Nuclear Field Theory predictions for {}^{11}Li and {}^{12}Be: shedding light on the origin of pairing in nuclei

Nuclear Field Theory predictions for Li and Be: shedding light on the origin of pairing in nuclei

Abstract

(August 13, 2019)

Recent data resulting from studies of two–nucleon transfer reaction on Li, analyzed through a unified nuclear–structure–direct–reaction theory have provided strong direct as well as indirect confirmation, through the population of the first excited state of Li and of the observation of a strongly quenched ground state transition, of the prediction that phonon mediated pairing interaction is the main mechanism binding the neutron halo of the 8.5 ms–lived Li nucleus. In other words, the ground state of Li can be viewed as a neutron Cooper pair bound to the Li core, mainly through the exchange of collective vibration of the core and of the pigmy resonance arizing from the sloshing back and forth of the neutron halo against the protons of the core, the mean field leading to unbound two particle states, a situation essentially not altered by the bare nucleon-nucleon interaction acting between the halo neutrons. Two-neutron pick-up data, together with data on Li, suggest the existence of a pairing vibrational band based on Li, whose members can be excited with the help of inverse kinematic experiments as was done in the case of reaction. The deviation from harmonicity can provide insight into the workings of medium polarization effects on Cooper pair nuclear pairing, let alone specific information concering the “rigidity” of the N=6 shell closure. Further information concerning these questions is provided by the predicted absolute differential cross sections associated with the reactions (gs) and ()(Be(p,t)(gs)). In particular, concerning this last reaction, predictions of can change by an order of magnitude depending on whether the halo properties associated with the orbital are treated selfconsistently in calculating the ground state correlations of the (pair removal) mode, or not.

I Introduction

At the basis of BCS theory of superconductivity one finds the condensation of strongly overlapping Cooper pairs. This model can be extended to the atomic nucleus, provided one takes pair fluctuations into account, fluctuations which renormalize in an important way the different quantities entering the theory, in particular the pairing gap (, see e.g. Brink:05 () Fig. 6.2 p. 152 and connected discussion). Not only pairing vibrations are more collective in the nuclear case than in condensed matter (see e.g. Anderson:58 ()) and in liquid He Wolfle:78 (), where they essentially correspond to a two quasiparticle–like state lying on top of twice the pairing gap. They also develop, in the nuclear case, as well defined vibrational bands around closed shell nuclei, where one can clearly distinguish between particle and hole degrees of freedom, in keeping with the relatively large gap existing between occupied and unoccupied single–particle states. Systematic evidence exists of the correlation and stability of the pair addition and pair subtraction modes (single Cooper pairs) in medium heavy nuclei (see e.g. Brink:05 () and Broglia:73 () and references therein). Making use of these building blocks a number of pairing vibrational bands have been identified, providing important insight into the mechanism in which nuclear superfluidity emerges from the condensation of many phonon pairing modes. In medium heavy nuclei, a large fraction of the pairing correlation energy ( 50–70%) is due to the strong bare interaction, medium polarization effects being important although not dominant (see Barranco:99 () as well as Brink:05 (), Ch. 10 and refs. therein). This is, among other things, one of the reasons why the multi–phonon pairing spectra seen to date are rather harmonic (see e.g. Flynn:72 ()). In other words, the interaction between pairing modes in heavy stable nuclei is weak and anharmonicities effects are, although very revealing of the nuclear structure around closed shell nuclei, not very important (note however Pauli principle effects Bortignon:78 ()).

