Improved strong lensing modelling of galaxy clusters using the Fundamental
Plane: detailed mapping of the baryonic and dark matter mass distribution of
Abell S1063

Giovanni Granata

Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16,
I-20133 Milano, Italy

Galaxy clusters are the most massive gravitationally bound structures in the
Universe, and around $85-90\%$ of their total mass is under the form of dark
matter (DM). As a consequence, they are excellent astrophysical laboratories
to test our hypoteses on the nature of DM itself. Thanks to several dedicated
photometric and spectroscopic surveys, strong gravitational lensing (SL) has
become the most accurate probe of the total mass distribution in the cores
(out to a few hundreds of kiloparsecs from the centre) of massive galaxy
clusters. SL can be combined with baryonic mass diagnostics to disentangle the
mass distribution of cluster- and galaxy-scale DM haloes from the total mass
distribution of the cluster. The resulting DM mass profiles can be compared to
the predictions of high-resolution cosmological simulations, based on the
$\Lambda$ cold dark matter (CDM) Cosmological Model.

The remarkable improvement in the accuracy of SL models, driven by recent
observational campaigns, has allowed us to map robustly the mass distribution
of the DM haloes hosting the member galaxies (usually referred to as sub-
haloes). On this scale, a significant discrepancy between the predictions of
SL models and high-resolution simulations has recently emerged: at a fixed
galaxy total mass, sub-haloes extracted from SL models are more compact than
their simulated counterparts [2].

The accuracy in the description of the cluster members in SL models is limited
by the degeneracies between the parameters defining their mass distribution.
These degeneracies have been reduced with the introduction of measured priors
on the velocity dispersion of the cluster galaxies. However, the usual choice
of adopting power-law scaling relations, with no scatter, to link the total
mass of members with their luminosity is still a simplified approach. To
obtain a more complex description of the cluster galaxies, we take advantage
of measured kinematics and structural parameters of the cluster galaxies to
calibrate the Fundamental Plane (FP) for the members of the massive galaxy
cluster Abell S1063 (AS1063). Our results are presented in [1]

Building the strong lensing model of Abell S1063

AS1063 is one of the six massive clusters included in the Hubble Frontier
Fields (HFF, [3]) photometric survey. We build a SL model of its total mass
distribution using the recent work by [4] as a starting point and reference.
Like [4], we model the diffuse DM and hot-gas mass distribution with
isothermal cored haloes. The parameters of the hot-gas mass distribution are
fixed from observations [5]. The parameters of the cluster-scale DM mass
distribution are optimised comparing the observed and model-predicted
positions of the same multiply-imaged background sources. Likewise, we model
the cluster galaxies with spherical, isothermal, truncated total mass
distributions. The total mass profile of each member is entirely defined by
two parameters: its velocity dispersion and its truncation radius. While in
[4] their values are derived with fixed power-law relations with respect to
their observed total luminosity, we choose to consider the FP, a more accurate
scaling law, which involves the magnitude, the half-light radius, and the
central velocity dispersion of early-type galaxies.

We use HFF images in the F814W band to measure the values of the magnitude and
of the half-light radius of all cluster members. Furthermore, for a sizeable
subset of them, MUSE-VLT integral field spectroscopy allows us to determine
the values of the line-of-sight central velocity dispersion. We calibrate the
FP relation, and then use it to obtain the values of the velocity dispersion
of all cluster members from their measured structural parameters. This
procedure allows for a more accurate determination of the velocity dispersion
values compared to the power-law approach. As for the truncation radii of the
galaxies, we calibrate a proportionality relation with their observed half-
light radii and use it to determine their values. Thanks to this procedure, we
can thus fix the total mass distribution of all member galaxies in a new SL
model of the cluster.

The more stringent constraints on the the mass distribution of the members
compared to previous works result in a reduction of the statistical
uncertainty on the parameters of the cluster-scale DM component, and in
particular on the value of the core radius of the main DM halo of the cluster.
The new procedure also allows for more realistic scaling laws between the
observables that describe the physical properties of the cluster members. For
instance, the relation between the values of their total mass and their
velocity dispersion is no longer a fixed power-law and now shows a significant
scatter and a shallower slope.

