∎

11institutetext: G. Grams, J. Margueron, R. Somasundaram 22institutetext: Univ
Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622,
Villeurbanne, France 33institutetext: S. Reddy 44institutetext: Institute for
Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA
JINA-CEE, Michigan State University, East Lansing, MI, 48823, USA

# Properties of neutron star crust with improved nuclear physics

Impact of chiral EFT interactions and experimental nuclear masses

Guilherme Grams  Jérôme Margueron  Rahul Somasundaram  Sanjay Reddy

(Received: date / Accepted: date)

###### Abstract

A compressible liquid-drop approach adjusted to uniform matter many-body
calculations based on chiral EFT interactions and to the experimental nuclear
masses is used to investigate the neutron star crust properties. Eight chiral
EFT hamiltonians and a representative phenomenological force (SLy4) are
confronted. We show that some properties of the crust, e.g. clusters mass,
charge, and asymmetry, are mostly determined by symmetric matter properties
close to saturation density and are therefore mainly constrained by
experimental nuclear masses, while other properties, e.g., energy per
particle, pressure, sound speed, are mostly influenced by low-density
predictions in neutron matter, where chiral EFT and phenomenological forces
substantially differ.

###### Keywords:

Neutron star crust chiral EFT interactions experimental nuclear masses

##  1 Introduction

Neutron stars (NS) are excelent laboratories to test nuclear physics models
under extreme conditions of density and isospin asymmetry Rezzolla2018 . They
are mainly divided into two regions: the core which contains about 99% of the
mass and the crust. In this study, we will focus on the properties of the
crust since it coincides with the density region where chiral EFT approach is
well defined. The upper bound in density of the crust is connected to the
saturation energy density of nuclear matter, $\rho_{\mathrm{sat}}\approx
2.6\times 10^{14}$ g cm-3. The outer crust contains a mixture of electrons and
nuclear clusters, complemented with a neutron fluid in the inner crust. This
neutron fluid is the unique physical realization of neutron matter (NM),
although it is not a pure uniform matter phase. From a nuclear theory view
point, NM presents several advantages which makes it easier to model and
accurate to predict. One may then wonder how the properties of NM, which are
predicted by several modelings, impact those of the crust, which can be
related to observations.

Several important and conceptual steps have recently been performed in the
prediction of NM from microscopic approaches based on the bare nuclear
interaction, i.e. the interaction between nucleons determined by free nucleon
scattering up to a few hundreds of MeV. The main one was carried by the
advance of nuclear interactions capturing low energy QCD symmetries in an
effective way (chiral effective field theory, $\chi_{EFT}$) and called chiral
NN interactions. This approach also generates many-body interactions
consistently with the order considered for the two-body interaction. They have
been applied to predict homogeneous nuclear matter properties up to the break-
down density, i.e. the maximal density at which $\chi_{EFT}$ approach can be
used, from various many-body approaches Carbone2013 ; Hebeler2015 ;
Drischler2019 ; Lynn2019 ; Rios2020 . At sufficiently low densities, NM is
well understood because three-body interactions are small, and the two-body
neutron-neutron interaction is strongly constrained by neutron-neutron
scattering phase shifts Heiselberg2000 ; Carlson2003 . The precise location of
the break-down density is still a matter of debate, and it is presently
expected to be at around saturation density ($n_{\mathrm{sat}}\approx 0.155$
fm-3) or slightly above Epelbaum2009 ; Tews2018 ; Drischler2020 . Note that
$\rho_{\mathrm{sat}}$ and $n_{\mathrm{sat}}$ represent the same quantity in
two different units.

Some properties of the inner crust are mainly constrained by those of NM,
which implies that the inner crust can be viewed as a uniform distribution of
neutrons and electrons modified by a few impurities (nuclear clusters), while
some other properties reflect the cluster properties, which makes the inner
crust viewed as a continuation of the nuclear chart, where isolated nuclei
becomes dilute nuclei surrounded by a shallow neutron fluid. The scope of this
study is to analyze which properties of the inner crust are mostly impacted by
the NM modeling, and can thus be considered as a way to probe the predictions
from $\chi_{EFT}$, and which ones mainly reflect the properties of isolated
nuclei, for which accurate experimental data exist.

