuclear matter properties (same as those
on SLy4) that are absent in these HFB mass fits might
be the deeper reason for this conflict.
Large tensor-term coupling constants straighten the
arches in the mass residuals in the heavy Sn and Pb iso-
topic chains, but the improvements are not completely
satisfactory. Large, combined proton-neutron and like-
particle coupling constants tend to transform the arch
for the tin isotopic chain into a an s-shaped curve, which
is not very realistic from the standpoint of expected cor-
rections through collective effects. It can again be as-
sumed that the deficiencies of the single-particle spectra
pointed out in Fig. 17 are responsible, where the ν 1h11/2
and π 1g9/2 are placed too high above the rest of the
single-particle spectra in heavy Sn isotopes. For Pb iso-
topes, large values of the tensor terms tend to overbind
the neutron-deficient isotopes. It is noteworthy that the
tensor terms seem to not much affect the mass residu-
als of the heavy Pb isotopes above N = 126, which are
on the flank of a very deep ravine that becomes visible
 50  60  70  80  90  100  110  120
FIG. 30: (Color online) Two-neutron separation energy along
the chain of isotopes (Z = 50).
when going towards heavier elements, cf. the SLy4 results
in Ref. [54].
It has been often noted that effective interactions that
give a similar satisfying description of masses close to the
valley of stability give diverging predictions when extrap-
olated to exotic nuclei. The standard example is the two-
neutron separation energy S2n(N,Z) = E(N,Z − 2) −
E(N,Z) for the chain of Sn isotopes. Results obtained
with a subset of our parameterizations are shown in Fig.
30. It is noteworthy that the differences for neutron-rich
nuclei beyond N = 82 are not larger than those for the
isotopes closer to stability. Around the valley of stabil-
ity, increasing the coupling constants of tensor terms, in
particular the like-particle ones, tilts the curve, pushing
it up for light isotopes and pulling it down it for heavy
ones, which reflects of course the position of the ν 1h11/2
level that is pushed into the N = 82 gap, see Fig. 17.
For the neutron-rich isotopes, small differences appear
around N = 90, which reflects the change of level struc-
ture above the ν 2f7/2 level and at the drip line, but they
are much smaller than the differences seen between pa-
rameterizations obtained with different fit protocols, see
Fig. 5 of Ref. [29].
2. Systematics
In the preceding section we showed how the J2 terms
in the energy functional modify the trends of mass resid-
uals along isotopic and isotonic chains, in particular the
amplitude of the arches between doubly-magic nuclei. In
this section, we want to examine how this translates into
quality criteria for the overall performance of the param-
eterizations for masses.
Figure 31 displays the root-mean-square deviation of
-60060120180240
∆Erms [MeV]
(T11)
β [MeV fm5]α [MeV fm
∆Erms [MeV]
FIG. 31: Root-mean-square deviation from experiment of the
binding energies of a set of 134 spherical nuclei, for each of
the forces TIJ , vs. α and β (The “(T11)” label indicates
the position of this parameterization in the (α, β)-plane).
Contour lines at ∆Erms = 2.0, 2.25, 2.5, 3.0, 3.5, 4.0 MeV. The
minimal value is found for T46 (∆Erms = 1.96 MeV).
the mass residuals for all our 36 parameterizations, eval-
uated for a set of 134 nuclei predicted to have spherical
mean-field ground states when calculated with the pa-
rameterizations SLy4 [54]. One observes a clear mini-
mum around T46, i.e. (α, β) = (240, 120), with (Eth −
Eexp)r.m.s. = 1.96 MeV, compared with 3.44 MeV for
T22 (α = β = 0). We found even slightly better values
with even more repulsive isoscalar and isovector coupling
constants, but the single-particle spectra of these inter-
actions turn out to be quite unrealistic, cf. Sect. IVB 1.
This already demonstrates that in the presence of the J2
terms a good fit of masses does not necessarily lead to
satisfactory single-particle spectra.