Mass profile decomposition

We decompose the total mass profile of the cluster derived from our new best-
fit SL model into all its baryonic and DM components. As anticipated, the hot-
gas mass profile is derived from X-ray observations. Combining instead the
measured stellar mass values of the cluster members, presented in [6], with
their surface brightness profile in the HST F814W band, we derive the stellar
mass profile of the cluster. We can thus disentangle the mass profiles of the
various cluster- and galaxy-scale DM haloes by subtracting the baryonic
component to the total mass distribution. This also allows us to derive the
cumulative projected gas-, stellar-, and baryonic-to-total mass fractions out
to a projected distance of $350\,\mathrm{kpc}$ from the cluster centre: at
this radius, we find a baryonic mass fraction of $0.147\pm 0.002$. The
profiles are presented in Figure 1.

Figure 1: Cumulative stellar-, gas-, and baryonic-to-total mass fractions from
the model presented in [1]. Error bars are derived from the observations.

Comparison with cosmological simulations

As anticipated, comparing the physical properties of the DM sub-haloes as
predicted by SL models to the most recent cosmological simulations is a test
of the foundations of the $\Lambda$ CDM cosmological model on which the
simulations are based, and of the micro-physics of DM. We first compare the
stellar-to-total mass fractions of the cluster members with the predictions of
recent HOD studies based on DM-only $N$-body simulations [7]. We find a
significant discrepancy: the stellar mass fraction values predicted by SL
models are almost an order of magnitude higher than those predicted by the HOD
procedure. This discrepancy is resolved if one considers, instead,
hydrodynamical simulations, which include gas particles and stars, as well as
the effects of the interaction between baryons and DM during the formation of
clusters. We consider high-resolution simulations of clusters with a mass
similar to that of AS1063 from [8]. We perform two-dimensional projections to
simulate the lensing observational conditions. In this case, we find
compatible stellar mass fraction values from the SL model and the simulation
suite.

Secondly, we examine how sub-haloes extracted from lensing models compare to
their simulated counterparts in terms of maximum circular velocity, which is a
proxy for their compactness. [2] recently found that hydrodynamical
simulations predict high-mass sub-haloes (total mass $M>10^{10}\,M_{\odot}$)
to be significantly less compact than forecast by a sample of state-of-the-art
SL models, including the model of Abell S1063 presented in [4]. The new
technique we adopt significantly impacts the relation between the total mass
and the maximum circular velocity of the sub-haloes, obtaining again a
different slope compared to [4] and allowing for the inclusion of a scatter.
However, as shown in Figure 2, our results agree with those from [4] in the
mass range considered, thus confirming the reported discrepancy. Several tests
to infer the origin of this discrepancy are being performed, focusing both on
SL modelling and on the implementation of the cosmological simulations.
However, no conclusive answer has been obtained so far. This leaves several
open questions, and could point towards a new fundamental challenge for the
$\Lambda$ CDM paradigm.

Figure 2: Maximum circular velocity of the cluster members as a function of
their total mass. The values predicted by the models presented in [1] and [4]
are shown in blue and in red, respectively. Only members with
$M>10^{10}\,M_{\odot}$ are shown.

References

## References

  * Granata [2022] Granata, G., et al., 2022, A&A, 659, A24
  * Meneghetti [2020] Meneghetti, M., et al., 2020, Science, 369, 1347
  * Lotz [2017] Lotz, J. M., et al., 2017, ApJ, 837, 97
  * Bergamini [2019] Bergamini, P., et al., 2019, A&A, 631, A130
  * Bonamigo [2018] Bonamigo, M., et al., 2018, ApJ, 864, 98
  * Mercurio [2021] Mercurio, A., et al., 2021, A&A, 656, A147
  * Girelli [2020] Girelli, G., et al., 2020, A&A, 634, A135
  * Planelles [2014] Planelles S., et al., 2014, MNRAS, 438, 195