To do so, one needs a modeling for the NS crust accurate enough in its
prediction, but which allows as well a clear understanding of the different
contributions to the equation of state (EoS). In their seminal paper in 1971,
Bethe, Baym, and Pethick Baym1971  have introduced the first version of the
compressible liquid-drop model (CLDM) with finite-size (FS) contribution
containing Coulomb and surface terms. Latter on Douchin and Haensel have
developed further this model by considering in 2001 the curvature term
contribution to the FS effects DouchinHaensel2001 , determined from many-body
methods DouchinHaensel2000 . In our study, we adopt the CLDM approach, as in
Ref. Carreau2019a , where the bulk term is fixed by the nuclear meta-model
Margueron2018a  and the FS term contains Coulomb, surface and curvature
contributions as described by the FS4 approximation presented in Ref.
Grams2021 . The FS terms may however still be fine tuned on experimental
nuclear masses, as suggested by Steiner Steiner2008  since they are derived in
the CLDM by neglecting the smoothing effect of the surface. The meta-model is
adjusted to reproduce the many-body perturbation theory predictions in uniform
matter from $\chi_{EFT}$ NN and 3N interactions, as first suggested by Tews
Tews2016 . With such a CLDM constrained by uniform matter calculations based
on several $\chi_{EFT}$ interactions on one side, and by the experimental
nuclear masses on the other side, the modeling of the crust properties is
qualitatively well under control and the connection between the different
ingredients in the model and the prediction is rather clear.

In the following, we first confront uniform matter calculations based on
$\chi_{EFT}$ interactions to the ones based on a representative
phenomenological force such as SLy4 Chabanat1997 . We then show that the fits
to experimental nuclear masses disfavor some of the $\chi_{EFT}$ Hamiltonians.
Finally, we compute NS crust properties, discuss their connection to NM
predictions and to the experimental nuclear masses, and confront our best
models to well-known EoS.

##  2 Uniform matter predictions based on $\chi_{EFT}$ interactions

Figure 1: Left: $\beta$-equilibrium energy per particle as function of the
baryon density for the $\chi_{EFT}$ Hamiltonians (the six hamiltonians H1-H7
Drischler2016 , except H6, and the two recent ones DHSL59 and DHSL69
Drischler2021 ) and the SLy4 force Chabanat1997  (NM: dashed line,
$\beta$-equilibrium: dotted line), pale orange band corresponds to the NM
energy per particle prediction for H1-H7. Right: $\beta$-equilibrium energy
per particle normalized to the NM Free Fermi Gas energy
$E_{\mathrm{FFG}}^{\mathrm{NM}}$ function of the Fermi momentum $k_{FB}$.

In a recent paper Grams2021 , we have presented in detail the CLDM that we use
in the present study and detailed how the bulk properties are connected to
uniform matter properties determined from $\chi_{EFT}$ interactions, by using
the meta-model approach Margueron2018a . The $\chi_{EFT}$ predictions that we
are analyzing are the many-body perturbation predictions based on the H1-H7
hamiltonians Drischler2016 , except H6 for which the 3H binding energy is not
well fit Drischler2016 , and the two DHS (DHSL59 and DHSL69) recent
predictions Drischler2021 .

The $\beta$-equilibrium properties in uniform matter, derived from the
$\chi_{EFT}$ approach as well as by the SLy4 force are shown in Fig. 1. The
energy per particle for H1-H7 (except H6), for DHS Hamiltonians and for SLy4
are shown first as function of the density (left panel) and then the energy
per particle normalized to the FFG energy
$E_{\mathrm{FFG}}^{\mathrm{NM}}=3\hbar^{2}k_{Fn}^{2}/(10m)$, with the neutron
Fermi momentum $k_{Fn}=(3\pi^{2}n_{B})^{1/3}$, as function of the isoscalar
Fermi momentum $k_{FB}=(3\pi^{2}n_{B}/2)^{1/3}$ (right panel). As expected, at
low densities the predictions of all $\chi_{EFT}$ converge to a narrow band,
while SLy4 predicts slightly larger values for the energy per particle. The
differences between SLy4 and chiral EFT predictions at low density are better
observed on the right panel where the energy per particle is normalized to the
Free Fermi Gas energy. SLy4 predicts up to 30% larger energy per particle
compared to the $\chi_{EFT}$ group for $0.1<k_{FB}<0.9$ fm-1, the largest
differences being observed for $k_{FB}\simeq 0.4$ fm-1. In comparison to this
difference, the $\chi_{EFT}$ predictions are much closer for $k_{FB}<0.9$
fm-1. The dispersion among the $\chi_{EFT}$ predictions is less or about a few
percent for $k_{FB}<0.9$ fm-1 and above the dispersion is larger than the
difference between SLy4 and chiral EFT.