Figure 32 demonstrates how the distribution of the
mass residuals Eth − Eexp affects the evolution of their
r.m.s. value for a subset of 9 parameterizations. For
T22 (α = β = 0), the distribution is centered at posi-
tive mass residuals, with only very few nuclei being over-
bound. Increasing β to 120 MeV fm5 (T42) or even 240
MeV fm5 (T62) shifts the median of the distribution to
smaller values, which yields more and more overbound
nuclei. For large values of β, the distribution spreads out
more, which diminishes the improvement from centering
the distribution closer to zero. For given β, increasing
α mainly shifts the median of the distribution without
spreading out its overall shape, which is preferable to
optimize the r.m.s. value.
These considerations, however, have to be taken with
caution. As said above, we aim at a model where certain
correlations beyond the mean-field are treated explicitly,
which asks for a distribution of mean-field mass residuals
with an asymmetric distribution towards positive mass
residuals, and a width that is similar to the difference
-10 -8 -6 -4 -2  0  2  4  6  8  10
Eth - Eexp [MeV]
FIG. 32: (Color online) Distribution of deviations from ex-
periment of the binding energies of a set of 134 spherical nuclei
(1 MeV bins) for a subset of parameterizations. Each panel
corresponds to a given value of β (from top to bottom: β = 0,
120, 240 MeV fm5).
between the maximum and minimum correlation energies
to be found.
D. Radii
The evolution of nuclear charge radii along isotopic
chains reflects how the mean field of the protons changes
when neutrons are added in the system. In the sim-
plistic liquid-drop model, it just follows the geometrical
growth of the nucleus ∼ A1/3, but data show that there
are many local deviations from this global trend. On
the one hand, radii are of course subject to correlations
beyond the mean field [54, 111, 112, 113, 114] On the
other hand, they are also sensitive to the detailed shell
structure, which, in turn, might be influenced by tensor
terms. We will concentrate here on two anomalies of the
evolution of charge radii, both of which are not much
influenced by collective correlations beyond the mean-
field (at least in calculations with the Skyrme interaction
SLy4) [54]: that the root-mean-square (r.m.s.) charge ra-
dius of 48Ca is almost the same as the one of the lighter
40Ca or possibly slightly smaller, and the kink in the iso-
topic shifts of mean-square (m.s.) charge radii in the Pb
isotopes, where Pb isotopes above 208Pb are larger than
what could be expected from liquid-drop systematics. In
-60  0  60  120  180  240
β [MeV fm5]
 40Ca 48Ca 
α = 0
α = 120
α = 240
(α = 120)
1s1/2
1p3/2
1p1/2
1d5/2
1d3/2
2s1/2
FIG. 33: (Color online) Middle panel: Difference of mean-
square charge radii between 40Ca and 48Ca as a function of
the proton-neutron tensor term coupling constant β for three
values of α. The experimental value (with error bar) is rep-
resented by the two horizontal black lines. Bottom panel:
Root-mean-square charge radii of 40Ca and 48Ca. Top panel:
Contribution of the single-particle proton states to the dif-
ference of the charge radii (mean square radius of the point
proton distribution, see Eq. (46)).
both cases it is plausible that shell effects are the de-
termining factor, although alternative explanations that
involve pairing effects have been put forward for the lat-
ter case as well [115, 116].
Charge radii have been calculated with the approxima-
tion used in Ref. [51]3 and derived from Ref. [117]
r2ch = 〈r
2〉p + r
r2n +
v2i µqi〈σ · ℓ〉i ,
where the mean-square (m.s.) radius of the point-proton
distribution 〈r2〉p is corrected by three terms: the first
two estimate the effects of the intrinsic charge distribu-
tion of the free proton and neutron (with m.s. radii r2p
and r2n) and the third adds a correction from the mag-
netic moments of the nucleons. Since we will consider
the shift of charge radii for different isotopes of the same
series, the actual value of r2p cancels out. For the second
correction term, which is independent from the interac-
tion, we take r2n = −0.117 fm
2 [29]. Finally, the magnetic
correction can only depend weakly on the details of the
interaction through the occupation factors v2i when non-
magic nuclei are considered. The same expressions had
been used during the fit of our parameterizations.
We begin with the Ca isotopes. Most parameteriza-
tions of Skyrme’s interaction are not able to reproduce
that the charge radius of 48Ca has about the same size as
that of 40Ca, see Fig. 11 in Ref. [29]. The middle panel of
Fig. 33 shows the difference of the the m.s. radii of 48Ca
and 40Ca in dependence of the tensor term coupling con-
stants α and β. First, this difference is almost indepen-
dent of α, the strength of the like-particle tensor terms.
