P Klaja
email: p.klaja@fz-juelich.de

P Moskal
email: p.moskal@fz-juelich.de

E Czerwiński

R Czyżykiewicz

A Deloff

D Gil

D Grzonka

L Jarczyk

B Kamys

A Khoukaz

J Klaja

W Krzemień

W Oelert

J Ritman

T Sefzick

M Siemaszko

M Silarski

J Smyrski

A Täschner

M Wolke

J Zdebik

M Zieliński

W Zipper

Two-proton correlation function for the pp → pp + η and pp → pp + pions reactions

Keywords: numbers: 13.60.Hb, 13.60.Le, 13.75.-n, 25.40.Ve

  L'archive ouverte pluridisciplinaire

Introduction

Momentum correlations of particles at small relative velocities are widely used to study the spatio-temporal characteristics of the production processes in relativistic heavy ion collisions [1]. This technique, called after Lednicky correlation femtoscopy [2], originates from photon intensity interferometry initiated by Hanbury-Brown and Twiss [3]. Implemented to nuclear physics [2,4,5] it permits to determine the duration of the emission process and the size of the source from which the particles are emitted [2]. A central role plays the correlation function which has been defined as the measured two-particle distribution normalized to a reference spectrum obtained by mixing particles from different events [2]. The importance of the correlation femtoscopy has been well established for investigations of the dynamics of heavy ion collisions with high multiplicity. However, as pointed out by Chajecki [6], in the case of low-multiplicity collisions the interpretation of correlation function measurements is still not fully satisfactory, especially in view of the surprising observation by the STAR collaboration indicating universality of the resulting femtoscopic radii for both, the hadronic (proton-proton), and heavy ion collisions [7]. One of the challenging issues in this context is the understanding of contributions from non-femtoscopic correlations which may be induced by the decays of resonances, global conservation laws [6], or by other unaccounted interactions. In contrast to heavy ion collisions, in the case of single meson production, the kinematics of all ejectiles may be entirely determined and hence a kinematically complete measurement of meson production in the collisions of hadrons gives access to complementary information which could shed light on the interpretation of the two-proton correlations observed in heavy ion reactions. It is also important to underline that the correlation of protons was never exploited till now in near threshold meson productions, and as an observable different from the distributions of cross sections, it may deepen our understanding of the dynamics of meson production. Particularly favourable are exclusive experiments conducted close to the kinematical threshold where the fraction of the available phase-space associated with low relative momenta between the ejectiles is large [8].

In this article we report on a η meson and multi-pion production experiment in which the mesons were created in collisions of protons at a beam momentum of 2.0259 GeV/c corresponding to an excess energy of Q = 15.5 MeV for the pp → ppη reaction. The measurement of the two-proton correlation function for these reactions is important not only in the context of studying the dynamics underlying the heavy ion physics. Such investigations are interesting by themselves because they offer a new promising diagnostic tool, still not exploited, for studying the dynamics of meson production in hadron collisions.

The correlation function carries information about the emitting source and, in particular, about the size of the interaction volume of the pp → ppη process. The knowledge of this size might be essential to answer the intriguing question whether the three-body ppη system is capable of supporting an unstable Borromean bound state. Borromean systems may be realized in a variety of objects on the macroscopic (e.g. strips of papers), molecular [9,10] and nuclear scale (e.g. 11 Li or 6 He nuclei [11][12][13]). According to Wycech [14], the large enhancement of the excitation function for the pp → ppη reaction observed close to the kinematical threshold may be explained by assuming that the proton-proton pair is emitted from a large (Borromean like) object whose radius is about 4 fm.

Experimental technique

The experiment was conducted using the proton beam of the cooler synchrotron COSY [15] and an internal hydrogen cluster target [16]. Momentum vectors of outgoing protons from the pp → ppX reaction were measured by means of the COSY-11 facility [17] presented schematically in Figure 1.

The two-proton correlation function R(k) was determined for the ppη and pp(pions) systems, respectively. Here, R(k) denotes a projection of the correlation function onto the momentum of one of the protons in the proton-proton center-of- [17]. Superimposed lines indicate trajectories of protons from the pp → ppη reaction. The positively charged particles were identified using the time of flight between the scintillator detectors S1 and S3 combined with a momentum reconstruction by tracking back the trajectories (reconstructed based on the signals from the drift chambers D1 and D2) through the magnetic field of the COSY dipole to the reaction point. The size of the detectors and their relative distances are not to scale. More detailed description can be found in [17][18][19][20]. mass system . It was calculated, by means of the well established technique, as a ratio of the momentum (k) dependent reaction yield Y (k) to the uncorrelated yield Y * (k) according to the formula (cf. [21])

R(k) + 1 = C Y (k) Y * (k) , ( 1
)
where C denotes an appropriate normalization constant. Y * (k) was derived from the uncorrelated reference sample obtained by using the event mixing technique introduced by Kopylov and Podgoretsky [5].

The separation of the correlations of the pp + η from the pp + pions system and corrections for the limited acceptance of the detection system constitute the two main challenges to be solved when deriving the correlation function from the experimental data.

Separation of events from the production of ppη and pp + pions systems

In the discussed experiment, the four-momenta of the two final state protons were measured only, and the unobserved meson was identified via the missing mass technique [18,20]. In such situations the entirely accessible information about the reaction is contained in the momentum vectors of the registered protons. Therefore, it is impossible to know whether in a given event the η meson or a few pions have Note, that some authors instead of k take as the independent variable the relative momentum of emitted particles q = | p 1 -p 2 | with k ≈ q/2. been created. However, statistically one can separate these groups of events using the missing mass spectra for each chosen region of the phase-space.

Thus, Y ppη (k) can be extracted for each studied interval of k by grouping the sample of measured events according to the value of k, next calculating the missing mass spectra of the pp → ppX reaction for each sub-sample separately, and counting the number of ppη events from these spectra. Examples of typical missing mass histograms for three different k intervals are presented in Figure 2. The statistics obtained in the considered measurement allowed to divide the kinematically available range of k into bins whose width (∆k = 2.5 MeV/c) corresponds approximately to the accuracy of the determination of k with (σ(k) ≈ 2 MeV/c). A clear signal of η meson production is observed in each spectrum on top of a continuous distribution originating from the multi-pion production. It is important to note that in the studied range of the missing mass the observed shape of the multi-pion distributions is well reproduced in corresponding simulation studies for each region of the phase space [18]. An extraction of Y * ppη (k) -unbiased by the multi-pion production -is, however, not trivial. Applying a mixing technique one can construct the uncorrelated reference sample, taking momentum vectors of protons corresponding to different real events. A real event is determined by the momentum vectors of two protons registered in coincidence, and an uncorrelated event will thus comprise momentum vectors of protons ejected from different reactions. Unfortunately, in such a sample of uncorrelated momentum vectors, due to the loss of the kinematical bounds, the production of the η meson will be not reflected on the missing mass spectrum and hence it cannot be used to extract a number of mixed-events corresponding to the production of the η meson. Therefore, in order to determine a background-free correlation function for the pp → ppη reaction we performed the following analysis: First, for each event, the probability ω was determined that this event corresponds to the pp → ppη reaction. The probability ω i , that the i th pp → ppX event with a missing mass m i , and a relative momentum of k i corresponds to a pp → ppη reaction was estimated according to the formula:

ω i = N η (m i , k i ) N η (m i , k i ) + N pions (m i , k i ) , ( 2
)
where N η stands for the number of the pp → ppη reactions and N pions is the number of events corresponding to the multi-pion production with the invariant mass equal to m i . The values of N η (m, k) and N pions (m, k) were extracted from the missing mass distributions produced separately for each of the studied intervals of k. An example of a missing mass spectrum with pictorial definitions of N pions and N η is presented in the middle panel of Figure 2. Now having introduced the weights we can calculate also the value of Y * (k) separately for the ppη and pp(pions) final states. We can achieve this by sorting an uncorrelated sample according to the k values similarly as in the case of the correlated events and next for each sub-sample we construct background free Y * ppη (k) distributions as a sum of the probabilities that both protons in an uncorrelated event originate from the reaction where the η meson was created. Specifically, if in a given uncorrelated event denoted by l, one momentum is taken from a real event say l1 and the second momentum from another real event l2, then the probability that both correspond to reactions where an η was created equals to ω l1 • ω l2 , and hence the uncorrelated yield Y * ppη (k) may be constructed from the sum l ω l1 • ω l2 , where l enumerates events in the uncorrelated sub-sample selected for a momentum range k.

Analogously one can calculate Y * pp(pions) assigning to the event a weight equal to 1 -ω.

Acceptance corrections

As the next necessary step in the data evaluation we corrected the determined yields to account for the finite geometrical acceptance and detection efficiency of the COSY-11 detectors [START_REF] Klaja | AIP Conf. Proc[END_REF]. Hereafter for simplicity by "acceptance" both the geometrical acceptance and detection efficiency will be denoted. The acceptance was calculated as a function of the proton momentum in the proton-proton rest frame k. It was obtained using Monte-Carlo simulations and evaluated according to formula:

A(k) = N acc (k) N gen (k) , ( 3
)
where N acc (k) and N gen (k) denote the number of accepted and generated events, respectively. Simulations were performed based on the GEANT-3 packages [23] including the realistic geometry of the detectors and a precise map of the field of the dipole magnet. The momentum and spatial beam spreads, multiple scattering, proton-proton final state interaction [START_REF][END_REF] and other known physical and instrumental effects were taken into account [17,18]. Knowing the acceptance [START_REF] Klaja | AIP Conf. Proc[END_REF] it would be straightforward to correct the nominator of equation 1, however the correction of the uncorrelated yield Y * (k) is not trivial since the momenta of protons in the uncorrelated event originate from two idependent real events which in general could correspond to different values of the detector acceptance. Therefore, in order to derive a correlation function corrected for the acceptance, we have created a sample of data that would have been measured with an ideal detector.

For this aim each experimental pp → ppη event was multiplied by 1/A(k). This means that a given reconstructed pp → ppη event with momentum of k was added to the experimental data sample 1/A(k) times.

Based on this corrected data sample we calculated the two-proton correlation function according to equation 1. In order to avoid mixing between the same events, a "mixing step" in the calculations was set to a value bigger than the inverse of the lowest acceptance value. The random repetition of identical combinations was also omitted by increasing correspondingly the "mixing step". In particular, a l th event, from the acceptance corrected data sample, was "mixed" with a (l+n) th event, where n > max(1/A(k)). If the (l+1) th event was the same as l th , then this was mixed with a (l+1+2n) event, etc.