Short CV 2018: BSc in Physics, University of Milan 2020: MSc in Physics,
University of Milan 2020–present: PhD in Physics, Astrophysics and applied
Physics, University of Milan


# Deformed Halo of ${}^{29}_{9}$F20

Ikuko Hamamoto1,2 1Division of Mathematical Physics, Lund Institute of
Technology at the University of Lund, Lund, Sweden
2Center for Nuclear Study, University of Tokyo, Bunkyo-ku, Tokyo 113-0033,
Japan

###### Abstract

Using a simple model based on the knowledge of spherical and deformed Woods-
Saxon potentials, it is shown that the recent observation of halo phenomena in
29F can be interpreted as evidence for the prolate deformation of the ground
state of 29F. The prolate deformation is the result of the shell structure,
which is unique in one-neutron resonant levels, in particular near degeneracy
of the neutron 1$f_{7/2}$ and 2$p_{3/2}$ resonant levels, together with the
strong preference of prolate shape by the proton number $Z$ = 9. On the other
hand, in oxygen isotopes spherical shape is so much favored by the proton
number $Z$ = 8 that the presence of possible neutron shell-structure may not
make the system deformed. Thus, the strong preference of particular shape by
the proton numbers 8 and 9, respectively, together with a considerable amount
of the energy difference between the neutron $1d_{3/2}$ and $2s_{1/2}$ orbits
in oxygen isotopes seems to play an important role in the phenomena of oxygen
neutron drip line anomaly, as was suggested by H.Sakurai et al. in 1999.

##  I INTRODUCTION

While the heaviest bound oxygen isotope that has the magic proton-number $Z$=8
is known as ${}^{24}_{8}$O16, the presence of bound fluorine nuclei,
${}^{29}_{9}$F20 and ${}^{31}_{9}$F22, is known. Namely, while the heaviest
bound isotope of oxygen accommodates only up to 16 neutrons, that of fluorine
($Z$=9), which has only one proton more, can accommodates up to 22 neurons.
Though having one proton more in fluorine isotopes makes the neutron potential
stronger so as to accommodate more neutrons, the increase of 6 (= 22-16)
neutrons in the heaviest bound fluorine isotope is unusual in the region of
light nuclei. For example, in Ref. DSA19 , the heaviest bound isotopes of
fluorine and neon are experimentally confirmed to be ${}^{31}_{9}$F22 and
${}^{34}_{10}$Ne24, respectively. Namely, the increase of the neutron number
in the heaviest bound isotope of neon ($Z$=10) from that of fluorine is only
2. The physics involved in the increase of 6 neutrons, sometimes called
”oxygen neutron drip line anomaly”, has been discussed in the literature. In
the abstract of Ref. HS99 , in which evidence for particle stability of 31F
was reported, it is stated that oxygen neutron drip line anomaly may
demonstrate the onset of deformation for neutron-rich fluorine isotopes. The
only direct experimental indication of the deformation of the ground state of
29F, which is known before 2020, is the result of the mass measurement of 29F
reported in Ref. LG12 . In a recent article SB20  it is reported that the
nucleus 29F shows a two-neutron halo by measuring the reaction cross section.
In the present paper I first concentrate on the interpretation of the recently
observed halo phenomena in 29F.

In order to study the halo phenomena, I try to use a simple model based on the
knowledge of spherical and deformed Woods-Saxon potentials. This is because
nuclear shape is anyway one-body property of the mean field, and the very
basic and simplest description of many-body systems is a (self-consistent)
mean-field approximation to the many-body problem. It is important to note
that in deformed nuclei one-particle picture is known to work BM75  much
better than in spherical nuclei because the major part of the residual
interaction in spherical mean field, quadrupole-quadrupole force, is already
included in the deformed mean field. The quantitative success in the analysis
of the data especially in some light odd-A nuclei such as ${}^{25}_{12}$Mg13
by using the one-particle picture in the deformed mean field can be found, for
example, in Ref. BM75 . In particular, in order to pin down the appearance of
halo phenomena, it is absolutely necessary to use the models in which radial
wave functions of weakly-bound particles are properly treated. In order to
obtain such wave functions of deformed halo nuclei with halo radial
dependence, the simple model used in the present paper seems to be more
appropriate than other model calculations available in the market.