Second, it is strongly correlated with β, the strength of
the proton-neutron tensor term, with large positive val-
ues of β bringing the difference of radii into the domain
of experimentally acceptable values [118] or even below,
with a best match obtained for β = 80 MeV fm5. This
effect can be explained by looking at the proton single-
particle spectra of 40Ca (Fig. 19) and 48Ca (Fig. 20). In-
deed, one observes that a positive neutron-proton tensor
coupling constant decreases the strength of the proton
spin-orbit field in 48Ca, which in turn lowers the π 1d3/2
level in 48Ca (compare the parameterizations TIJ in
Fig. 20 with increasing I for given J). As a consequence,
the m.s. radius of this state decreases as it sinks deeper
into the potential well of 48Ca. At the same time, this
level is pushed up in 40Ca, which slightly increases the
contribution of this state to the charge m.s. radius of this
nucleus. This effect is demonstrated in the top panel of
Fig. 33, which displays the degeneracy-weighted and nor-
malized change of the m.s. radii of proton hole states be-
tween 40Ca and 48Ca as a function of the proton-neutron
tensor term coupling constant β for forces with a like-
particle tensor term coupling constant α = 120 MeV fm5.
Indeed, the decreasing contribution from the π1d3/2 state
to the m.s. radius significantly decreases the isotopic shift
3 There is a typographical error in Eq. (4.2) in Ref. [51], that
was copied to Eq. (110) in Ref. [29]: the ~/mc factor should
be squared, as is trivially found by dimensional analysis and
confirmed by Ref. [117].
-60  0  60  120  180  240
α [MeV fm5]
β = 0
β = 120
β = 240
FIG. 34: Change of slope in the m.s. charge radii ∆2r2ch
around 208Pb, Eq. (47), in fm2 as a function of α for three
values of β. The experimental value is about one and a half
times as large as the largest theoretical value shown here, see
text.
between both Ca isotopes. It has to be noted that the
m.s. value of the charge radii of 40Ca and 48Ca are al-
most independent of alpha and that their absolute values
are not reproduced for any of our parameterizations.
The latter study demonstrates the correlation between
the isotopic shift of m.s. charge radius between 40Ca and
48Ca and the absolute single-particle energy of the pro-
ton 1d3/2 state. This level can be moved around within
the single-particle spectrum with the J2 terms. However,
the agreement of the calculated single-particle energy of
the proton 1d3/2 state in both nuclei with experiment is
not necessarily improved for the parameterizations that
reproduce the isotopic shift of the m.s. charge radius.
Furthermore, a good reproduction of the isotopic shift
does not guarantee that the absolute values of the charge
radii are well reproduced, see the bottom panel in Fig. 33.
In fact, they are predicted too large for all of our pa-
rameterizations, which again points to deficiencies of the
central field. Altogether, this suggests that in spite of
its sensitivity to the coupling constants of the J2 terms,
the isotopic shift of m.s. charge radius between 40Ca and
48Ca should not be used to constrain them before one
has gained sufficient control over the central interaction.
A few further words of caution are in place. The charge
radii of all light nuclei are significantly increased by dy-
namical quadrupole correlations, see Fig. 23 of Ref. [54].
Correlations beyond the static self-consistent mean field
are also at the origin of the arch of the ms charge radii
between 40Ca and 48Ca that is neither reproduced by
any pure mean-field model, see again Fig. 11 in Ref. [29],
nor by the beyond-mean-field calculations with SLy4 of
Ref. [54], while the shell model allows for a satisfactory
description [119].
Many explanations have been put forward to explain
the kink in the isotopic shifts of Pb radii. As it qual-
itatively appears in relativistic mean-field models, but
not in non-relativistic ones using the standard spin-orbit
interaction (16), it has been used as a motivation to gen-
eralize the isospin mix of the standard spin-orbit energy
density functional, Eq. (18), to simulate the isospin de-
pendence of the relativistic Hartree models [78, 79]. The
resulting parameterizations are not completely satisfac-
tory, as the price for the improvement of the radii is a fur-
ther deterioration of spin-orbit splittings [14], while the
relativistic mean field gives a satisfactory description of
both. Some standard Skyrme interactions that take the
tensor terms from the central Skyrme force into account
also give a kink, but it is by far too small to reproduce
the experimental values [52].