Results and conclusions

The two-proton background-free correlation functions for the pp → ppη and pp → pp + pions reactions corrected for the acceptances are presented in Figure 3 (full squares). As mentioned in the introduction the shape of the obtained correlation function reflects not only the space-time characteristics of the interaction volume but it may also be strongly modified by the conservation of energy and momentum and by the final state interaction among the ejectiles. In order to estimate the influence on the shape induced by the kinematical bounds we have constructed the correlation functions for both, the pp → ppη and pp → pp+pions reaction assuming a point-like source using a Monte-Carlo simulation. The results of the simulations are presented in Figure 3 (open squares) and it is apparent that they differ significantly from the experimental correlation function. At the origin the Coulomb repulsion between protons brings the experimental correlation function down to zero. Next, in order to extract from the experimental data the shape of the correlation function free from the influence of energy and momentum conservation we constructed a double ratio:

R(k) + 1 = Const Y exp (k) Y * exp (k) / Y MC (k) Y * MC (k) , ( 4
)
where Const denotes the normalization constant, and indices 'exp' and 'MC' refer to the experimental and simulated samples, respectively. The determined double ratios are given in Table 1 and presented in decays of Z boson [26][27][28] or W-pairs [29]. In Figure 4 a significant discrepancy between the two-proton correlation functions determined from inclusive heavy ion reactions (triangles) [25] and from exclusive proton-proton measurements (full squares) is clearly visible. The data from the kinematically exclusive measurement do not reveal a peak structure at 20 MeV/c. Although a similar enhancement is also seen in the invariant mass of the proton-proton distributions [20] it disappears in the correlation function for the pp → ppη and pp → pp + pions reactions. At present it is not possible to draw a solid quantitative conclusion about the size of the system since e.g. in the case of the pp → ppη reaction it would require to solve a three-body problem where pp and pη [30] interactions are not negligible and both contribute significantly to the proton-proton correlation. However, based on semiquantitative predictions [START_REF] Deloff | AIP Conf. Proc[END_REF] one can estimate that the system must be unexpectedly large with a radius in the order of 4 fm. This makes the result interesting in context of the predicted quasi-bound ηN N state [START_REF] Ueda | [END_REF] and in view of the hypothesis [14] that at threshold for the pp → ppη reaction the proton-proton pair may be emitted from a large Borromean like object whose radius is about 4 fm.

Figure 1 .

 1 Figure 1. Schematic view of the COSY-11 detection setup[17]. Superimposed lines indicate trajectories of protons from the pp → ppη reaction. The positively charged particles were identified using the time of flight between the scintillator detectors S1 and S3 combined with a momentum reconstruction by tracking back the trajectories (reconstructed based on the signals from the drift chambers D1 and D2) through the magnetic field of the COSY dipole to the reaction point. The size of the detectors and their relative distances are not to scale. More detailed description can be found in[17][18][19][20].

2 kFigure 2 .

 22 Figure 2.Examples of missing mass spectra measured[20] for the pp → ppX reaction at k ∈ (27.5; 30.0) MeV/c, k ∈ (55.0; 57.5) MeV/c and k ∈ (82.5; 85.0) MeV/c. Points represent experimental data. Dotted histograms depict simulations of the pp → pp2π, pp → pp3π and pp → pp4π reactions, while solid line histograms present the sum of simulations for the pp → ppη reaction and for the pp → pp2π, pp → pp3π and pp → pp4π reactions, respectively. The simulated spectra were fitted to the data adjusting only the amplitude[20].

Figure 3 .

 3 Figure 3.Acceptance corrected experimental proton-proton correlation functions for the production of the η meson (left panel) and multi-pions with the invariant mass equal to the mass of the η meson (right panel). Full squares denote the experimental data and open squares represent results of simulations determined for a point-like source assuming the homogeneous population of the phase space for the reaction products.

Figure 4 .

 4 The double ratio is a well established measure of correlations used e.g. by the ALEPH, OPAL and DELPHI collaborations for studying the Bose-Einstein or Fermi-Dirac correlations e.g. in the

Figure 4 .

 4 Figure 4. The two-proton acceptance corrected correlation functions normalized to the corresponding simulated correlation function for a point-like source. Results for the pp → ppη (full squares) and pp → pp + pions (open squares) are compared to the two-proton correlation function determined from heavy ion collisions (triangles) [25].

Table 1 .

 1 The double ratios determined for the production of the η meson and multi-pions with the invariant mass equal to the mass of the η meson. In the first column of the table k denotes the center value of a 2.5 MeV/c interval.

	k	( R + 1)η	(R + 1) pions
	[MeV/c]
	1.25	--	--
	3.75	--	0.09 ± 0.03
	6.25	0.10 ± 0.04	0.13 ± 0.04
	8.75	0.28 ± 0.08	0.15 ± 0.04
	11.25	0.42 ± 0.09	0.29 ± 0.05
	13.75	0.54 ± 0.10	0.39 ± 0.06
	16.25	0.74 ± 0.12	0.50 ± 0.07
	18.75	0.91 ± 0.14	0.63 ± 0.08
	21.25	0.94 ± 0.13	0.70 ± 0.08
	23.75	1.11 ± 0.15	0.79 ± 0.10
	26.25	1.08 ± 0.14	0.74 ± 0.09
	28.75	1.34 ± 0.17	0.85 ± 0.10
	31.25	1.24 ± 0.15	0.87 ± 0.10
	33.75	1.12 ± 0.14	0.92 ± 0.10
	36.25	1.23 ± 0.15	0.86 ± 0.09
	38.75	1.14 ± 0.13	0.89 ± 0.10
	41.25	1.18 ± 0.14	0.91 ± 0.10
	43.75	1.06 ± 0.13	0.91 ± 0.10
	46.25	1.04 ± 0.12	0.87 ± 0.10
	48.75	1.04 ± 0.12	0.90 ± 0.10
	51.25	1.04 ± 0.12	1.00 ± 0.11
	53.75	1.10 ± 0.13	0.92 ± 0.11
	56.25	0.95 ± 0.11	1.02 ± 0.11
	58.75	0.96 ± 0.12	0.93 ± 0.11
	61.25	1.07 ± 0.13	1.00 ± 0.13
	63.75	0.97 ± 0.12	1.13 ± 0.14
	66.25	0.98 ± 0.13	1.01 ± 0.13
	68.75	0.88 ± 0.12	1.11 ± 0.14

Acknowledgements:

The work was partially supported by the European Community-Research Infrastructure Activity under the FP6 and FP7 programmes (Hadron Physics, RII3-CT-2004-506078, PrimeNet No. 227431), by the Polish Ministry of Science and Higher Education under grants No. 3240/H03/2006/31, 1202/DFG/2007/03, and 0084/B/H03/2008/34, by the German Research Foundation (DFG), and by the FFE grants from the Research Center Jülich, and by the virtual institute "Spin and strong QCD" (VH-VP-231).


Ikuko Hamamoto

Change of shell structure and magnetic moments of odd-N deformed nuclei towards neutron drip line

Examples of the change of neutron shell-structure in both weakly-bound and resonant neutron one-particle levels in nuclei towards the neutron drip line are exhibited. It is shown that the shell-structure change due to the weak binding may lead to the deformation of those nuclei with the neutron numbers N ≈ 8, 20, 28 and 40, which are known to be magic numbers in stable nuclei. Nuclei in the "island of inversion" are most easily and in a simple manner understood in terms of deformation. As an example of spectroscopic properties other than single-particle energies, magnetic moments of some weakly-bound possibly deformed odd-N nuclei with neutron numbers close to those traditional magic numbers are given, which are calculated using the wave function of the last odd particle in deformed Woods-Saxon potentials.

INTRODUCTION

Some neutron-rich nuclei towards the neutron drip line, which have the neutron numbers N ≈ 8 and 20, are now widely recognized to be deformed and often called nuclei in the "island of inversion" [1,2,3]. In this paper the change of shell structure in weakly-bound and one-particle resonant neutron levels coming from a unique behaviour of smallneutron levels compared with large-levels is exhibited. The behaviour comes from the strong -dependence of the heights of centrifugal barrier, which are proportional to ( + 1)/R 2 b where R b is slightly larger than the nuclear radius. The centrifugal barrier is absent for s neutrons which can thus freely extend to the outside of nuclei as the binding energy approaches zero. Consequently, the energy of weakly-bound s neutron levels is relatively insensitive to the strength of the potential. In contrast, large-neutrons are confined to the inside of the potential and, thus, sensitive to the strength of the potential. The barrier becomes increasingly higher both for larger neutrons and for smaller nuclei. As an example, in figure 1 we sketch the relative energies of three oneneutron levels in the sd shell depending on binding energies. The figure illustrates how the levels with smaller -values shift downwards relative to those with larger -values, as the potential strength becomes weaker. The cases of the strongly bound (such as by 30 MeV) levels and those bound by about 10 MeV (approximately equal to the Fermi level of stable sd shell nuclei) can be found from figure 2-30 of [4], while the level order of very weakly bound or one-particle resonant neutrons can be found for example in figure 1 of [5,6]. The level scheme being bound by about 10 MeV is similar to that of the conventional modified-oscillator potential plus a spin-orbit potential. This similarity is accidental. Namely, this never means that "the one-particle level scheme of weaklybound neutrons becomes similar to that of the harmonic oscillator potential (plus a spin-orbit potential)", which is unfortunately often stated in some publications. Indeed, the harmonic oscillator potential has no surface and, consequently, using the potential it is impossible to express some phenomena which are unique in weakly-bound particles.

The similar change of the relative level order and the shell structure, which is sketched in figure 1 for the sd shell, occurs for other shells. For example, in the pf shell the p 3/2 and p 1/2 levels are lowered relative to the 1f 7/2 and 1f 5/2 levels as the potential becomes weaker or the binding energies become smaller, since the centrifugal barrier for =1 neutrons is six times lower than that for =3 neutrons. This change of the shell structure can lead to the deformed ground state of some nuclei with the traditional magic number for neutrons. See examples in section 3.