Taking some simple one-body potentials the relation between the preferred
shape and the degree of shell filling can be analytically studied. In a single
$j$-shell oblate (prolate) shape is favored in the beginning (at the end) of
the shell filling, while in the harmonic-oscillator potential prolate (oblate)
shape is favored in the beginning of (at the end of) the filling of a major
shell. The characteristic feature of the shell structure expressed by the
harmonic oscillator potential is modified in realistic potentials decisively
by the presence of the spin-orbit potential as well as the feature of finite-
well potentials. Nevertheless, it is found that some important gross feature
except some phenomena such as halo can be seen to remain in simple but
slightly more realistic one-body potentials, for example, in a modified
oscillator potential BM75 .

The shape of a given nucleus is determined by the shell structure of both
protons and neutrons. The reason why fluorine isotopes can survive up to
$N$=22 might be guessed because the proton number $Z=9$ strongly favors
prolate deformation, as seen from the strong deformation-dependence of the
energy of the [220 1/2] orbit, which the 9th proton should occupy. The
dependence can be seen, for example, in Fig. 5-1 of Ref. BM75 . It is noted
that for prolate deformation the [220 1/2] orbit is the lowest-lying orbit
among the orbits coming from the $sd$ major shell. However, it has been so far
difficult to establish experimentally evidence for the deformation of those
fluorine isotopes close to the neutron drip line. Besides the mass measurement
of 29F in LG12 , it is reported in Ref. PD17  that the only bound excited
state of 29F is observed at $E_{x}$ = 1080 (18) keV. This rather low
excitation energy may be certainly interpreted as an indication of the
deformed ground state in 29F. The spin assignment of the ground and first
excited states of 29F in Ref. PD17  is based on shell-model calculations and
is so far not experimentally fixed. On the other hand, studying neutron
removal from 29F beams, in Ref. AR20  it is proposed that the ground state of
${}^{28}_{9}$F19 has $p$-wave valence neutron configuration. This may indicate
that 28F, the neighbor nucleus of 29F, is deformed.

Immediately after Ref. HS99  the structure of neutron-rich F isotopes was
studied in detail by means of large-scale shell-model calculations UT01 . The
calculated result of 29F in Ref. UT01 , in particular, shows large
probabilities of 2p-2h and 4p-4h components in the wave-function indicating
the deformation of the ground state of 29F. However, harmonic-oscillator wave-
functions used in the shell-model calculation and np-nh expansions make
impossible to pin down properly the properties depending on halo of deformed
nuclei.

In the present paper I show that the nucleus 29F is presumably prolately
deformed because, then, the presence of the neutron halo phenomena can be very
naturally understood. In other words, the observation of halo phenomena in the
nucleus 29F can be taken as evidence for prolate deformation of the ground
state of 29F. The proton number $Z$ = 9 in fluorine may definitely favor the
prolately deformed bound system and, consequently, the bound isotopes of
fluorine with $N$ = 20 and 22 may be present.

Deformed (neutron) halo expresses the halo phenomena to be observed in some
deformed nuclei close to the (neutron) drip line. Neutron shell structure and
the resulting possible deformation in neutron-drip-line nuclei are discussed
in Refs. IH07 ; IH10 ; IH12 . The change of the shell structure in weakly-
bound neutrons is described in p.238-240 of Ref.BM69 . Deformed halo has been
indeed observed in several odd-N nuclei: for example, in 31Ne TN09 ; TN14 , in
37Mg NK14  and in 29Ne NK16 . The degeneracy of the $f_{7/2}$\- and
$p_{3/2}$-shells and the consequence of Jahn-Teller effect JT37  play a
crucial role in the formation of deformed halo in these odd-N nuclei as well
as in the present nucleus 29F.