Plotting the m.s. radii along the chain of Pb isotopes as
a function of N , the slopes are nearly linear when looking
separately at the isotopes below and above 208Pb. We
will concentrate on the change in the slope at 208Pb that
is brought by the tensor terms, which can be quantified
through the second finite difference of the m.s. radii at
208Pb
∆2〈r2ch〉(
208Pb) (47)
r2ch(
206Pb)− 2 r2ch(
208Pb) + r2ch(
210Pb)
There are two conflicting values to be found in the lit-
erature, either 46.4± 1.4 fm2 [118] and the significantly
larger 59 ± 3 fm2 [120]. Figure 34 shows the change of
slope around 208Pb as defined through Eq. (47) as a func-
tion of the like-particle tensor coupling constant α and
for three different values of β. It is striking to see that
this quantity is almost independent of the neutron-proton
tensor coupling constant β, so the change is mainly in-
duced by the tensor interaction between particles of the
same kind. It has been noted before that the kink in the
isotopic shift of the charge radii in Pb isotopes is corre-
lated to the single-particle spectrum of neutrons above
N = 126, in particular the position of the 1i11/2 level.
(This has to be contrasted with the Ca isotopic chain
discussed above, where the difference of charge radii be-
tween 40Ca and 48Ca appears to be particularly sensi-
tive to the single-particle spectrum of the protons.) The
closer the 1i11/2 level is to the 2g9/2 level that is filled
above N = 126, the more the 1i11/2 becomes occupied
through pairing correlations. Through the shape of its
radial wave function, the partial filling of the nodeless
1i11/2 increases the neutron radius faster than filling only
the 2g9/2, and in particular faster than for the isotopes
below N = 126. As the protons follow the density distri-
bution of the neutrons, the charge radius grows rapidly
beyond N = 126. This offers an explanation why the
kink increases with the like-particle tensor term coupling
constant α: for large values of the weight α of the neu-
tron spin-orbit current in the neutron spin-orbit poten-
tial, Eq. (35), the spin-orbit splitting of the ν 1i levels is
reduced such that the 1i11/2 approaches the 2g9/2 level
in 208Pb, see Fig. 18.
While the kink is clearly sensitive to the tensor terms,
they cannot be responsible for the entire effect, as
even for extreme parameterizations that give unrealistic
single-particle spectra the calculated kink hardly reaches
about three quarters of its experimental value.
V. SUMMARY AND CONCLUSIONS
We have reported a systematic study of the effects of
the J2 (tensor) terms in the Skyrme energy functional for
spherical nuclei. The aim of the present study was not to
obtain a unique best fit of the Skyrme energy functional
with tensor terms, but to analyze the impact of the tensor
terms on a large variety of observables in calculations
at a pure mean-field level and to identify, if possible,
observables that are particularly, even uniquely, sensitive
to the J2 terms. To reach our goal, we have built a set
of 36 parameterizations that cover the two-dimensional
parameter space of the coupling constants of the J2t terms
that does not give obviously unphysical predictions for a
wide variety of observables we have looked at. The fits
were performed using a protocol very similar to that of
the SLy parameterizations [51, 52]. The 36 actual sets of
parameters can be found in the Physical Review archive
[85].
We use a formalism that explicitly relates the tensor
terms in the energy functional to underlying effective
density-dependent central, spin-orbit and tensor forces
(or vertices) in the particle-hole channel. As has been
known for long, a zero-range tensor force gives no qual-
itatively new terms for spherical mean-field states when
combined with a central Skyrme force, but solely modifies
the coupling constants of the J2 terms that are already
present. The contribution from the central Skyrme force
to the coupling constants of the J2 terms depends on the
same parameters t1, x1, t2 and x2 that determine the ef-
fective mass and contribute to the surface terms. As the
latter terms are much more important for the description
of bulk properties than the J2 terms, the coupling con-
stants of the J2 terms are confined to a very small region
of the parameter space. From this point of view, adding
a tensor force is necessary to explore it fully.