The question which then arises is whether this systematic difference between
the group of $\chi_{EFT}$ predictions and SLy4, which is a typical prediction
for phenomenological models, impacts the NS crust properties. And if yes, for
which specific properties?

Figure 2: Left: Residuals for $N=Z$ nuclei for SLy4 and $\chi_{EFT}$
Hamiltonians. Right: Residuals for Pb isotopes ($Z=82$). Light grey band shows
the selected Hamiltonians H1-H4, dark grey band shows the two best Hamiltonian
(H2 and H3) when confronted to the experimental nuclear masses. For clarity,
the experimental data are corrected by the odd-even mass staggering as:
$E_{\mathrm{cl}}^{\mathrm{oem}}(A,I)=a_{\mathrm{oem}}A^{-1/2}\delta_{np}$,
where $a_{\mathrm{oem}}=-12$ MeV and $\delta_{np}=1$ if $N$ and $Z$ odd, $0$
if $N$ or $Z$ odd, and $-1$ if $N$ and $Z$ even BohrMottelson1969 .

##  3 Application of the CLDM to finite nuclei

Before addressing the links between $\chi_{EFT}$ predictions in NM and NS
crust properties, we briefly illustrate the role played by finite nuclei to
constrain the CLDM.

We confront in Fig. 2 the CLDM prediction for finite nuclei energies with the
experimental masses AME2016  for symmetric nuclei (left panel) and for neutron
rich Pb isotopes fixing $Z=82$ (right panel). The ability of the models H1-H4
to reproduce low- and high-A is quite uniform, while the models H5, H7 as well
as DHSL59 and DHSL69 depart from the trend given by the best hamiltonians, at
low-$A$ and for asymmetric nuclei (neutron rich or neutron poor). The
confrontation with the experimental nuclear masses allows us to select H1-H4
hamiltonians (with light-gray band) as being our optimal models, see Ref.
Grams2021  for more details, with H2-H3 (dark-gray band) being the best. Note
that the phenomenological SLy4 force reproduces experimental nuclear masses
with an accuracy equivalent to our best models.

Figure 3: Energy per particle as function of the baryon density around
saturation density $n_{\mathrm{sat}}\approx 0.16$ fm-3 in symmetric matter.
There is correlation between the energy per particle in SM at
$n_{\mathrm{sat}}$ and the residuals shown in Fig. 2.

It is interesting to connect these results with the energy per particle
predicted by these models in symmetric matter (SM) shown in Fig. 3. Even if
the contribution from the FS terms (Coulomb and surface) are absent in uniform
matter, one could observe a clear correlation between the CLDM predictions for
$N=Z$ nuclei shown in Fig. 2 and the energy per particle from Fig. 3 at around
$n_{\mathrm{sat}}$. This shows the large impact of the saturation properties
in uniform matter on the experimental nuclear masses, as expected.

In the following, the different CLDM constructed upon $\chi_{EFT}$ are
weighted according to their reproduction of the experimental nuclear masses:
H2 and H3 are the best $\chi_{EFT}$ models and define the dark-gray bands
plotted for NS crust predictions, then come H1 and H4 which define the light-
gray bands, and include the dark-gray one as expected. The others (H5, H7,
DHSL59 and DHSL69) are represented but they will depart from the best-model
bands.

##  4 Neutron star crust constrained by $\chi_{EFT}$ and experimental nuclear
masses

We now come to the description of the crust, where the CLDM model for finite
nuclei is completed by the contributions of the neutron fluid and of the
electrons gas, which also interact with the protons by the Coulomb
interaction. At variance with isolated nuclei, nuclear systems in NS crust are
at $\beta$-equilibrium and the Wigner-Seitz cell components respect charge
neutrality. Details of the model are given in Ref. Grams2021 .