In Sec. II the main points of the model used are briefly summarized, and the
neutron one-particle levels as a function of axially-symmetric quadrupole
deformation are shown for the standard Woods-Saxon potential, of which
parameters are adjusted for 29F. Examining the figure obtained one finds that
in the $N=20$ system with weakly-bound neutrons the prolate deformation of
$\beta\approx 0.4$ may be energetically favored as much as spherical system.
On the other hand, since the proton number $Z$ = 9 would absolutely favor
prolate deformation, it may be expected that the nucleus 29F may be prolately
deformed. Then, the structure of the possible deformed halo for the prolate
deformation will be examined. Conclusion and discussions are given in Sec.
III.

##  II MODEL AND NUMERICAL CALCULATIONS

In Fig. 1 I show the neutron one-particle energies as a function of axially-
symmetric quadrupole deformation. Negative energies are eigenvalues of the
deformed Woods-Saxon potential, while positive energies express one-particle
resonant energies which are defined in terms of the eigenphase
$\delta_{\Omega}$ IH05 . Some neutron numbers, which are obtained by filling
all lower-lying levels, are indicated with open circles.

Since the model used is explained in detail in Ref. IH19 , here I summarize
only the important points. As a spherical one-body potential which is an
approximation to the (self consistent) mean field, I take the phenomenological
Woods-Saxon potential described in p.238-240 of Ref. BM69 . Then, I examine
the shell structure of one-particle energy spectra of neutrons in the axially-
symmetric quadrupole-deformed potential which is constructed in a standard
way. I write the single-particle wave-function in the body-fixed (intrinsic)
coordinate system as

$\Psi_{\Omega}(\vec{r})=\frac{1}{r}\,\sum_{\ell j}R_{\ell j\Omega}(r)\,{\bf
Y}_{\ell j\Omega}(\hat{r}),$ (1)

which satisfies

$H\,\Psi_{\Omega}=\varepsilon_{\Omega}\,\Psi_{\Omega}$ (2)

where $\Omega$ expresses the component of one-particle angular-momentum along
the symmetry axis and is a good quantum number. The values of the parameters
in the potential are taken from Ref. BM69 . For example, the diffuseness $a$ =
0.67 fm and the radius $R$ = $r_{0}A^{1/3}$ with $r_{0}$ = 1.27 fm. The depth
of the potential depends on the mass number $A$, the proton number $Z$ and the
neutron number $N$. For example, if one proton is added to a system, the depth
of the neutron potential becomes deeper reflecting the strong attractive
neutron-proton interaction.

The coupled differential equations for the radial wave functions of one-
particle orbits $R_{\ell j\Omega}(r)$, which are obtained from the Schrödinger
equation, are integrated in coordinate space with the correct asymptotic
behavior in respective $(\ell,j)$ channels for $r\rightarrow\infty$. In this
way, the $r$ dependence of low $(\ell,j)$ components of weakly-bound
low-$\ell$ neutrons expresses properly possible halo effect. See, for example,
Fig. 2, in which the $r$-dependence of various components of the weakly-bound
neutron orbit [330 1/2] that is the only $\Omega^{\pi}$ = 1/2- orbit expressed
by a long-dashed curve in Fig. 1, is shown. Indeed, in Fig. 2 it is seen that
$\ell=1$ components exhibit clear halo phenomena, while the $r$-dependence of
$\ell=3$ components is not so much different from that of well-bound case. For
comparison, see Fig. 6 in Ref. IH19  for a well-bound case of the [330 1/2]
orbit. In Fig. 2 one sees a typical example of the $r$-dependence of radial
wave functions of various $\ell$ components of weakly-bound neutrons in the
case of deformed halo.