There is, however, the alternative interpretation of the
Skyrme energy functional from the density matrix ex-
pansion, which in the absence of ab-initio realizations so
far is used as a motivation to set up energy functionals
with independent, and phenomenologically fitted, cou-
pling constants of all terms not constrained by symme-
tries. In particular, this can be used to set unwanted or
underconstrained terms to zero, as it is done for many
existing parameterizations of the (central) Skyrme in-
teraction. For the ground states of spherical nuclei, as
discussed here, the frameworks cannot be distinguished.
For deformed nuclei and, in particular, polarized nuclear
matter, this choice will make a difference.
As a result of our study, we have obtained a long list
of potential deficiencies of the Skyrme energy functional,
most of which can be expected to be related to the prop-
erties of the central and spin-orbit interactions used. In
fact, these deficiencies become more obvious the moment
one adds a tensor force, as it appears that the presence of
a tensor force unbalances a delicate compromise within
various terms of the Skyrme interaction that permits to
get the global trend of gross features of the shell structure
right.
Our conclusions, however, have to be taken with a
grain of salt. On the one hand, some might depend on
the fit protocol; and on the other hand, we have to stress
that (within the framework of our study – and all others
available so far using mean-field methods) the compari-
son between calculated and empirical single-particle en-
ergies is not straightforward and without the risk of being
misled. However, without even looking at single-particle
spectra, we find that
1. The presence of the tensor terms leads to a strong
rearrangement of the other coupling constants,
most notably that of the spin-orbit force. In fact,
we find that the variation of the spin-orbit strength
W0 provoked by the presence of tensor terms has
a larger impact on the global systematics of single-
particle spectra than the tensor terms themselves.
The rearrangement of the parameters of the central
and spin-orbit parts of the effective interaction sug-
gests that perturbative studies of the tensor terms,
in which they are added to an existing parameter-
ization without readjustment, allow only very lim-
ited conclusions.
2. In the Skyrme energy functional, the combined cou-
pling constants of the spin-orbit and tensor terms
are nearly exclusively fixed by the mass differences
between 40Ca, 48Ca and 56Ni. This correlation ap-
pears to be (at least partly) spurious, the rapidly
varying spin-orbit and tensor terms being misused
to simulate missing physics in the standard Skyrme
functional.
3. The cost function χ2 used in our fit protocol prefers
parameterizations with β = 0, i.e. pure like-particle
tensor terms ∼ (J2n + J
p), without giving a clear
preference for a value of the corresponding coupling
constant α. By contrast, the mass residuals of 134
spherical even-even nuclei are minimized for inter-
actions with large α and β. However, and as we
will discuss in [41], the deformation properties of
many nuclei obtained with the latter parameteriza-
tions are unrealistic, which disfavors this region of
the parameter space.
4. The difference of the charge radii of 40Ca and 48Ca
turns out to be particularly sensitive to the abso-
lute single-particle energy of the proton 1d3/2 level,
which can be moved around by the J2 terms. As
the parameterizations that give the best agreement
for the absolute placement of this level do not nec-
essarily give the best overall single-particle spectra
for these two nuclei, this quantity should not be
used to constrain the J2 terms.
Concerning the global properties of the spin-orbit current
J and its contribution to the spin-orbit potential, we have
shown that
1. The spin-orbit current J in non-spin-saturated
doubly-magic nuclei as 56Ni, 100Sn, 132Sn or 208Pb
is dominated by the nodeless intruder orbitals.
Through the contribution of the tensor terms to
the spin-orbit field, the feedback effect on their own
spin-orbit splitting is maximized.
2. In light nuclei, J and consequently the contribu-
tion of the J2 terms to the binding energy and the
spin-orbit potential, vary rapidly between near-zero
and very large values when adding just a few nucle-
ons to a given nucleus. In heavy spherical nuclei,
the variation becomes much slower and smoother
as on the one hand one does not encounter spin-
saturated configurations anymore, and on the other
hand there are more and more high-ℓ states with
large degeneracy that require more nucleons to be
filled.