The impact of nuclear physics uncertainties on the NS crust has already been
analyzed in various ways. Steiner Steiner2008 , for instance, has constructed
several NS crust using inputs from current experimental information while
allowing exploration of the EoS uncertainties, in particular the one induced
by the symmetry energy. Tews Tews2016  has analyzed the impact of $\chi_{EFT}$
predictions in NM introducing bands reflecting $\chi_{EFT}$ uncertainties.
Other approaches have been constructed from currently available EoSs, which
may or may not consider the actual predictions for low-density neutron matter
Baym1971 ; DouchinHaensel2001 ; Carreau2019a ; Newton2012 ; Fantina2013 ;
Vinas2017 ; Carreau2020 .

In the present work, we explore both the theoretical uncertainties in the
nuclear matter equation of state from many-body approach based on $\chi_{EFT}$
NN and 3N interactions, as well as the confrontation with the experimental
nuclear masses. This is the first approach which investigates in a
systematical way these two types of constraints, considering the latest
$\chi_{EFT}$ calculations and using the confrontation with experimental
nuclear masses to select among the best models for NS crust predictions.

Figure 4: Composition (top), asymmetry and $2Y_{e}$ (bottom left) and volume
fraction occupied by nuclei (bottom right) on the neutron star crust as
function of the Fermi momentum.

We display in Fig. 4 some properties of the NS crust determined by the
$\chi_{EFT}$ Hamiltonians and the SLy4 phenomenological force. On the top
panels we show the predictions for the composition: the cluster mass number
$A_{\mathrm{cl}}$ (left) and proton number $Z_{\mathrm{cl}}$ (right) through
the crust. Note that $A_{\mathrm{cl}}$ and $Z_{\mathrm{cl}}$ are the masses
and proton numbers associated to the nuclear cluster only, excluding the
contribution of the neutron fluid. Here we prefer for consistency and easy
comparison among the results to show all our results in terms of the Fermi
momentum $k_{FB}$. On the bottom panels we show the cluster asymmetry
$I_{\mathrm{cl}}=(N_{\mathrm{cl}}-Z_{\mathrm{cl}})/A_{\mathrm{cl}}$ and
electron fraction $Y_{e}$ (left panel) and the volume fraction
$u=V_{\mathrm{cl}}/V_{\mathrm{WS}}$ (right panel).

Concerning the predictions for $A_{\mathrm{cl}}$, $Z_{\mathrm{cl}}$,
$I_{\mathrm{cl}}$ and $Y_{e}$, SLy4 is compatible with the predictions of the
$\chi_{EFT}$ models (inside the light-gray band). It is remarkable to observe
that the predictions of SLy4 are located between the ones of H1 and H2, as
already observed in Fig. 3 for SM. One can then conclude that these
predictions are tightly connected to the SM properties of these models, itself
closely related to the experimental nuclear masses, and only loosely related
to the NM predictions.

The volume fraction $u$ however shows a marked difference between the
prediction of SLy4 and the ones from the $\chi_{EFT}$, which are well grouped
together. The volume fraction $u$ predicted by SLy4 is larger that the chiral
EFT group by about 20-30%.

Figure 5: Left: energy per particle in the crust for the $\chi_{EFT}$
Hamiltonians and SLy4 phenomenological force. Right: sound speed in the crust
predicted by the same models. Figure 6: EoS (pressure versus energy density)
in cgs units for the $\chi_{EFT}$ Hamiltonians and SLy4 phenomenological
force.

Other quantities impacted by the NM properties are the energy per particle in
the crust, the pressure and the sound speed. The energy per particle and the
sound speed are plotted in Fig. 5 and the EoS, i.e. the pressure as a function
of the energy density, in Fig. 6. We observe that in the density range where
$k_{F}<1.0$ fm-1 the prediction for the energy per particle from SLy4 force is
about 10-30% different from the predictions of the group of $\chi_{EFT}$
Hamiltonians, see Fig. 5. The pressure being defined as the density derivative
of the energy per particle is also impacted by this marked effect, see Fig. 6.
Finally, the sound speed is obtained as the derivative of the pressure with
respect to the energy density, which for the considered densities is almost
identical to the density $n_{B}$. The sound speed predicted by SLy4 is also
markedly different from the group of $\chi_{EFT}$ Hamiltonians. Note however
that for $k_{FB}>0.6$ fm-1 there is a large spreading in the prediction of
$\chi_{EFT}$ Hamiltonians which reduces a bit the effect observed for the
energy per particle.