We expect the situation to be rather different in the case of weakly bound, strongly polarizable light halo nuclei like Li and, to some extent also Be. In both cases, the N–N bare interaction provides a small contribution ( keV) to the two-neutron binding energy. Thus a major fraction of the glue holding the neutron Cooper pair (halo) to the Li and to the Be cores arizes from the exchange of collective vibrations Barranco:01 (); Gori:04 (). To further clarify the origin of pairing in nuclei, it would then be important to look for the pairing vibrational band associated with the closed shell systems Li and Be. Larger or smaller anharmonicities as compared with other closed shell systems (like e.g. Pb) would provide a welcome input to shed light on the relative contributions to the nuclear pairing force, both in terms of the strength as well as of the radial shape (form-factors), of the bare force and of the effective pairing force arising from the exchange of collective vibrations between pairs of nucleons moving in time reversal states close to the Fermi energy. In fact, if all of the correlations came from the short range, bare interaction, many processes leading to anharmonic effects would not be present in the pairing vibrational nuclear spectrum. In particular, a special group of those related to Pauli principle corrections reflecting the overcompleteness of the NFT basis of elementary excitations, i.e. single-particle, particle-hole-like and pair vibrational modes.

Ii Nuclear Field Theory

Nuclear Field Theory (NFT) provides a systematic method for describing both nuclear structure Bes:76a (); Bes:76b (); Bes:76c (); Bes:75 (); Mottelson:76 (); Bortignon:77 (); Broglia:76 () and nuclear reactions Broglia:05c (). In particular, it allows to deal with the problem of overcompleteness of the degrees of freedom arising from the identity of particles appearing explicitly and the particles participating in the collective motion. The non–orthogonality of the variety of degrees of freedom must be expected quite generally in descriptions that: a) from the nuclear structure point of view exploit simultaneously the single-particle degrees of freedom, as well as the quanta of particle–hole type together with those involving pairing vibrations Bes:66 () made out of two correlated particles (pair addition modes) or of two correlated holes (pair subtraction modes); b) from the reaction point of view, make use of mean fields of target and of projectile which are assumed to be independent of each other, with the added possibility, in the case of the study of pairing correlations through two–particle transfer reactions, that a member of a Cooper pair Cooper:56 (), Leggett:06 () is in one system (e.g. target) and the second partner in the other one (i.e. projectile). This is in keeping with the fact that the typical value of the pairing gap in superfluid nuclei (or of correlation energies in the case of normal systems) is 1 MeV, implying that Cooper pair partners are correlated over distances of the order of the associated coherence length, 20-30 fm. This result, together with the fact that , being the average mean field potential, implies, as a rule, that successive transfer, where acts twice, is dominant as compared to the simultaneous transfer of two nucleons, a process in which acts once, the second nucleon being transferred partially making use of the interaction energy between the partners of the Cooper pair, and partially exploiting the non–orthogonality of the single–particle wavefunctions associated with the mean field of target and projectile. It is of notice that this argumentation is at the basis of one of the cornerstones of superconductivity in metals, namely the Josephson effect Josephson:62 (); Cohen:62 () (see also Potel:09a () and references therein).

Tailored after Feynman’s version of Quantum Electrodynamics (QED) Feynman:61 (), that is in terms of Feynman diagrams, NFT is essentially only restricted by the limitations in the (experimental) knowledge one has of the particle–vibration coupling vertices , where indicates the quantum numbers of the phonon but associated with gauge space, as well as whether the mode in consideration is the first, second, etc. in energy, while , is the transfer quantum number ( implies particle-hole excitations, while pair-addition and pair-removal modes respectively, see e.g. Broglia:73 (), Bortignon:77 () and Bohr:75 () and refs. therein). At variance with QED for which an essentially definitive value of the fine structure constant exists, uncertainties are to be ascribed to the variety of values. This is due to the combined effect of the fragmentation of the modes, as well as the fact that to extract from a reaction measurement one needs to use optical potentials, with all the associated limitations of such an approach (see Fig. 1).

In any case, progress which has taken place during recent years has ushered studies of nuclear pairing with two-nucleon transfer reactions into the quantitative era, with uncertainties well within experimental errors (see e.g. Potel:10 (); Potel:11a (); Potel:11b (); Potel:12a () and refs. therein).

Iii The Cooper pair problem and Li

The basis for the understanding of superconductivity in metals worked out by Bardeen, Cooper and Schrieffer (BCS theory Bardeen:57a ()) was provided by Cooper Cooper:56 () who solved the problem of a pair of electrons interacting above a noninteracting Fermi sea of electrons via a two–body potential. Li can be viewed as a rather accurate realization of Cooper’s model, within the framework of nuclear physics: a pair of neutrons weakly bound to the Li core.