Whether a given nucleus with weakly-bound neutrons will deform or not depends, of course, also on the proton number, since some proton number may definitely favor spherical shape. For example, the proton number Z=8, oxygen isotope, has been known to prefer spherical shape. Nevertheless, it was recently reported [7] that the unbound nucleus 12 8 O 4 outside the proton drip line is deformed in the way similar to its mirror nucleus 12 4 Be 8 . It is important to notice that the notion of one-particle states in deformed nuclei can be much more widely, in a good approximation, applicable than that in spherical nuclei. This is because in the deformed mean field the major part of the long-range two-body interaction in the spherical mean field is already taken into account in the mean field. Thus, the spectroscopy of deformed nuclei is often much simpler than that of spherical (vibrating) nuclei. For example, we note that the analysis of observed spectroscopic properties of low-lying states in light mirror nuclei, 25 12 Mg 13 and 25 13 Al 12 , in terms of one-particle motion in deformed (Nilsson) potential is very successful [8]. As seen in those examples, the numerical results of the intrinsic configurations, which are obtained in the present work for the ground state of nuclei with the neutron number of N magic ± 1, can be equally applicable to the low-lying excited states (or isomeric states) of the neighboring nuclei.

As the one-particle energy ε Ω < 0 approaches zero in a deformed potential, the probability of = 0 component in the wave function approaches unity in all Ω π =1/2 + bound neutron orbits. However, the energy, at which the s-dominance shows up, depends on both deformation and individual orbits. On the other hand, in the case of Ω π =1/2 -and Ω π =3/2 -the p-components increases as ε Ω (< 0) → 0, but the probability of = 1 components at ε Ω ⇒ 0 is less than unity and depends on individual levels and deformations.

Except in very light nuclei a significant change of shell structure in weakly-bound protons is not expected because of the presence of the Coulomb barrier. For this reason the shell structure and magnetic moments of proton-drip-line nuclei are not included in this work. On the other hand, the magnetic moment of deformed odd-Z nuclei towards the neutron drip line is a very useful and interesting quantity, since it may clearly show possible deformation of those nuclei. However, the proton separation energy of those neutron-rich nuclei is so large that the shell structure as well as magnetic moments can be reasonably evaluated using some old traditional models, for example, the modifiedoscillator model. Therefore, the numerical calculations are not included here.

When possible rotational spectra or strongly-enhanced quadrupole moments, which are the direct sign of deformation, are difficult to be measured in neutron drip line nuclei, an indication of deformation is to observe the unusually low-lying 2 + state in even-even nuclei. On the other hand, one-particle motion in the mean field that shows the shape of nuclei can be easily recognized by studying the low-energy spectra of odd-A nuclei [8]. In particular, the spin-parity of low-lying states or magnetic moments of the ground or isomeric states of odd-A drip-line nuclei may be sometimes more easily measured and can be used as a clear indication of possible deformation. Therefore, in the present paper we survey the Nilsson diagrams that can be applicable to weakly-bound nuclei with N ≈ 8, 20, 28 and 40, and give the estimated values of magnetic moments of possibly deformed odd-N nuclei with the neutron number close to those traditional magic numbers.

In section 2 the essential points of the model used in the present work are briefly summarized. Numerical examples for nuclei in the N ≈ 8 region is given in section 3.1, the N ≈ 20 region in section 3.2, the N ≈ 28 region in section 3.3, and the region N ≈ 40 region in section 3.4. Conclusions and discussions are given in section 4.

MODEL

Recognizing the great usefulness of the Nilsson diagram, in which one-particle energies are plotted as a function of deformation parameter for a given potential, in the present article we apply the model and idea presented in [9]. The quadrupole deformation parameter β is defined in [10]. The coupled equations derived from the Schrödinger equation are solved in coordinate space with the correct asymptotic behaviour of wave functions for r → ∞, both for bound [10] and resonant [11,12] levels. In particular, one-particle resonant levels in a deformed potential are estimated using the eigenphase formalism [START_REF] Newton | Scattering Theory of Waves and Particles[END_REF]. Namely, one-particle resonance is obtained if one of calculated eigenphases increases through π/2 as one-particle energy increases. Oneparticle resonance is not obtained if none of the calculated eigenphases do not increase through π/2 as energy increases. The definition is a natural extension of the definition of one-particle resonance for spherical potentials in terms of phase shift, which can be found in standard textbooks. One-particle resonance with Ω π =1/2 + is not at all obtained if for ε Ω (< 0) → 0 the =0 component of the wave function inside the potential exceeds a certain probability [12]. On the other hand, for Ω π = 1/2 + one-particle resonant levels can be always found at least for small positive energies. Compared with the Nilsson diagram based on modified oscillator potentials, the striking difference of the level scheme exhibited in the present work comes from the behaviour of levels with low values (in particular, =0 and 1) for β=0 and those with small Ω values (in particular, Ω π = 1/2 + , 1/2 -, and 3/2 -) for β = 0, in both the weakly-bound and positive-energy regions.

We use the parameters of Woods-Saxon potentials taken from the standard ones [4] for stable nuclei except the depth, V W S . The potential depth for neutrons is adjusted so that the energy of the one-neutron level last occupied in a given odd-N nucleus is close to the measured neutron separation energy. The diffuseness, the strength of the spin-orbit potential and the radius parameter are, for simplicity, taken from those on p.239 of [4]. Considering the possible contribution by weakly-bound neutron(s) to the self-consistent potential, a slightly larger diffuseness might be appropriate for presently studied nuclei. However, it is noted that the major part of the nuclear potential is provided by well-bound nucleons of the core. Moreover, a larger diffuseness leads to the degeneracy of the 2s 1/2 and 1d 5/2 (2p 3/2 and 1f 7/2 ) levels already at a larger binding energy than the one exhibited in the present article. It is also remarked that for a given deformation the N-th deformed one-particle orbit filling-in all lower-lying Nilsson levels is almost uniquely determined for a given one-particle energy and within a reasonable variation of Woods-Saxon potential parameters.

The magnetic dipole moments of odd-N deformed nuclei are calculated using equations (4-86), (4-87), and (4-88) of [8], while the matrix elements of ν and s ν are evaluated using one-particle wave-functions in the deformed Woods-Saxon potential. See equations (5-86) and (5-87) of [8]. The tabulated magnetic moments are calculated for the band-head states for given intrinsic configurations. The spin of the band-head state is I = Ω for Ω = 1/2. For Ω=1/2 configurations the decoupling parameter a is calculated, and the spin of the band-head state is different from I=1/2 for both a < -1 and a > 4. See [8].

The major part of the reduction of the effective g s factor, g ef f s , from g free s in the present model is supposed to come from the spin polarization of deformed even-even core. For large deformation where the asymptotic quantum numbers [N n z Λ Ω] become good quantum numbers, the spin polarizations of ∆K=0 type vanish, since the quantum numbers Λ and Σ (= Ω -Λ) become constants of the motion. Consequently, the longitudinal g ef f s factor may approach g free s , while the transverse matrix element (g K -g R )b in equation (4-86) of [8] is still affected by the presence of ∆K=1 polarizations of deformed even-even core. The latter contributes only to the magnetic moments of K=1/2 bands. The quenching of the spin fluctuations in the large deformation limit seems to be only partially achieved for the equilibrium deformations of stable deformed nuclei [14]. (A further reduction in the spin polarization of nuclei presently studied may come from the fact that loosely-bound neutrons couple weakly with strongly-bound core nucleons.) In the present numerical calculations we use g ef f s =g free s in the lightest mass (N ≈ 8) region where deformed nuclei have larger deformation, while in heavier mass regions g ef f s =(0.7)g free s is used. We do not try any detailed evaluation of the spin polarization effect in individual deformed nuclei, since our present aim is to show the spin-parity as well as magnetic moments for deformed shape of nuclei studied, which are very different from those for spherical shape.

In the present coupled channel calculation the j = 1/2, 3/2, 5/2, 7/2, 9/2, 11/2 and 13/2 channels are included for positive-parity levels, while the j = 1/2, 3/2, 5/2, 7/2, 9/2 and 11/2 channels for negative-parity levels.

NUMERICAL EXAMPLES

Odd-N neutron-rich nuclei in N ≈ 8 region

As it is known that the shape of the Be isotope can be very different for the A and A+1 nuclei, the notion of the mean field is not so well established in nuclei of this very light mass region. Nevertheless, in figure 2 the Nilsson diagram is shown, of which the parameters are adjusted to the nucleus 17 6 C 11 , since in this mass region the spinparity and/or magnetic moments are measured for some odd-N nuclei with weakly-bound neutron(s). In figure 2 the [200 1/2] level can be obtained as a one-particle resonant level as far as d components are dominant. In contrast, for β > 0.46 the resonant level is not obtained because the s component becomes dominant in the one-particle wavefunction. Similarly, due to the s-component dominance the extension of the [220 1/2] level to the positive-energy region on the oblate side as a resonant level is not possible for β < -0.12.

In figure 2, it is seen that in the spherical limit (β=0) the very weakly bound 1d 5/2 and 2s 1/2 levels are almost degenerate. Namely, as the binding becomes very weak, the energy eigenvalue of the s orbit shifts downward relative to that of the d level, as schematically exhibited in figure 1. The calculated and measured magnetic moments of the ground state of 17 6 C 11 and 11 4 Be 7 are compared in table 1. It is remarked that for a much weaker strength of the potential such as the one applicable for 11 Be, in which the [220 1/2] and [101 1/2] orbits at β ≈ 0.6 are bound by several hundreds keV, the calculated bound one-particle level scheme and magnetic moments around β=0.6 are found to remain nearly the same. When in a given region of deformation the curve of the Nilsson level is almost a straight line as a function of deformation, the one-particle wavefunction depends very little on deformation and, consequently, the calculated magnetic moment remains almost independent of deformation. We notice that measured magnetic moment of 15 C 9 (I=1/2 + ) that can be a spherical nucleus, |µ| = 1.720 ± 0.009 µ N [15], is nearly equal to that of 11 Be 7 (I π =1/2 + ) that is most easily interpreted as a deformed nucleus, assuming that the sign of the former is minus. This is theoretically expected because the [220 1/2] wave-function of the 7th neutron in 11 Be consists exclusively of 1d 5/2 and 2s 1/2 components, of which non-diagonal matrix element of magnetic dipole operator is zero, and, furthermore, the neutron one-particle magnetic moments of 1d 5/2 and 2s 1/2 are the same.