In summary, the present study illustrates that different NS crust observables
are related to different properties of uniform matter. For instance, the
cluster mass $A_{\mathrm{cl}}$, charge $Z_{\mathrm{cl}}$ and asymmetry
parameter $I_{\mathrm{cl}}$ are essentially fixed by the SM or close to SM
properties where the NM does not play an important role. These properties are
controlled by the nuclear masses, and the models can be sorted according to
their reproduction of the experimental data. NM properties impact however
other observables, such as the volume fraction $u$, the energy per particle,
as well as the internal pressure and the sound speed. These latter observables
are the ones which are greatly influenced by the NM predictions based on
chiral EFT, and their accurate prediction can therefore be explained by the
relatively well known properties of NM.

##  5 Comparison to predictions from other EoS

Figure 7: Crust composition $A_{\mathrm{cl}}$, $Z_{\mathrm{cl}}$ as function
of the baryon density. Comparison among $\chi_{EFT}$ H1-H4, SLy4MM and
previous works.

Finally, we compare our predictions for the NS crust properties to previously
determined EoS in Fig. 7. Our dark-grey contour is given by H2 and H3, while
the light-grey contour is defined by H1 and H4. The SLy4 crust properties are
shown in dotted red cruve, and can be compared to the Douchin-Haensel
predictions in green squares, which are based on the same interaction model.
As a consequence, the SLy4 and Douchin-Haensel EoS are very close, and the
small differences which are observed are due to small differences in the CLDM
model.

For completeness, we represent in Fig. 7 the seminal BBP Baym1971  and Negele-
Vautherin Negele1973  EoS which are the first realistic predictions for the NS
crust properties. Our light-grey band is compatible with these other EoS shown
in Fig. 7. The dark-grey band is however narrower and represents the actual
uncertainties in the NS crust properties by combining together $\chi_{EFT}$
calculation and the experimental nuclear masses.

##  6 Conclusions

In this study, we have shown that combining together the latest predictions
for the uniform matter EoS from $\chi_{EFT}$ approaches with the experimental
nuclear masses allows predictions of the NS crust properties which are
narrower than the actual uncertainty resulting from the confrontation of
various EoS previously known. These predictions were indeed made, for most of
them, before the recent achievement from the $\chi_{EFT}$ approach. One can
therefore attribute the reduction of the uncertainties in the predictions of
the NS crust properties to the advances in the modeling of low-density nuclear
matter made possible by $\chi_{EFT}$.

In addition, the prediction for the NM EoS at low density, where the unitary
limit plays an important role in constraining the matter properties, impacts
some NS observables, such as the volume fraction $u$, the energy per particle,
the pressure and the sound speed. In the future, it will be interesting to
study more closely the role of the unitary limit constrain in NM on these
properties.

###### Acknowledgements.

G.G., J.M. and R.S. are supported by CNRS grant PICS-08294 VIPER (Nuclear
Physics for Violent Phenomena in the Universe), the CNRS IEA-303083 BEOS
project, the CNRS/IN2P3 NewMAC project, and benefit from PHAROS COST Action
MP16214. S.R. is supported by Grant No. DE-FG02-00ER41132 from the Department
of Energy , and the Grant No. PHY-1430152 (JINA Center for the Evolution of
the Elements), and PHY-1630782 (Network for Neutrinos, Nuclear Astrophysics,
and Symmetries (N3AS)) from the National Science Foundation. This work is
supported by the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the
Université de Lyon for its financial support within the program
Investissements d’Avenir (ANR-11-IDEX-0007) of the French government operated
by the National Research Agency (ANR).