Now, it was found in a NFT study of this halo nucleus Barranco:01 (), that the bare nucleon–nucleon interaction does not bind the Cooper pair (see Fig. 2 (a)). It is the exchange of the vibrations of the halo field and of the core (dipole and quadrupole) between the halo neutrons which, similar to phonons in metals, binds the Cooper pair (see Fig. 2 (b)). The NFT results of ref. Barranco:01 () provide a rather accurate description of the nuclear structure properties of Li. The calculations were carried out following all what is well known about NFT and the particle–vibration coupling mechanism (see, e.g. Bes:76a (); Bes:76b (); Bes:76c (); Bes:75 (); Mottelson:76 (); Bortignon:77 (); Broglia:76 (), Bohr:75 () and refs. therein). A single mean field potential (Saxon–Woods) was used for all single–particle (…) states (cf. Bohr:69 () pp. 236-240). Parity inversion of the and states of Li results from the coupling of these resonances to vibrations of the Li core 1, when the coupling to the continuum is taken properly into account. Making use of the resulting single-particle (dressed) states, of the bare N-N interaction and of the interaction induced by the exchange of collective modes between the dressed single-particle halo states, including the pigmy resonance of Li (cf. Fig. 2 (b)), a quantitative account of the experimental findings was obtained, in particular concerning the extremely low two–neutron separation energy MeV Bachelet:08 (); Smith:08 () (see also Caption to Fig. 2). Notice that if the interaction is not included, the absolute value of is decreased by only about 100 keV.

The NFT description of the structure of Li also contained precise predictions (see e.g. Brink:05 () Ch. 11, in particular Fig. 11.6) concerning how to force virtual correlation processes, to become real. Namely, through two–neutron pick–up reactions, like e.g. . In this way it was found that one can excite, among others, the multiplet of states (see Fig. 2 (c), see also Figs. 3 and 4). These predictions had to wait short of ten years to be tested (see ref. Tanihata:08 ()), and proved not to be wrong. 2 In fact, making use of a detailed nuclear reaction theory of two-particle transfer which considers successive, simultaneous and non–orthogonality processes, together with inelastic and break up channels and thus accurately treats the coupling to the continuum concerning both the structure and reaction aspects of the calculations, one has been able Potel:10 () to check the validity of such predictions, and reproduce in a quantitative fashion, the experimental findings of ref. Tanihata:08 (). In particular the observation of the member of the () multiplet of Li which is the first excited state of this nucleus (see Fig. 4). This result provided the first, direct confirmation of the central role played by vibrational states in nuclear pairing.

Iv Pairing vibrational band of Li

The unified NFT description of structure and reactions used in refs. Barranco:01 (); Potel:10 (), provides a number of suggestions concerning the consequences the findings reported in Tanihata:08 () has. In particular, the existence of a many–phonon pair vibrational spectrum (see Fig. 5), similar to that observed in and reactions in the Pb region3 (see e.g. Brink:05 (); Broglia:73 (); Flynn:72 ()).

The energy of the two–phonon state () mode of Li, where indicates the number of pair addition (; ground state of Li) and pair removal (; ground state of Li) modes is expected to be, in the harmonic approximation Bes:66 () (see also App. A),

(1)

where is the binding energy of the Li–isotope with mass number (see Fig.6, see also App. A and Fig. 21 below).

The excitation energy of a state of the vibrational band (again in the harmonic approximation), can be written as

(2)

The absolute value of the different transitions between these states can be expressed in terms of the basic cross–sections

(3)

An embodiment of the above definitions is provided by the values Tanihata:08 (); Young:71 () (see also AjzenbergSelove:78 ()),

(4)

and

(5)

and the principle of detailed balance. It is of notice that while and are intrinsic properties of the pairing vibrational modes (it is reminded that the Fermi energy within the framework of the pairing vibrational model is determined by the value of the energy associated with the minimum of the RPA dispersion relation, see Apps. A and C), (4) and (5) depend also on the bombarding conditions (bombarding energy, scattering angle, etc), as well as on the reaction (, (O,O), etc). Within this context it is of notice that the values reported in (4) and (5) corresponds to tritons of MeV and of 5 MeV per nucleon respectively (see also App. D).