3. The contribution from the zero-range spin-orbit
force to the spin-orbit potential is peaked at the
nuclear surface, as it is proportional to the gradi-
ent of the density. By contrast, the contribution
from the zero-range tensor terms is peaked further
inside of the nucleus, modifying the width of the
spin-orbit potential with varying nucleon numbers.
As shown in Ref. [48], experimental data tend to
dislike such a modification.
4. Large negative coupling constants of the tensor
terms will lead to instabilities, where a nucleus
gains energy separating the levels from many spin-
orbit partners on both sides of the Fermi energy.
This process leads to unphysical single-particle
spectra and rules out a large part of the parameter
space. In particular cases, one might even obtain
a (probably spurious) coexistence of two spherical
configurations with different shell structure in the
same nucleus, which are separated by a barrier.
The main motivation to add J2 terms is of course to im-
prove the single-particle spectra. All observations and
conclusions concerning those have to be taken with care,
as in this study we compare the eigenvalues of a spher-
ical single-particle Hamiltonian with the separation en-
ergy to low-lying states in the odd-A neighbors of doubly
and semi-magic nuclei (as was done in all existing earlier
studies). When looking at the single-particle spectra in
doubly-magic nuclei (or semi-magic nuclei combined with
a strong subshell closure of the other species) we find that
1. The relative error of the spin-orbit splittings de-
pends strongly on the principal quantum number of
the orbitals within a given shell, such that for pa-
rameterizations without the tensor terms the split-
tings of the intruder state (without nodes in the ra-
dial wave function) is tentatively too small, while it
becomes too large with increasing number of nodes.
Adding the tensor terms further increases the dis-
crepancy. This problem can only be resolved by
an improved control over the shape of the spin-
orbit potential. Indeed, the size of the spin-orbit
splittings is related to the overlap of the radial
wave function of a given single-particle state with
the spin-orbit potential. The tensor terms mod-
ify the width of the spin-orbit potential, but to
cure this deficiency calls for a large negative like-
particle tensor coupling constant α, which is not
consistent with the evolution of spin-orbit splittings
along chains of semi-magic nuclei, and will lead to
instabilities.
2. We also find that, in a given nucleus, the predicted
spin-orbit splittings of neutron levels are larger
than those of the protons when both are compared
to experiment, which hints at an unresolved isospin
trend in the spin-orbit interaction.
3. For spin-saturated doubly-magic nuclei as 16O and
40Ca, the spin-orbit splittings of the spin-saturated
species of nucleons depends strongly on the cou-
pling constants of the J2 terms, although they do
not contribute to the spin-orbit field. This is a con-
sequence of the strong correlation between the spin-
orbit and tensor term coupling constants, which try
to compensate each other in spin-unsaturated nu-
clei. For parameterizations with strong tensor-term
coupling constants, the resulting spin-orbit force
leads to unrealistic single-particle spectra of spin-
saturated configurations.
4. The centroid of the spin-orbit partners that give
the intruder state is tentatively too high compared
to the major shell below.
The main effect of the tensor terms, that most of the
recent studies concentrate on, is the evolution of spin-
orbit splittings with N and Z. Unfortunately, there are
no data for the splittings themselves, such that one re-
lies on data for the evolution of the distance of two lev-
els with different ℓ. The comparison is compromised by
the global deficiencies of the single-particle spectra listed
above. Still, a careful comparison of calculations and
experiment suggests that
1. The evolution of the proton 1h11/2, 1g7/2 and 2d5/2
levels in the chain of Sn isotopes and that of the
proton 1f5/2 and 2p3/2 levels in Ni isotopes call for
a positive proton-neutron tensor coupling constant
β with a value around 120 MeV fm5, consistent
with the findings of Refs. [48, 49, 50].
2. The evolution of the neutron 1d3/2 and 2s1/2 levels
between 40Ca and 48Ca calls for a like-particle ten-
sor coupling constant α with a similar value around
120 MeV fm5. This it at variance to the findings of
Refs. [48, 49, 50], but in qualitative agreement with
the parameterization skxta of Brown et al. [48] for
which the tensor terms were derived from a realis-
tic interaction but disregarded thereafter because
of its poor description of spin-orbit splittings.