## References

  * (1) L. Rezzolla, P.A.M. Pizzochero, D.I. Jones, N. Rea, I. Vidaña, _The Physics and Astrophysics of Neutron Stars_ , vol. 457 (Springer International Publishing, 2018).  DOI 10.1007/978-3-319-97616-7.  URL https://www.springer.com/gp/book/9783319976150
  * (2) A. Carbone, A. Cipollone, C. Barbieri, A. Rios, A. Polls, Phys. Rev. C 88, 054326 (2013)
  * (3) K. Hebeler, J. Holt, J. Menéndez, A. Schwenk, Annual Review of Nuclear and Particle Science 65(1), 457 (2015)
  * (4) C. Drischler, K. Hebeler, A. Schwenk, Phys. Rev. Lett. 122, 042501 (2019)
  * (5) J. Lynn, I. Tews, S. Gandolfi, A. Lovato, Annual Review of Nuclear and Particle Science 69(1), 279 (2019)
  * (6) A. Rios, Frontiers in Physics 8, 387 (2020)
  * (7) H. Heiselberg, V. Pandharipande, Annual Review of Nuclear and Particle Science 50(1), 481 (2000).  DOI 10.1146/annurev.nucl.50.1.481.  URL https://doi.org/10.1146/annurev.nucl.50.1.481
  * (8) J. Carlson, J. Morales, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 68, 025802 (2003).  DOI 10.1103/PhysRevC.68.025802.  URL https://link.aps.org/doi/10.1103/PhysRevC.68.025802
  * (9) E. Epelbaum, H.W. Hammer, U.G. Meißner, Rev. Mod. Phys. 81, 1773 (2009)
  * (10) I. Tews, J. Carlson, S. Gandolfi, S. Reddy, The Astrophysical Journal 860(2), 149 (2018).  DOI 10.3847/1538-4357/aac267.  URL https://doi.org/10.3847/1538-4357/aac267
  * (11) C. Drischler, R.J. Furnstahl, J.A. Melendez, D.R. Phillips, Phys. Rev. Lett. 125, 202702 (2020)
  * (12) G. Baym, H.A. Bethe, C.J. Pethick, Nucl. Phys. A175(2), 225 (1971).  DOI 10.1016/0375-9474(71)90281-8
  * (13) F. Douchin, P. Haensel, Astron. & Astrophys. 380, 151 (2001)
  * (14) F. Douchin, P. Haensel, J. Meyer, Nucl. Phys. A 665, 419 (2000)
  * (15) T. Carreau, F. Gulminelli, J. Margueron, Eur. Phys. J. A 55, 188 (2019)
  * (16) J. Margueron, R. Hoffmann Casali, F. Gulminelli, Phys. Rev. C 97, 025805 (2018)
  * (17) G. Grams, R. Somasundaram, J. Margueron, S. Reddy, To be submitted (2021)
  * (18) A.W. Steiner, Phys. Rev. C 77, 035805 (2008).  DOI 10.1103/PhysRevC.77.035805
  * (19) I. Tews, Phys. Rev. C 95, 015803 (2017).  DOI 10.1103/PhysRevC.95.015803.  URL https://link.aps.org/doi/10.1103/PhysRevC.95.015803
  * (20) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys. A 627, 710 (1997)
  * (21) C. Drischler, K. Hebeler, A. Schwenk, Phys. Rev. C 93, 054314 (2016)
  * (22) C. Drischler, J. Holt, C. Wellenhofer, ArXiv:2101.01709v1 [nucl-th] (2021)
  * (23) A. Bohr, B.R. Mottelson, _Nuclear structure vol I_ (Addison-Wesley, 1969)
  * (24) M. Wang, G. Audi, F.G. Kondev, W.J. Huang, S. Naimi, X. Xu, Chinese Physics C 41(3), 030003 (2017).  DOI 10.1088/1674-1137/41/3/030003
  * (25) W.G. Newton, M. Gearheart, B.A. Li, The Astrophysical Journal Supplement Series 204(1), 9 (2012)
  * (26) A.F. Fantina, N. Chamel, J.M. Pearson, S. Goriely, Astron. & Astrophys. 559, A128 (2013)
  * (27) X. Vinas, B.K. Sharma, C. Gonzales-Boquera, M. Centelles, Bulg. J. Phys. 44, S81 (2017)
  * (28) T. Carreau, F. Gulminelli, N. Chamel, A.F. Fantina, J.M. Pearson, Astron. & Astrophys. 635, A84 (2020)
  * (29) J.W. Negele, D. Vautherin, Nucl. Phys. A207(2), 298 (1973).  DOI 10.1016/0375-9474(73)90349-7