For spherical shape ($\beta$ = 0) in Fig. 1 the large energy gap at $N$ = 20
is kept, namely one may say that the magic number $N$ = 20 remains for
spherical shape. On the other hand, the energy of the neutron Fermi level of
the $N$ = 20 system is nearly equal for $\beta$ = 0 and $\beta$ $\approx$
0.44. Noting that the shape of a nucleus is determined by both protons and
neutrons and, furthermore, the $Z$ = 9th proton outside the magic number $Z=8$
favors prolate deformation, from Fig. 1 it may be natural to expect that the
prolately-deformed state of 29F is lower in energy than the spherical state.
Namely, the ground state of 29F is not spherical but presumably prolately
deformed with $\beta\approx 0.4$. Now, if the ground state of 29F has such
prolate deformation, the neutron negative-parity level with
$\Omega^{\pi}=1/2^{-}$, which comes from the $pf$ major shell and is often
called [330 1/2], is just occupied.

I note that the reason why in Fig. 1 the [330 1/2] level, which is the lowest-
lying level among the levels coming from the $pf$-shell, comes down so steeply
as $\beta(>0)$ increases from zero is that in the spherical shape ($\beta=0$)
the resonant $2p_{3/2}$ level lies energetically close to and even lower than
the resonant $1f_{7/2}$ level. Consequently, as the deformation sets in, the
$2p_{3/2}$ and $1f_{7/2}$ levels immediately mix strongly. In other words, the
possible prolate deformation of 29F comes from the shell structure unique in
one-particle resonant neutrons IH16 .

In Fig. 2 the components of the radial wave function $R_{\ell j\Omega=1/2}(r)$
of the neutron [330 1/2] orbit at $\beta$ = 0.38 are shown as a function of
radial coordinate. It is seen that the occupation of the weakly-bound [330
1/2] orbit exhibits halo phenomena due to the components of weakly-bound $p$
neutrons, while no weakly-bound negative-parity orbits are occupied in the
$N=20$ system around spherical shape as seen from Fig. 1. In short, it can be
said that the observed halo phenomena in 29F indicate the prolate deformation
of the nucleus 29F.

One may wonder whether or not there is a considerable correlation in the two
weakly-bound neutrons. In the present paper I do not go into the correlation
problem. However, around $\beta$ = 0.4 in Fig. 1 one finds that just below $N$
= 20 the neutron $\Omega^{\pi}$ = 1/2- and 1/2+ levels are present while just
above $N$ = 20 the neutron $\Omega^{\pi}$ = 3/2- and 3/2+ levels are present.
Since those negative-parity weakly-bound neutron levels, $\Omega^{\pi}$ = 1/2-
and $\Omega^{\pi}$ = 3/2-, are expected to be the considerable components of
the wave function of the possibly correlated two neutrons and contain
considerable amounts of $2p_{3/2}$ component, the appearance of halo phenomena
is expected irrespective of the presence or absence of a considerable amount
of two-neutron correlation.

##  III CONCLUSION AND DISCUSSIONS

Inspired by the recent experimental work reporting the halo phenomena in 29F
SB20 , I have tried to understand the origin of the halo, using one-particle
neutron spectra obtained from a standard Woods-Saxon potential. The appearance
of the neutron halo can be naturally understood if the nucleus 29F is
prolately deformed. The prolate deformation is expected from combining the
proton number $Z=9$ with the one-particle energy spectra of weakly-bound
neutrons around $N=20$ as a function of deformation. In short, the presence of
the neutron halo indicates the prolate deformation of the nucleus 29F.
Furthermore, noting that the oxygen isotopes ($Z=8$) strongly favor spherical
shape and one expects a considerable energy difference between the neutron
$1d_{3/2}$ and $2s_{1/2}$ orbits for spherical shape, the shape difference
between the isotopes of oxygen and fluorine may be an important element in the
understanding of so-called oxygen neutron drip line anomaly, as was suggested
in Ref. HS99 . I note that the large energy difference between the neutron
$1d_{3/2}$ and $2s_{1/2}$ levels in oxygen isotopes has been already
recognized, for example, from the fact that the nucleus ${}^{24}_{8}$O16 has
been sometimes called a doubly magic nucleus. For example, see Ref. RVFJ09 .