3. Combined this leads to a dominantly isoscalar ten-
sor term with a coupling constant CJ0 around 120
MeV fm5, while the isovector coupling constant will
have a small, near-zero, value.
Our study is obviously only a stepping stone towards
improved parameterizations of the Skyrme energy den-
sity functional. There are a number of necessary further
studies and future theoretical developments
1. The deformation properties of selected parameter-
izations TIJ from this study will be discussed in a
forthcoming paper [41].
2. The influence of the terms depending on time-odd
densities and currents in the complete energy func-
tional (23) on nuclear matter and finite nuclei (rota-
tional bands etc) is under investigation as well. The
existing stability criteria of polarized matter have
to be generalized as the tensor force introduces new
unique terms, for example in the Landau parame-
ters [121].
3. It is well known that the strength of the spin-
orbit force has to scale with the effective mass
of an interaction, which in turn determines the
average density of single-particle levels. All pa-
rameterizations discussed here have a similar effec-
tive mass close to m∗0/m = 0.7 that was already
used for the SLy parameterizations. This value is
somewhat smaller than the one obtained from ab-
initio calculations. We have checked that increas-
ing the effective isoscalar mass to the more realistic
m∗0/m = 0.8 (which within our fit protocol requires
to use two density dependent terms [84]) does not
significantly affect any of our conclusions.
4. It is evident that improvements of the central and
spin-orbit parts of the energy density functional are
necessary, which will require a generalization of its
functional form. Other motivations were found re-
cently to perform such a generalization [84].
5. The only quantity that we found sufficiently sen-
sitive to the tensor terms is the evolution of the
distance between single-particle levels in isotopic
or isotonic chains of semi-magic nuclei. The dis-
tance between the levels that can be used for such
studies is so large, that it might be compromised
by their coupling to collective excitations. Reliable
calculations including pairing, polarization as well
as particle-vibration coupling effects [89, 90] along
isotopic and isotonic chains are needed to test the
quality, reliability and limits of the simplistic iden-
tification of the eigenvalues of the spherical mean-
field Hamiltonian in an even-even nucleus with the
separation energy to or from low-lying states in the
adjacent odd-A nuclei.
Acknowledgments
We thank P. Bonche, H. Flocard, P.-H. Heenen and
B. A. Brown for stimulating and encouraging discus-
sions. Work by M. B. and K. B. was performed within
the framework of the Espace de Structure Nucléaire
Théorique (ESNT). T. L. acknowledges the hospitality
of the SPhN and ESNT on many occasions during the
realization of this work. This work was supported by
the U.S. National Science Foundation under Grant No.
PHY-0456903.
APPENDIX A: COUPLING CONSTANTS OF
THE SKYRME ENERGY FUNCTIONAL
The coupling constants of the central Skyrme energy
density functional in terms of the parameters of the cen-
tral Skyrme force are given by
0 (r)
1 = −
ρα0 (r)
As0 = −
ρα0 (r)
As1 = −
0 (r)
Aτ0 =
Aτ1 = −
AT0 = −
AT1 = −
0 = −
A∆s0 =
A∆s1 =
t2 . (A1)
The coupling constants of the spin-orbit energy density
functional in terms of the parameters of the spin-orbit
force are given by
A∇J0 = −
A∇J1 = −
W0 . (A2)
The coupling constants of the tensor energy density func-
tional in terms of the parameters of Skyrme’s tensor force
are given by (Table I in [59])
BT0 = −
(te + 3to) B
(te − to) (A3)
BF0 =
(te + 3to) B
1 = −
(te − to) (A4)
B∆s0 =
(te − to) B
1 = −
(3te + to) (A5)
B∇s0 =
(te − to) B
1 = −
(3te + to) . (A6)
APPENDIX B: PHASE TRANSITIONS
The densities ρ and τ entering the energy functional
(28) vary smoothly with nucleon numbers as they fol-
-1025
-1020
-1015
-1010
 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7
FIG. 35: Total binding energy of 120Sn as a function of C =
r Jn ·∇ρn in a constrained calculation. The dashed curve
shows results obtained with the parameterization mentioned
in the text, while the solid curve shows results obtained with
SLy5.
low the geometric growth of the nucleus. As a result, a
functional depending only on ρ and τ usually shows a
unique minimum for given N , Z and shape. The situ-
ation is quite different when the tensor terms are taken
into account. Indeed, the amplitude of the spin-orbit
current density J (27) depends on the number of spin-
unsaturated single-particle states in the nucleus; it varies
from (almost) zero in spin-saturated nuclei to large finite
values as a consequence of shell and finite-size effects, see
Fig. 5.