The spin-parity (3/2 + ) of the ground state of 17 C 11 is in a very natural way interpreted as the band-head state of the N=11th neutron occupying the [211 3/2] level. The near degeneracy of the 1d 5/2 and 2s 1/2 levels at β=0 can well be the reason why the nucleus 17 C is deformed (Jahn-Teller effect). The measured magnetic moment is in good agreement with the calculated value based on the deformation, as seen in table 1.

Odd-N neutron-rich nuclei in N ≈ 20 region

In figure 3 the Nilsson diagram is shown, of which the parameters are approximately adjusted to the nucleus 31 10 Ne 21 . For a slightly stronger potential which may be applicable for 31 12 Mg 19 (the neutron separation energy S n = 2.38 MeV) and 33 12 Mg 21 (S n = 2.22 MeV), the calculated bound one-particle level scheme and magnetic moments remain nearly the same as those presented here. See figure 3 of [9]. Neither the 2p 3/2 nor 2p 1/2 levels are obtained as one-particle resonant levels for the present potential. Nevertheless, in figure 3 the approximate positions of those levels, which are extrapolated from the resonance energies obtained for slightly stronger spherical Woods-Saxon potentials, are indicated in figure 3 with the question mark. It is noted that when both the 2p 3/2 and 1f 7/2 levels appear as very low-lying resonant levels, the 2p 3/2 level may lie lower than the 1f 7/2 level.

The second lowest Ω π =1/2 -level for β > 0 denoted by the dotted curve in figure 3 cannot continue for β < 0.33 as a one-particle resonant level, because one-particle resonant levels with the major component of =1 cannot survive for higher ε Ω values. The complicated behaviour of the resonant level expressed by the dotted curve for β < 0 indicates clearly the influence of the 2p 3/2 and 2p 1/2 levels that do not explicitly appear in figure 3. It is noted that for a pure 1f 7/2 shell the dotted curve increases almost linearly as |β| increases for β < 0, just as a smooth extension of the curve for β > 0. Around β = -0.4 the major component of the Ω π =1/2 -level is =1. As β(< 0) increases ε Ω increases, and the one-particle resonant level with the =1 major component can hardly survive for ε Ω > 1.2 MeV, where both the width and ε Ω rapidly increase. The major component of the one-particle resonant level expressed by the dotted curve for -0.2 < β < 0.07 is =3, and consequently the resonant level is well defined with the reasonably small width.

The near degeneracy of 1f 7/2 , 2p 3/2 and 2p 1/2 levels at β=0 can be the origin of possible deformation that appears in the system with a few weakly bound neutrons occupying the 1f 7/2 -2p 3/2 -2p 1/2 shell (Jahn-Teller effect). Namely, using the degeneracy the energy of a particular combination of the pf components can be made lower when deformation sets in.

In this region of "island of inversion" there are two odd-N nuclei of which magnetic moments of the ground state are measured. The measured magnetic moment [18] of the ground state (I π =1/2 + ) of 31 12 Mg 19 is -0.88355(15) µ N , while it is reported in [20] that the magnetic moment of 33 12 Mg 21 is -0.7456(5) µ N and the spin is 3/2. Though the parity of the ground state of 33 Mg is currently still under debate (for example, see the recent β-decay study in [19]), from table 2 the measured negative magnetic moment seems to have no other choice than negative parity for the ground state. It is remarked that the Nilsson orbit that can be occupied by the 21st neutron for 0 < β < ∼ 0.6 is either [330 1/2] or [202 3/2] or [321 3/2] within the reasonable variation of potential parameters. All of the three Nilsson orbits lead to I=3/2 for the band-head spin. Considering that we have fixed parameters g ef f s and g R in the numerical calculation of magnetic moments, of which some standard values are used, the agreement of the calculated and measured values in table 2 is very good. In table 2 calculated magnetic moments are tabulated for a given β value, however, it should be noted that calculated values remain approximately constant, as far as the Nilsson level is almost a straight line as a function of β.

Odd-N neutron-rich nuclei in N ≈ 28 region

Odd-N nuclei with weakly bound neutrons in this region are currently under study and, to our knowledge, no measurements of magnetic moments are reported. Examining E(2 + 1 ) of even-even nuclei with N=28, possible candidates for the deformed ground state are 42 14 Si 28 and 44 16 S 28 , of which E(2 + 1 ) is 770 (19) [21] and 1297 (18) keV [22], respectively. There have been both some shell-model calculation and more elaborated many-body calculation, which suggested an oblate deformation for the nucleus 42 Si. Thus, in this region we show magnetic moments of odd-N nuclei evaluated for both prolate and oblate shape. Though 40 Mg lies inside the neutron drip line, Mg isotope is not discussed here, since none of the neighboring odd-N nuclei, 41 Mg and 39 Mg, is bound.

In figure 4 we show the Nilsson diagram, of which the parameters are approximately adjusted to 41 14 Si 27 . The reported S n value [23] of this nucleus is 1.34 MeV with a rather large ambiguity. It is seen that at β=0 the 2p 3/2 level lies only 1.2 MeV higher than the 1f 7/2 level when the former is bound only by 0.28 MeV. Then, there may be a good chance for N=27 nuclei being deformed, since the neutron number N=28 may not work as a magic number. Calculated magnetic moments are shown in table 3. It is interesting to see that the magnetic moment of 41 Si can be very different if it has an oblate shape. This is because the (positive or) almost vanishing value of the calculated magnetic moment comes from a subtle balance between the p 3/2 and p 1/2 components in the one-particle wave-function of the [301 1/2] orbit. That means, the calculated magnetic moments of oblate nuclei with N=27 depend on the size of the oblate deformation, as can be guessed from the curved (and not straight-line) Nilsson level in figure 4 as a function of β.

In the g-factor measurement of 43 16 S 27 , which is an isotone of 41 Si, it is reported [24] that the 320 keV isomeric state has 7/2 -and g=-0.317(4). This observation of the excited state indicates that the ground state of 43 S is deformed.

In figure 5 the Nilsson diagram for neutrons is shown, of which the parameters are approximately adjusted to 45 16 S 29 . The comparison between figure 4 and figure 5 clearly shows that the N=28 shell-gap at β=0, namely the distance between the 2p 3/2 and 1f 7/2 levels, becomes increasingly smaller as the binding energy of the 2p 3/2 level approaches zero. This appreciable change of the N=28 shell-gap indicates that for a given neutronnumber N≈28 the Si isotope has a better chance to be deformed than the S isotope. The nucleus 43 14 Si 29 is known to lie inside the neutron drip line, however, to our knowledge no spectroscopic information is yet available. In figure 5 the asymptotic quantum numbers, [N n z Λ Ω], are written on the Nilsson levels which may be occupied by the N=29th neutron, while calculated magnetic moments, in which the last odd neutron is placed in respective Nilsson orbits, are given in table 4.

Odd-N neutron-rich nuclei in N ≈ 40 region

Neutron-drip-line nuclei in the region of N=40 are still far away from reaching experimentally. Moreover, N=40 is not one of magic numbers in the j-j coupling shell model. Nevertheless, we include this subsection, because; (a) while there are no deformed N=40 nuclei along the stability line, recent experimental studies show relatively low energies (≈ 500 keV) of the first excited 2+ state of even-even nuclei of both 26 Fe and 24 Cr isotopes in the region of N ≈ 40. Indeed, it is stated [25] that the nucleus 64 24 Cr 40 may be a center of deformation in this region; (b) Some magnetic-moment measurements of isomeric states, the presence of which is not rare in this region, have already been reported.

Odd-N nuclei of Fe and Cr isotopes with N≈40 around 64 24 Cr 40 have measured S n values of a few MeV. In figure 6 the Nilsson diagram is shown, which may be useful for the Fe and Cr isotopes with N≈40. It is noted that the 2p 1/2 and 1f 5/2 levels in figure 6 are nearly degenerate in contrast to the level scheme shown in figures 4 and 5. This is because in the weaker potentials of the latter figures the 2p 1/2 level becomes increasingly lower relative to the 1f 5/2 level. Calculated magnetic moments are given in table 5, in which those given for β=0.25 remain nearly the same for β=0.35. It may be speculated that the observed 387 keV isomeric state in 67 26 Fe 41 [26] may be the I π =1/2 -or 5/2 - state coming from the [301 1/2] or [303 5/2] intrinsic configuration if the isomeric state is deformed.

CONCLUSIONS AND DISCUSSIONS

Examples of the change of neutron shell-structure for weakly bound neutrons are illustrated for neutron-rich nuclei with N≈8, 20, 28 and 40. Both weakly-bound and resonant one-particle levels are properly calculated by directly solving the Schrödinger equation in mesh of space coordinate with the appropriate boundary condition. Magnetic moments of possibly deformed odd-N nuclei in the region are calculated using one-particle wave-functions in deformed Woods-Saxon potentials.

Among the examples taken in the present paper, the near degeneracy of the weaklybound 1d 5/2 and 2s 1/2 levels (figure 2), that of the one-particle resonant 1f 7/2 , 2p 3/2 and 2p 1/2 levels (figure 3), and that of the 1f 5/2 and 2p 1/2 levels bound by several MeV (figure 6) are the most clear-cut examples of the unique shell-structure, which is very different from what is known for traditional stable nuclei. It is hoped that more experiments will pin down the unique shell structure. In the present paper we have argued that the shell structure may lead to, among others; (a) the appearance of the N=16 magic number [27]; (b) possible deformed shapes of 11-14 Be [28], 17 C, and 19 C; (c) deformed nuclei in the island of inversion around N ≈ 20; (d) The disappearance of the magic number N=28 and possible deformation of some neutron-drip-line nuclei with N≈28.

Calculated magnetic moments of the deformed ground states of 11 Be, 17 C, 31 Mg, and 33 Mg are in good agreement with already measured ones, while the values of g s and g R have not been adjusted to individual nuclei. Except the case of 11 Be where the unique magnetic moments of the d 5/2 -s 1/2 mixed neutron wave-functions play a role, both measured and calculated values of magnetic moment are significantly different from what is expected from spherical shape and, thus, can be used as a good identification of deformation. The measurement of magnetic moments of other neutron-drip-line nuclei is strongly wanted.