In some available publications, for example Refs. RVFJ09  and TIT20 , it is
suggested that the oxygen neutron drip line anomaly comes from the fact that
due to the presence of a $1d_{5/2}$ proton in fluorine isotopes the proton-
neutron tensor interaction lowers the energy of the neutron $1d_{3/2}$ orbit
so as to make a long fluorine neutron drip line. However, this suggestion may
explain the drip line extended to $N$ = 20 but not to $N$ = 22, 31F.
Furthermore, a question is whether the strength of the tensor force, which is
needed to extend the neutron drip line, is acceptable or not.

From Fig. 1 it is seen that the nucleus 31F can be also prolately deformed due
to the proton number $Z=9$, however, whether or not it shows neutron halo
phenomena depends in a subtle way on the size of the deformation, as seen from
Fig. 1. If the size of the prolate deformation is such that the last two
weakly-bound neutrons in 31F occupy mainly the $\Omega^{\pi}$ = 3/2+ orbit
coming from the $1d_{3/2}$ orbit, the halo phenomena could be rather weak,
even if the neutron $\Omega^{\pi}=1/2^{-}$ level is occupied.

What is described in the present article is based on the simple
phenomenological one-body potential, Woods-Saxon potential. Though simple
models can be expected to show some essential points of physics in a simple
way, it is hoped that advanced microscopic models, which can explain also halo
phenomena of 29F in an acceptable way, will quantitatively show the physics
that is described in the present article.

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Figure captions

Figure 1 :

Calculated neutron one-particle energies in the potential produced by 29F as a
function of quadrupole deformation parameter $\beta$. Some neutron numbers,
which are obtained by filling all lower-lying levels, are indicated with open
circles. One-particle bound and resonant energies at $\beta=0$ are $-$5.956,
$-$3.713, $-$1.041, and $+$3.002 MeV for the $1d_{5/2}$, $2s_{1/2}$,
$1d_{3/2}$, and $1f_{7/2}$ levels, respectively. The width of one-particle
resonant level of $1f_{7/2}$ is 0.422 MeV. The one-particle resonant level of
$2p_{3/2}$, which is expected below that of $1f_{7/2}$ at $\beta=0$, is not
obtained when the one-particle resonance is defined in terms of eigenphase.
The unusual behavior of the $\Omega^{\pi}=1/2^{-}$ curve in the region of
$0.1<\beta<0.25$, which is connected to $1f_{7/2}$ at $\beta=0$, is due to the
fact that the one-particle resonant level cannot be obtained for
$\varepsilon_{\Omega}>$ 1 MeV when the major component is $\ell=1$.
Nevertheless, figure indicates that at $\beta=0$ the $p_{3/2}$ one-neutron
resonant level lies lower than the $f_{7/2}$ one-neutron resonant level. Some
one-particle resonant levels ($\varepsilon_{\Omega}>0$) for $\beta\neq 0$,
which are defined in terms of the eigenphase $\delta_{\Omega}$ IH05 , are not
plotted if they are not relevant to the present interest.

Figure 2 :

An example of deformed halo wave-functions: Components of the radial wave
function $R_{\ell j\Omega=1/2}(r)$ of the neutron $\Omega^{\pi}=1/2^{-}$
orbit, so-called [330 1/2], expressed by a long-dashed curve in Fig. 1, at
$\beta$ = 0.38 are shown as a function of radial coordinate. Respective $(\ell
j)$ quantum numbers are denoted without writing the radial node quantum-number
$n$, because the wave functions are not eigenfunctions of any spherical
potential. The probabilities of respective $(\ell j)$ components in the wave
function are 0.070, 0.537, 0.009, and 0.376 for $p_{1/2}$, $p_{3/2}$,
$f_{5/2}$, and $f_{7/2}$, respectively. Since the probabilities of $h_{9/2}$
and $h_{11/2}$ are only 0.000 and 0.008, respectively, the $h_{9/2}$ and
$h_{11/2}$ wave functions are not plotted. The thin vertical line at $r$= 3.90
fm denotes the radius of the Woods-Saxon potential used.