This behavior poses the risk of an instability, which
was already reported in [5]: multiplying J with a large
coupling constant in the spin-orbit potential (35) might,
for certain combinations of the signs of the coupling con-
stant and the spin-orbit currents of protons and neutrons,
increase the spin-orbit splittings. In some nuclei, this will
cause two levels originating from different ℓ shells to ap-
proach the Fermi energy, one from above and the other
from below, or even to cross. In that situation, their
occupation numbers will change such that J increases
further, which feeds back onto the spin-orbit potential
and ultimately leads to a dramatic rearrangement of the
single-particle spectrum.
We faced this problem when attempting to fit param-
eter sets with large negative CJ0 and C
1 . During the fit,
some nuclei sometimes fell into the instability, depending
on the values of the other coupling constants. As this is
a highly nonlinear threshold effect that results in a very
large energy gain from tiny modifications of the coupling
constants, the corresponding fits did not, and could not,
converge.
In special cases, one might even run into a situation
with two coexisting minima, where as a function of a
suitable coordinate the configuration with regular shell
structure is separated from a configuration with unphys-
ical large spin-orbit splittings by a barrier. In such a
120Sn
SLy5 (a)
SLy5 (a) (b)
1g9/2
2d5/2
1g7/2
3s1/2
2d3/2
1h11/2
2f7/2
3p3/2
3p1/2
2p3/2
2p1/2
1f5/2
1g9/2
2d5/2
3s1/2
1h11/2
2d3/2
1g7/2
FIG. 36: Single-particle spectra corresponding to the mini-
mum found with SLy5 and (a) the secondary minimum found
with TXX , (b) the absolute minimum (see Fig. 35; left: neu-
tron levels, right: proton levels).
case, a calculation of the ground state might converge
into one or the other minimum depending on the initial
conditions chosen for the iterative solution of the HFB
equations. In a calculation along an isotopic or isotonic
chain, the coexistence will reveal itself through a large
scattering of the mass residuals, which will fall on two
distinct curves. We illustrate this phenomenon in Fig. 35
for 120Sn using a parameter set denoted “TXX” with
CJ0 = −157.57 MeV fm
5 and CJ1 = −114.88 MeV fm
which is located outside the parameter space shown in
Fig. 1, to its lower left. Among the various possible
recipes for a constraint on the spin-orbit current density,
we chose to minimize the following quantity:
E[ρ]− µ
d3r Jn · ∇ρn − C
where µ is a Lagrange parameter and C is a constant used
to tune the constraint. The energy curve exhibits two
minima denoted (a) and (b). The corresponding single-
particle spectra are shown in Fig. 36 along with those
obtained for SLy5. The minimum (a) corresponds to an
almost spin saturated neutron configuration where both
spin partners are either occupied or empty,4 which is very
similar to what is found using SLy5. In the minimum (b),
which is deeper by more than 7 MeV, the single-particle
spectrum is completely reorganized in order to maximize
the spin-orbit current density and take advantage of its
contribution in the functional. In this situation the neu-
tron spin doublets 2d, 1g and 1h split on both sides of
the Fermi surface and generate a large spin-orbit current
density.
This clearly shows that the parameter sets with large
and negative coupling constants of the J2 terms must
be discarded since for many nuclei they lead to ground
states with unrealistic single-particle structure.
Note that this kind of instability does not appear for
the spin-orbit term: although its contribution to the en-
ergy functional (28) also varies between small, sometimes
near-zero, and very large values, see Figs. 28 and 29, it
is only linear in J. As a consequence, its contribution to
the spin-orbit potential (35) lacks the feedback mecha-
nism outlined above as it does not scale with J. Still, its
contribution to the total energy is usually much larger
than that of the J2 terms, so it plays a decisive role for
the absolute energy gained when varying J.
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