Magnetic moments estimated in the present work are applicable not only to the ground state of the odd-N nuclei studied but also to excited or isomeric states in the neighboring nuclei. The presence of isomeric states is often expected in the region of N≈28 and N≈40 neutron-rich nuclei due to the coexistence of spherical and deformed shape or the unique shell-structure. Magnetic moments presently estimated for a given deformation remain nearly the same when one-particle energies in the Nilsson diagram vary almost linearly as a function of deformation, because the structure of single-particle wave-functions in those cases remains nearly independent of deformation.

It should be remarked that the change of the shell structure described in the present paper is different from (and independent of) the one coming from the neutron-proton tensor force [29], which is currently very fashionable for being an origin of shell-structure change. The latter depends on the proton number of respective nuclei when the neutron shell-structure is discussed. Moreover, the effect of the tensor force is usually estimated using harmonic oscillator wave-functions in the shell model and, thus, the effect of weakly binding is not included. One-particle levels are denoted by the asymptotic quantum numbers, [Nn z ΛΩ], on the prolate side (β > 0), while on the oblate side (β < 0) the quantum number Ω is shown only in the case that it is difficult to see as a continuation of the curve on the prolate side. The Ω π = 1/2 -levels are denoted by dotted curves, while positive-parity levels are plotted by solid curves. The neutron numbers 8, 10 and 16, which are obtained by filling in all lower-lying levels, are indicated with circles. See the text for details. Positive-parity levels are plotted by solid curves. The neutron number 20, which is obtained by filling in all lower-lying levels, is indicated with a circle. The 2p 3/2 and 2p 1/2 neutron levels at β=0 are not obtained as one-particle resonant levels, and the next low-lying one-particle resonant level at β=0 is the 1f 5/2 level found at 9.50 MeV. The approximate positions of the 2p 3/2 and 2p 1/2 levels at β=0 are indicated with "?", which are extrapolated from the resonant energies obtained by using a slightly stronger Woods-Saxon potential. See the text for details. Positive-parity levels are plotted by solid curves. The neutron numbers 20 and 28, which are obtained by filling in all lower-lying levels, are indicated with circles. The Ω π =1/2 -level originating from 2p 1/2 at β=0 does not survive as one-particle resonance for ε Ω > 1 MeV and β < -0.05. On the other hand, for simplicity of the figure, only in the neighborhood of β=0 we have plotted resonant levels originating from f 5/2 . See the text for details.

Ω π =1/2 - Ω π =3/2 - Ω π =5/2 - Ω π =7/2 - [321 1/2] [312 5/2] [330 1/2] [321 3/2] [301 1/2]

=Figure 1 .

 1 Figure 1. Qualitative sketch of relative energies of three one-neutron levels in the sd shell depending on the binding energy or the potential strength. The level order sketched in the middle figure is the one known from stable sd shell nuclei.

  4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

Figure 2 .

 2 Figure 2. Neutron one-particle levels as a function of axially-symmetric quadrupole deformation. Parameters of the Woods-Saxon potential are designed approximately for the nucleus 17 6 C 11 , of which S n = 730 keV. The diffuseness, the radius and the depth of the Woods-Saxon potential are 0.67 fm, 3.27 fm, and -40.0 MeV, respectively.One-particle levels are denoted by the asymptotic quantum numbers, [Nn z ΛΩ], on the prolate side (β > 0), while on the oblate side (β < 0) the quantum number Ω is shown only in the case that it is difficult to see as a continuation of the curve on the prolate side. The Ω π = 1/2 -levels are denoted by dotted curves, while positive-parity levels are plotted by solid curves. The neutron numbers 8, 10 and 16, which are obtained by filling in all lower-lying levels, are indicated with circles. See the text for details.

Figure 3 .

 3 Figure 3. Neutron one-particle levels as a function of axially-symmetric quadrupole deformation. Parameters of the Woods-Saxon potential are designed approximately for the nucleus 31 10 Ne 21 , of which the measured S n value is (0.3±1.6) MeV. The diffuseness, the radius and the depth of the Woods-Saxon potential are 0.67 fm, 3.99 fm, and -39.0 MeV, respectively. Some one-particle levels are denoted by the asymptotic quantum numbers, [Nn z ΛΩ].Positive-parity levels are plotted by solid curves. The neutron number 20, which is obtained by filling in all lower-lying levels, is indicated with a circle. The 2p 3/2 and 2p 1/2 neutron levels at β=0 are not obtained as one-particle resonant levels, and the next low-lying one-particle resonant level at β=0 is the 1f 5/2 level found at 9.50 MeV. The approximate positions of the 2p 3/2 and 2p 1/2 levels at β=0 are indicated with "?", which are extrapolated from the resonant energies obtained by using a slightly stronger Woods-Saxon potential. See the text for details.

  particle levels in Woods-Saxon potential V WS = -39.0 MeV R = 4.38 fm a = 0.67 fm

Figure 4 .

 4 Figure 4. Neutron one-particle levels as a function of axially-symmetric quadrupole deformation. Parameters of the Woods-Saxon potential are designed approximately for the nucleus 41 14 Si 27 , of which S n = 1.34 MeV. The diffuseness, the radius and the depth of the Woods-Saxon potential are 0.67 fm, 4.38 fm, and -39.0 MeV, respectively. Some one-particle levels are denoted by the asymptotic quantum numbers, [Nn z ΛΩ].Positive-parity levels are plotted by solid curves. The neutron numbers 20 and 28, which are obtained by filling in all lower-lying levels, are indicated with circles. The Ω π =1/2 -level originating from 2p 1/2 at β=0 does not survive as one-particle resonance for ε Ω > 1 MeV and β < -0.05. On the other hand, for simplicity of the figure, only in the neighborhood of β=0 we have plotted resonant levels originating from f 5/2 . See the text for details.

Figure 5 .Figure 6 .

 56 Figure 5. Neutron one-particle levels as a function of axially-symmetric quadrupole deformation. Parameters of the Woods-Saxon potential are designed approximately for the nucleus 45 16 S 29 , of which S n = 2.21 MeV The diffuseness, the radius and the depth of the Woods-Saxon potential are 0.67 fm, 4.52 fm, and -41.0 MeV, respectively. Some one-particle levels possibly occupied by the N=29th neutron are denoted by the asymptotic quantum numbers, [Nn z ΛΩ]. Positive-parity levels are plotted by solid curves. The neutron numbers 20 and 28, which are obtained by filling in all lowerlying levels, are indicated with circles. Neutron resonant levels originating from 1f 5/2 at β=0 are plotted as far as they are obtained following the definition of the eigenphase formalism. On the other hand, not all neutron resonant levels originating from 1g 9/2 are plotted, for simplicity of the figure. See the text for details.

Table 1 .

 1 Calculated magnetic dipole moments of the ground states of very light odd-N nuclei with one weakly-bound neutron, in comparison with observed ones. Measured neutron separation energies are expressed by S n . Values of g R =0.35 (for β =0) and g ef f s =g f ree s are used. Corresponding one-particle levels in figure 2 are those labeled as [211 3/2], [220 1/2] and 2s 1/2 for 17 C, 11 Be and 15 C, respectively.

	Nucleus	S n	I π	µ obs	Reference µ calc (at β, [N n z Λ Ω])
		(keV)		(µ N )		( µ N )
	17 C 11	727 3/2 + ±0.758(4)	[16]	-0.75 (β=0.4, [211 3/2])
	11 Be 7	504 1/2 + -1.6816(8)	[17]	-1.7 (β=0.6, [220 1/2])
	15 C 9	1218 1/2 + ±1.720(9)	[15]	-1.9 (β=0, 2s 1/2 )

Table 2 .

 2 Calculated magnetic dipole moments of N=21 and 19 nuclei, in which the last odd neutron is placed in corresponding Nilsson orbits for prolate shape, in comparison with available experimental data. See the Nilsson diagram in figure3. Values of g R =0.38 and g ef f Note that for spherical shape one obtains µ calc (f 7/2 ) = -1.3 µ N and µ calc (d 3/2 ) = +0.80 µ N , respectively, using g ef f

					s	= (0.7)g f ree s	are used. s	=
		(0.7)g f ree s	.
	Nucleus	S n	(I π ) obs	µ obs		Reference (I π ) cal	µ calc (at β, [N n z Λ Ω])
		(keV)			(µ N )		( µ N )
	33 Mg 21	2222	3/2	-0.7456(5)	[20]	3/2 --0.88 (β=0.25, [330 1/2])
							3/2 + +0.91 (β=0.35, [202 3/2])
							3/2 --0.39 (β=0.45, [321 3/2])
	31 Mg 19	2378	1/2 + -0.88355(15)	[18]	1/2 + -1.00 (β=0.45, [200 1/2])
							3/2 --0.91 (β=0.35, [330 1/2])

Table 3 .

 3 Calculated magnetic dipole moments of N=27 nuclei, in which the last odd neutron is placed in corresponding Nilsson orbits. See the Nilsson diagram in figure4. Values of g R =0.38 and g ef f Note that for spherical shape one obtains µ calc (p 1/2 ) = +0.4 µ N , µ calc (p 3/2 ) = -1.3 µ N and µ calc (f 7/2 ) = -1.3 µ N , respectively, using g ef f

		s are used. s = (0.7)g f ree s = (0.7)g f ree s .
	Nucleus	S n	(I π ) cal	µ calc (at β, [N n z Λ Ω])
		(MeV)		(µ N )
	41 Si 27	1.34	3/2 -+0.07 (β=-0.4, [301 1/2])
			3/2 --0.66 (β=0.25, [321 1/2])
			5/2 --0.58 (β=0.45, [312 5/2])

Table 4 .

 4 Calculated magnetic dipole moments of N=29 nuclei, in which the last odd neutron is placed in corresponding Nilsson orbits. See the Nilsson diagram in figure5. Values of g R =0.38 and g ef f Note that for spherical shape one obtains µ calc (p 1/2 ) = +0.4 µ N , µ calc (p 3/2 ) = -1.3 µ N and µ calc (f 7/2 ) = -1.3 µ N , respectively, using g ef f

		s are used. s = (0.7)g f ree s = (0.7)g f ree s .
	Nucleus	S n	(I π ) cal	µ calc (at β, [N n z Λ Ω])
		(MeV)		(µ N )
	45 S 29	2.21	7/2 --0.74 (β = 0.25, [303 7/2])
			1/2 -	+0.59 (β=0.45, [310 1/2])
			1/2 -+0.59 (β=-0.40, [310 1/2])
			3/2 -+0.16 (β=-0.40, [312 3/2])

Table 5 .

 5 Calculated magnetic dipole moments of N ≈ 40 nuclei, in which the last odd neutron is placed in corresponding Nilsson orbits. See the Nilsson diagram in figure 6. Values of g R =0.38 and g ef f

Acknowledgments

The author is grateful to Dr. H. Ueno for introducing measured magnetic moments of neutron-rich nuclei in the presently-studied mass region to her.


R A Broglia
email: broglia@mi.infn.it

G Potel
email: potel@mi.infn.it

F Barranco
email: barranco@us.es

E Vigezzi
email: vigezzi@mi.infn.it

Difference between stable and exotic nuclei: medium polarization effects

The 1 S 0 phase shift is large and positive at low densities (relative momenta), while it vanishes and eventually becomes negative at densities of the order of the saturation nuclear density. The bare N N -potential, parametrized so as to reproduce these phase shifts leads to a sizable Cooper pair binding energy in nuclei along the stability valley. It is a much debated matter whether this value accounts for the "empirical" value of the pairing gap or whether a similarly important contribution arises from the exchange of collective vibrations between Cooper pair partners. In keeping with the fact that two-particle transfer reactions are the specific probe of pairing in nuclei, and that exotic halo nuclei like 11 Li are extremely polarizable (representing, as far as this property is concerned, almost a caricature of stable nuclei), we find that the recently studied reaction, namely 11 Li+p → 9 Li+t, provides, for the first time, direct evidence of phonon mediated pairing in nuclei.

Introduction

Arguably, one of the greatest achievements of many-body physics has been that of providing a complete description and a thorough understanding of superconductivity. At the basis of it one finds BCS theory and the Josephson effect. The first recognized the central role played by the appearance of a macroscopic coherent field usually viewed as a condensate of strongly overlapping Cooper pairs, the quasiparticle vacuum. The second made it clear that a true gap is not essential for such a state of matter to exist, but rather a finite expectation value of the pair field. Consequently, the specific experiments to study the superconducting state is Cooper pair tunneling. Such experiments allowed for a detailed probing of the phonon mediated pairing interaction leading to a thoroughly quantitative "exact" era in the study of pairing in metals, with uncertainties well below the 10% level.

From this vantage point of view, it is not difficult to argue that major progress in the understanding of pairing in atomic nuclei -a subject lying at the forefront of nuclear research but still far away from having entered the "exact" era-will arise from a systematic study of two-particle transfer reactions on the drip line, exotic nuclei like e.g. 11 Li, stabilized by the pairing correlations associated with a single Cooper pair.

While in the infinite system the existence of a bound state of the Cooper pair happens for an arbitrarily weak attractive interaction, in the nuclear case this phenomenon takes place only if the strength of the interaction is larger than a critical value connected with the discreteness of the nuclear spectrum around the Fermi energy. In fact, in the case of 11 Li the pairing interaction arising from the bare nucleon-nucleon potential seems not able to bind the halo neutrons, and one can posit that the exchange of collective vibrations between the Cooper pair partners is the main source of pairing in the low density systems [1]. At the basis of this result is the low momenta (large neutron mean square radius) of the two halo neutrons and associated small two neutron separation energy S 2n (≈ 380 keV) and thus: a) the low angular momentum content (essentially s-, p-and d-waves) of the phase space available to the halo neutrons to correlate, and b) the high polarizability of the halo field extending far away from the 9 Li core. In such a low-angular momentum phase space the two neutrons are not able to fully profit from the strong force-pairing, known to receive important contributions from many high-l partial waves [2], [START_REF] Mottelson | Proceedings of the Ecole d'été de Physique Théorique[END_REF], [START_REF] Mottelson | International School of Physics "Enrico Fermi" Course XV , Nuclear Spectroscopy[END_REF]. On the other hand, the collective L = 0, 1 and 2 fluctuations of the medium strongly renormalize the neutron motion leading to the barely bound Cooper pair [1].

Minimal mean field approximation

Classically, particles have definite positions, waves well defined momenta. Because of the complementary principle, the results of momentum and position measurements must fulfill the relation ΔxΔp x ≥ , tantamount to saying that particles can be described equally well as waves and viceversa [START_REF] Bohr | [END_REF], and that nothing is gained by discussing basic problems in terms of one rather than the other picture [START_REF] Heisenberg | The physical principles of quantum mechanics[END_REF]. One of the most important consequences of this shift of paradigm in the description of the physical world is the fact that, contrary to the classical picture where particles in empty space (vacuum) have well defined masses and charges, regardless of the rest of the universe, in quantum mechanics all physical properties can be viewed as collective properties [START_REF] Laughlin | A different universe[END_REF] or, more accurately, many-body properties. Quoting from Laughlin, "A nice example of a collective effect . . . is emitted light from dilute atomic vapors with special wavelengths so insensitive to outside influences that they can be used to make clocks accurate to one part in one hundred trillion. But these wavelengths have a detectable shift at one part in ten million -ten trillion times larger than the timing errors of the clock-which should not have been present in an ideal world containing nothing but the atom [START_REF] Lamb | [END_REF], [9] . . . calculations then revealed this shift to be an electrical effect of the vacuum of space . . . The ostensible empty vacuum space, in other words, is not empty at all but full of "stuff" " (see also [START_REF] Broglia | Procs. of the International Meeting "Why is there something[END_REF]).

From this vantage point of view, it is particularly illuminating for the subject of the present article to quote what Ben Mottelson [START_REF] Mottelson | International School of Physics "Enrico Fermi" Course XV , Nuclear Spectroscopy[END_REF] wrote at the beginning of the modern era of nuclear structure: ". . . in a many-body system such as the nucleus every feature is in some sense a collective phenomenon -every property depends on the total organization of the system and reveals the (collective) contribution of all nucleons. Indeed the most striking and fundamental collective picture in all nuclear phenomena is the existence of an average field in which the nucleons move approximately independently." The impact of these statements becomes even stronger by remembering the fact that the Hartree-Fock vacuum corresponds to a system in which all fermion levels are occupied up to the Fermi energy, similar to the Dirac vacuum (negative energy solutions of Dirac equation). The zero point fluctuations (ZPF) associated with the collective vibrations displayed by nuclei, renormalize in an important way the single-particle motion, giving rise to effective masses (density of levels around F ) as well as to finite lifetimes for levels removed from the Fermi energy, as required by the experimental data.

This phenomenon is nothing else but another example of the fact that, as stated by Feynman [START_REF] Feynman | Theory of fundamental processes[END_REF] ". . . nothing is really free. For an electron going from X to Y (see Fig. 1), the pole of the propagator for a free particle is at p 2 = m 2 . However, making measurements at X and Y we could not tell if the electron had emitted and absorbed any number of photons" (tantamount to saying: being affected any number of times by the ZPF of the vacuum). "Such processes, the simplest of which is shown in Fig. 1, cause a shift in the position of the pole. Physically, this means that what we measure (the "experimental" mass, m exp ) is not the "bare" mass, but something else which includes the effect of the virtual process mentioned above. . . This discussion shows that the "bare" mass. . . is in fact not directly observable." In other words, one can state that the different process which renormalize the single-particle motion, namely correlation (CO) and polarization (PO) processes, arise from ZPF (see Fig. 2).

Within this context one can mention that quantal ZPF are also having profound consequences not only at the level of the very small, like an atomic nucleus, but also at the level of the space time description of the Universe as embodied by the theory of general relativity [START_REF] Krauss | Geometry and destiny in General Relativity and Gravitation[END_REF]. At the heart of this theory one finds the field equation G μν = 8πGT μν which states that the geometry of spacetime, embodied in the curvature tensor G μν , is determined by the distribution of matter and energy T μν (stress-energy tensor), where G is Newton's constant. To create a static model of the universe, Einstein introduced in his equation a term proportional to the spacetime metric tensor g μν of strength Λ (cosmological constant), to counterbalance gravity's attraction on cosmic scales. This cosmological term was added to the left side of the field equation, implying that it was a property of space itself. It was abandoned once it became clear that the universe was expanding. Recent work on quantum gravity has shown the need for a new cosmological term ρ vac (the energy density vacuum) associated with the quantal ZPF of the vacuum, again proportional to g μν . This term is now to be added to the right hand term of the field equation, implying a form of energy which arises from virtual particle-antiparticle pairs (see Fig. 3).

In keeping with the above discussion, but now within the field of nuclear physics, one can posit that minimal mean field theory should contain, aside from the Hartreeand the exchange (Fock-) potential, the (complex) self-energy contributions associated with ZPF renormalization processes (dynamical shell model, see [START_REF] Mahaux | [END_REF]).

In other words, in the simplest version of the nuclear shell model, it is assumed that the nucleons move independently of one another in a static mean field. This is necessarily an oversimplification of the physical reality, but many experiments indicate that it has some degree of validity. In fact, Hartree-Fock theory provides essentially the right sequence of single-particle levels and thus predicts, as a rule, the correct magic numbers for nuclei along the valley of stability. However, close to the Fermi energy (within a range of ≈ ±5 MeV) it predicts a density of levels which is too low as compared with the experimental data. Furthermore, away from the Fermi energy (±10 MeV), while the predicted HF density of levels is correct, it fails to account for the finite lifetime (width) experimentally observed. The above shortcomings underscore the need for a better theory. Not surprisingly, and using again an analogy from the physics of nature developed by Feynman (as well as by Schwinger and Tomonaga [14]) the whole of the physics of such a theory corresponds to taking into account vacuum fluctuations as has been done in going from the standard (Schrödinger, Dirac) description of the atom, to QED.

In fact, in HF theory, the vacuum (ground state) corresponds to a determinant with all levels below the Fermi energy occupied, those above empty. Now, such a system can vibrate with particle-hole as well as particle-particle like collective modes which, as a rule behave like quasi harmonic modes. The associated ZPF correspond to virtual excitations, a process described, in lowest order, by an oyster type diagram (see Figs. 2 and4(c'),see also [14]). Such a process renormalizes the single-particle levels leading to the correct density around the Fermi energy, and to a breaking of the singleparticle states far away from the Fermi energy. In other words a minimal, self-consistent theory of independent particle motion is achieved when one considers the fact that in their motion, nucleons interact with the rest of the nucleus and drag core excitations, leading to dispersion corrections of the single-particle motion known as polarization and correlation effects (see Fig. 2). One then speaks of the motion of quasiparticles.

In other words, quasiparticles based on single-particle levels with energies close to the Fermi energy display a group velocity smaller than that of particles with the same wavelength. This effect is measured by both the ω-mass m ω (E), which is a function of the energy E of the quasiparticle, and by Z ω (E) = m/m ω (E), the discontinuity of the Fermi energy, closely connected with the spectroscopic factor as measured in singleparticle transfer reactions. For quasiparticles associated with single-particle levels displaying energies far removed from the Fermi energy, medium polarization effects give rise to inhomogeneous damping. Such a phenomenon is characterized by the FWHM Γ ω (E), which provides the range of energies over which, due to the particle-vibration coupling mechanism, the single-particle strength is distributed. Of notice that this phenomenon does not give rise to a proper lifetime of the quasiparticle state but to a fractionation of the single-particle strength and thus to a dephasing.

Because the main contributions to medium polarization effects arise from the coupling of nucleons to low-lying surface collective vibrations ( ω λ ≈ few MeV), one expects that m ω (E) displays a well defined peak around F , in keeping with the fact that once E -F is larger than ω λ , the collective excitations are decoupled from the particle, whose group velocity therefore increases. There is strong experimental evidence which testifies to this scenario. Now, in many-particle systems, virtual phonons can, not only be emitted and reabsorbed by the same fermion, but also be exchanged between two fermions, giving rise to an induced interaction (medium polarization effects).

We know that such effects play a central role in the phenomenon of superconductivity in metals at low temperatures. Being consistent with the analogy employed in ref. [15] to justify the use of BCS theory in explaining the energy gap observed in the intrinsic excitation spectrum of spheroidal nuclei -an analogy which started the field of nuclear superfluidity -it seems fair to ask how important medium polarization effects are in nuclear pairing. In fact, in keeping with the dynamical shell model discussed above, the question is not whether one has to consider the pairing interaction arising from the exchange of vibrations between nucleons moving in time reversal states close to the Fermi energy, but what the values of the associated matrix elements are as compared with those associated with the bare nucleon-nucleon interaction.

A change in paradigm

Although not explicitly stated, the last section provides evidence for a change of paradigm in the treatment of physical systems, ranging from the Universe to the atomic nucleus. From one of a static, symmetry dominated scenario (geometry of space time, mean field of atoms and nuclei), to another in which the properties of a system do not depend only on the particles which form it, nor on the forces acting among them but also, and primarily, on the medium in which they are embedded (and thus on the ω-and k-dependent generalized dielectric function). And by (external) medium it is meant both that felt by the two weakly bound neutrons of 11 Li, namely the halo field created by themselves, as well as the vacuum (field) felt by an atom or by the whole universe, field which its own expansion creates. To these fields are associated ZPF. It is similar to a quantal harmonic oscillator, in which each degree of freedom contributes a quantity (1/2) ω 0 to the ground state energy, the associated ZPF being measured by ( ω/2C) 1/2 .

In the case of 11 Li, these quanta are essentially the dipole (pigmy) resonance and the quadrupole mode. In the case of the electromagnetic vacuum, the photon. More generally, on the very small scales where quantum effects become important, even empty space is not really empty. Instead virtual particle-antiparticle pairs pop up of the vacuum, travel for short distances and then disappear again on timescales such that one cannot observe them directly (see Fig. 3). Yet their indirect effects are very important, and can be measured. For example, in the Lamb shift, as well as in the Casimir effect to name but two.

Returning to the nuclear case, mean field defines, among other things, a surface. This surface can vibrate collectively, its instantaneous distortion, associated with the ZPF of the collective modes, pointing at each instant of time, into a different direction, thus averaging out the dynamic breaking of symmetry they introduce. Within this scenario, the ground state mean field properties (mean square radius, binding energy, density of levels at the Fermi energy, etc.) of the system under study will be renormalized by the ZPF. In fact, a finite system which fluctuates (both radius and diffusivity) will effectively display varied properties from those associated with the original static, mean field solution ( see refs. [16], [17], and Figs. 4(a)-(e)). Of notice that the same picture can be used to describe rotations, by keeping inertia finite and making the restoring force goes to zero. In this case, the ZPF correspond to an averaging of the angles between the laboratory reference frame, and the privileged orientation defined by the nuclear deformation. Because these angles vary over the whole range of possible values (in keeping with the fact that the restoring force associated with orientation is zero) in this case the ZPF diverge. The associated rotational mode is the equivalent, in the case of finite nuclei, of the Anderson-Goldstone-Nambu mode of field theories displaying spontaneous symmetry breaking. Thus, it is intimately connected with symmetry restoration. Neglecting ZPF implies, in this case, violation of angular momentum invariance, a possibility not contemplated by quantum mechanics. In keeping with this fact one can posit that the same is true in the case of finite (as opposite to divergent) values of the ZPF (vibrations with finite restoring force). In fact, no new physics is introduced in going from finite to vanishing restoring force. If one cannot neglect ZPF in deformed systems (angular momentum projection), one cannot do it either in the case of finite amplitude vibrations.

The same can be argued concerning pairing modes. In fact, mean field not only determines a surface in 3D-space, but also in k-space (Fermi surface), which defines a number of particles (Fermi energy). Fluctuations in it lead to pair addition and to pair substraction modes ( see ref. [18] and Figs. 4(f)-(j)) and eventually pairing rotations when the associated restoring force for pairing vibrational modes vanishes. The coupling of pairing vibrations to single-particle motion can affect alignment in deformed rotating nuclei (see Fig. 4(i)) as well as the value of the pairing gap in an important way (see [19], [20]).

Of notice that when one calculates, on a mean field basis, the nuclear linear response, that is, collective p-h like modes, the only consistent way to avoid talking about ZPF, is to use the Tamm-Dancoff approximation (see e.g. [START_REF] Rowe | Nuclear Collective Motion. Models and Theory[END_REF] and Fig. 4(b)). Now, we know that such an approximation is not physically correct, as it violates the energy weighted sum rule (EWSR), which, in the case of the (electric) dipole mode is tantamount to saying violation of (charged) particles (Thomas-Reiche-Kuhn sum rule). This is the main reason why linear response is calculated in the RPA (or QRPA for superfluid nuclei) taking into account ground state correlations. As seen from Fig. 4(c) this is equivalent to saying that one can excite collective modes equally well through a direct promotion of a particle from a level lying below to Fermi energy (Fig. 4 (b)) to one above it than by forcing, with the help of an external field, the virtual ZPF (oyster diagram, see Fig. 4(c')) from being virtual to become real. As seen from Figs. 4(d) and (e) (see also Fig. 2), a nucleon in presence of ZPF becomes dressed, that is becomes a real nucleon, whose properties can be confronted with experiment (effective-mass (ωmass m ω ), -occupation (Z ω = (m/m ω )), -charge, etc.). In other words, the same process which is needed to fulfill the EWSR (Figs. 4(b),(c)), implies that nucleons should be dressed (Figs. 4 and4(d) and (e)). Consequently the RPA used in the calculation of collective modes is intrinsically wrong, as the backwards going amplitude Y λ ph requires the consideration of ZPF which inescapably leads to effective mass process not considered in RPA (nor in QRPA). Consequently, the minimal description of collective modes is that shown with the diagram of Fig. 5(a). Now, this diagram is one of the many which describes the coupling of one-with two-phonon states. Because of Furry's theorem (see e.g. [14]), it necessarily requires that vertex correction processes (see Fig. 5(b)) be taken into account on equal footing than self-energy ones. More appropriate within the framework of Nuclear Field Theory (NFT) (see refs. [START_REF] Bes | [END_REF], [23], [START_REF] Mottelson | Elementary Modes of Excitation in Nuclei[END_REF]) context, the diagrams shown in Figs. 5(a) and 5(b) are both necessary to satisfy generalized Ward identities (see e.g. [START_REF] Schrieffer | Superconductivity[END_REF]).

Stating the same concept but in an even simpler way one can posit that of all the three possible particle-vibration coupling vertices (see Figs. 5(c),(d) and (e)) RPA (or QRPA) selects only two, namely 5(c) and 5(d). This is because RPA is a harmonic approximation. In other words, one assumes the collective vibration to be a phonon state of an harmonic oscillator Hamiltonian. Thus the coupling between one-and twophonon states must be zero. This is guaranteed if one does not consider the scattering diagram shown in Fig. 5(e). Now, this is a contradiction in terms, in keeping with the fact that it is through an inelastic scattering experiment, as displayed in Fig. 6(a) that one can measure the particle-vibration coupling vertex, or better the transition density of the RPA mode (the same is true for pairing vibrations, see Fig. 6 (b)). But if one has eliminated graph 5(e) in the calculation of the mode (of notice that single arrowed lines pointing upwards represent bound particles), one can hardly imagine that the vertex becomes operative only because the nucleon is now in a scattering state (single (curved) arrowed line pointing upwards, Fig. 6(a)).

Summing up, it is certainly very important to try to use the best available fourpoint vertex (including also three-body interactions) in the calculation of the process displayed in Fig. 5(f), as well as those associated with the pairing interaction (Fig. 5 (g) and (h)) taking properly into account, aside from the standard central term, the isovector, the spin-spin, tensor, etc. components. However, without medium polarization effects of the type displayed in Figs. 5(a) and (b) as well as (h), the corresponding theoretical description is not consistent, as e.g. QED is in the description of electromagnetic processes. Even worse, one may be able to fit some data. However sooner or later the need to include fluctuations on par with mean field properties will be forced by experiment. As we shall see below in connection with the (pairing) particleparticle channel (see Fig. 5), by the direct observation of individual quanta of the glue acting among Cooper pairs. Surprisingly, while any modern calculation of the pairing gap in neutron and nuclear matter considers not only the bare NN-interaction (or whatever effective force which plays its role), but also the induced pairing interaction, a clear resistance to consider medium fluctuation effects in the case of pairing in finite nuclei is apparent.

On the other hand, this resistance of today practitioners has in some sense a touch of déjà vu. In fact the surprise and, to some extent resistance caused by the advent of the nuclear shell model (Goeppert Meyer-Jensen) which, apparently, so directly contradicted the liquid drop and compound nucleus model developed by Niels Bohr and collaborators, triggered a number of (static mean field like) theoretical explanations mostly based on the Pauli principle. The limitations of such an approach have been forcefully argued by Ben Mottelson [START_REF] Mottelson | Trends in Nuclear Physics[END_REF] making use of the quantality parameter Q = ( /M a 2 )/|v 0 | which provides a measure of the validity of independent particle motion (Q 1 implies localization, while Q 1 is tantamount to delocalization, an example of the fact that, while potentials prefer definite relations among particles, fluctuations, quantal or classical, favour symmetries). Typical values of the parameter defining the bare NN-potential (a = 1 fm, v 0 = -100 MeV) lead to Q ≈ 0.4. That associated with the induced interaction medium polarization effects arising from the exchange of collective vibrations between nucleons moving close to the Fermi energy (a ≈ 10 fm, v 0 ≈ -1 MeV) are associated with Q ≈ 4. While the value Q ≈ 0.4 can be assigned to stable nuclei, the second, largest value is more representative of halo nuclei like 11 Li and 12 Be. Consequently, one expects exotic, halo, nuclei to provide an excellent testing ground to study the role virtual processes play in the renormalization of single-particle motion and of NN-interactions, in particular pairing ( 1 S 0 ) interaction (see Fig. 5 (h)). One-and two-particle transfer reactions which make real such virtual processes, can be used as the specific probes to learn about the highly polarizable fragile objects known as halo nuclei, thus shedding light on what, arguably, can be considered the essence of finite, quantum many-body systems: zero point fluctuations.

Let us conclude this section by reminding that there is an essential difference between the concepts of symmetries and that of spontaneous symmetry breaking (and of its intrinsically connected modes leading to symmetry restoration). The first one provides a static, geometric view of the phenomenon. The second one not only considers paramount all and each one of the symmetries of the Hamiltonian describing the system, but at the same time and on equal footing the fluctuations of the mean field responsible for the dynamic maintenance of the associated invariance (or symmetry restoration). In other words, spontaneous symmetry breaking not only takes fluctuations into account but is, as a rule, associated with a boson (Anderson-Goldstone-Nambu mode) for which the fluctuations are not only large, but diverge.

Of notice that the static → dynamic change of paradigm in the study of complex systems is being observed in other fields of physical research, like that associated with the study of proteins and protein inhibitors (see e.g. [START_REF] Broglia | [END_REF]). Proteins, the hardware of life (metabolism and structure), are linear chains of amino acids produced by the ribosome. They fold in short times (typically milliseconds) to their native, biologically active structure. In particular enzymes which, in the native conformation display an active (catalysis inducing) site. In other words, proteins to work have to be folded. The static, symmetry driven picture of nature, see proteins as folded proteins. Not surprisingly, conventional drugs are designed to bind to the active site thus blocking activity, and consequently the ability of the patogen agent (virus, bacteria, etc.) for which the target enzyme plays a central role, to mature and eventually reproduce. Mutations, as a rule, distorting the active site, create resistance. In certain cases (e.g. in connection with inhibitors of the viral hepatitis C) within days from the beginning of therapy.

Studying the mechanism through which a protein, starting from its denatured state folds into the native conformations, one can develop inhibitors which block folding, that is the dynamic process, by binding to those segments of amino acids which, playing a central role in the folding process, cannot be mutated. As a consequence, folding inhibition is likely not to create resistance, opening new venues to deal with infectious diseases. In other words, also in the field of biological physics and, as a result of it, in the design of drugs, a shift of paradigm is taking place, recognizing that proteins are dynamic systems folding as well as displaying conspicuous fluctuations, also in the native (ground state).

Medium polarization pairing interaction in nuclei

There are a couple of facts which have made the discussion of bare versus induced pairing interactions in nuclei difficult: 1)The bare NN-interaction leads, as a rule, to a solution of the BCS (or to the HFB) equations with a finite value of the pairing gap (at variance with the case of metal superconductivity, where the (screened) Coulomb repulsion among electrons leads to values of Δ = 0). 2)Although the pairing gap is an important quantity in the discussion of superfluidity and superconductivity, it is not the specific observable of these phenomena. In fact, the specific probe of pairing correlations in nuclei are two-particle transfer reactions, the associated absolute cross section being indirectly related to the pairing gap. Not surprisingly, we have found [28] that a recent two-particle transfer experiment provides direct, quantitative evidence of the role played in pairing correlations by the exchange of collective modes between Cooper pair partners as had been predicted few years ago (see [29], Ch. 11).

A multipole expansion of the NN-pairing interaction reveals that none of the multipole terms is more important the other, and that high multipoles have to be considered to achieve convergence. One can thus quench the importance of this interaction by concentrating on a nucleus in which the phase space allowed to them to correlate has small angular momentum content. A second condition is that of choosing a system which is very polarizable. It is argued in the next section that such a system are light halo nuclei in general and 11 Li in particular.

Exotic versus stable nuclei, direct observation of phonon mediated pairing

In ref. [30] it is stated that exotic nuclei, being much less bound than stable nuclei, offer a unique framework to study mean field properties without the complications of medium polarization effects. Paraphrasing the paper's arguments with the help of a cartoon representation of 210 Pb and of 11 Li (Fig. 7), one could argue that valence neutrons in Pb can exert a stronger polarization of the core than in Li, because they are closer to it. Now, nucleons in a nucleus resents not only of the bare NN-force, but also of medium polarization effects. In other words, the NN-force is modified by the nuclear dielectric function. Microscopically, this means that nucleons in a nucleus not only exchange pions but also nuclear vibrations. The fact that the full Thomas-Reiche-Kuhn (dipole) sum rule is concentrated in Pb at about 14 MeV, while as much as 20% of it is found around 1 MeV in 11 Li (pigmy resonance), constitutes a sobering warning concerning which of the two systems is more or less polarizable. In fact, microscopic calculations suggest that while the relative contribution to the pairing interaction associated with the bare NN-potential and with the exchange of collective vibrations is fifty-fifty in the case of stable nuclei [31], [32], it is more 20%-80% in the case of light exotic, halo nuclei ( 11 Li, 12 Be) [1], [33]. At this point it is natural to ask about the experimental evidence which, specifically, can test these predictions.

Recently the reaction 11 Li(p, t) 9 Li has been studied [34]. Aside from the ground state, the first excited state of 9 Li(1/2 -, E x = 2.69 MeV) was populated. The 2.69 MeV state of 9 Li is the 1/2 -member of the multiplet (2 + ⊗ p 3/2 (π)), the |2 + being the quadrupole, particle-hole like vibration of 8 He. Assuming a direct reaction process, it can be seen from Fig. 8, that the reaction 11 Li+p → 9 Li(1/2 -) + t provides direct evidence of the exchange of quadrupole phonons between the halo neutrons [28]. The absolute value of the associated cross section is consistent with an amplitude of 0.1 of the | ((s, p) 2 + ⊗ 2 + ) 0 + p 3/2 (π); 3/2 -configuration in the ground state wavefunction 11 Li, in overall agreement with the prediction of ref. [1].

In the last sentence of their paper [34] the authors state two things, namely: 1) that the population of the first excited state of 9 Li suggests a 1 + or 2 + configuration of the halo neutrons; 2) that this shows that a two-nucleon transfer reaction as they study may give new insight in the halo structure of 11 Li. We cannot emphasize strongly enough our support for such statements. Concerning the first one, because NFT studies of the structure of 11 Li [1] indicated this to be the only mechanism for 11 Li to be bound, at the risk of questioning all what has been learned concerning the workings of the particle-vibration coupling mechanism (see e.g. [START_REF] Bohr | Nuclear structure[END_REF], [START_REF] Mahaux | [END_REF], [START_REF] Bes | [END_REF], [23], [START_REF] Mottelson | Elementary Modes of Excitation in Nuclei[END_REF]). In reference to the second one, because it is high time to rediscover that pair transfer is the specific probe of pairing in nuclei (see [START_REF] Bohr | Nuclear structure[END_REF], [START_REF] Broglia | [END_REF], see also [37]).

Conclusions

The essence of finite many-body systems, like e.g. the atomic nucleus, are fluctuations. The interweaving of single-particle motion and the collective vibrations is required by sum rule arguments. The less bound a system is, the more important polarization effects are expected to be. It is then not surprising that by studying a very fragile object like 11 Li through a two-particle transfer experiment, specific probe of pairing, one has obtained the first direct evidence of phonon mediated Cooper pair binding in nuclei. Such a result indicates that while medium polarization effects pervasively renormalize all of the observables, they may be directly observed only in specific experiments. On the other hand, their effect, virtually, is present at all times. The sooner one comes to terms with this fact the better. 11 Li viewed as a system with two neutrons (solid dots) moving around a core (dashed area). The distance of the valence neutrons from the core reflects the fact that the two neutron separation energy of these nuclei is S 2n = 15.2 MeV and 0.380 MeV respectively.

Figure 1 .Figure 2 .

 12 Figure 1. Lowest order self energy process of a particle with its own field.

Figure 3 .

 3 Figure 3. Schematic representation of quantal vacuum ZPF. Particle-antiparticle pairs are spontaneously (virtually) excited and annihilated.

Figure 4 .

 4 Figure 4. Schematic representation of the surface (a) and of pairing (doubled arrowed line) (f) vibrations.

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. Self-energy processes of particle hole vibration ((a) and (b)) RPA ((c),(d)) and scattering (e) particle-vibration coupling vertices. In (f) we display the RPA diagrams in terms of p-h bubbles. (g) Four point vertex, (h) particle-particle (pairing) interaction mediated by collective p-h vibrations.

Figure 8 .

 8 Figure 8. (a) NFT-Feynman diagram associated with the process 1 H (11 Li(gs), 9 Li(1/2 -;2.69 MeV) 3 H), which treats on equal footing the nuclear structure ( 2 + , 2+ , ) and the reaction mechanism ( , ). Arrowed lines indicate bound particles, curved arrowed lines, scattering states. Curly brackets indicate angular momentum coupling, while horizontal dashed lines indicate magnetic quantum number conservation In (b) and (c) a schematic representation of the initial ( 11 Li) and final ( 9 Li(1/2 -, M f ; 2.69 MeV)) nuclear states is given, respectively.

The authors want to acknowledge important discussions with Thomas Duguet concerning the pairing NN-interaction. His work and that of his collaborators on the subject are likely to be instrumental in ushering the field of nuclear superfluidity into its really quantitative era.