With such caveat in mind one expects that (see also Flynn:72 ()), at equal bombarding conditions as those corresponding to (4) and (5) (see Tanihata:08 (); Young:71 ())

(6)

and

(7)

in a similar way in which the transition is expected to display a –value twice as large as that associated with the transition of the one phonon state to the ground state i.e. .

In Fig. 5 the one– and two–phonon quadrupole spectrum of Li expected in the harmonic approximation is also shown. In Fig. 6 we display the pairing vibrational spectrum of Li and the associated absolute two–particle transfer differential cross sections of the pair addition and pair removal modes, in comparison with the experimental data Tanihata:08 (),Young:71 ().

In the case of the reaction LiLi(gs) the absolute differential cross section was calculated making use of the two-nucleon spectroscopic amplitudes obtained from the term of the neutron component of the Li ground state (, cf. Potel:11b ()),

(8)

with and , and

(9)

describing the motion of the two halo neutrons around Li. Successive, simultaneous and non-orthogonality contributions to the two-particle transfer process were taken into account (see Potel:10 (); Potel:12a (); Tanihata:08 () and refs. therein). The optical parameters used to describe the Li+p, Li+d and the Li+t channels were taken from refs. Tanihata:08 (), An:06 () .

In the case of Li(t,p)Li(gs) reaction, the optical parameters were taken from refs. Young:71 (),An:06 () while the two-neutron spectroscopic amplitudes, reported in Table I, were calculated within the RPA (cf. e.g. Brink:05 () and Broglia:73 (), see App. A), making use of a Saxon-Woods potential with standard parametrization (see Bohr:69 () pp. 236-240) to describe the single-particle energies, and of the experimental correlation energy of the two-neutron holes in Li (as an example (in this case regarding Be-isotopes), cf. Caption to Fig. 23). As seen from Fig. 6 and Table II, theory provides an overall account of the experimental findings. While it is the component (9) which is directly involved in the LiLi(gs) transfer process, the amplitudes and play a central role concerning the absolute value of the cross section. In fact, incorrectly normalizing the state (9) to 1, that is , leads to an integrated cross section in the angular range - of 12.1 mb. This result is a factor of 2 larger than that predicted making use of (9), and lies outside the errors of the experimental value (see Table II as well as Potel:10 () and Tanihata:08 ()); see also Fig. 7(a), result labeled TDA).

Within this context, one can also play the same game concerning the two-nucleon spectroscopic amplitudes associated with the reaction Li(t,p)Li(gs), by setting and normalizing the RPA pair removal wavefunction according to . In other words, neglecting ground state correlations (TD approximation) which leads to and . The resulting absolute cross section for the reaction Li(t,p)Li(gs) integrated in the range becomes 9.1 mb, a value which lies outside the experimental errors of the observed value (see Table II and Fig. 7 (b)).

As shown in Figs. 8 and 9, anharmonicities in the pairing vibrational spectrum can arise due to a number of physical effects. For example: 1) Overcompletness (see Fig. 9 (b) and (c)); 2) Pauli principle correction to multiphonon states (see e.g. Figs. 8(a), 8(b) and 9 (c)) leading, in connection with phonon mediated pairing (see e.g. Fig. 8(b)), to correlation (CO)-like processes Mahaux:85 () (see inset (a) of Fig. 10). Within this context, it is of notice that the parity inversion observed to take place between the and states of a number of isotones (see Fig.10) is due, to a large extent, to CO processes Barranco:01 (); Gori:04 () (see inset (a) in Fig. 10). This is in keeping with the fact that among the largest components of the vibration of He and of Be one finds the particle-hole component.

It is then expected that the contributions to the anharmonicity of the pairing vibrational band in light halo nuclei (like Li but also Be, see Sect. V), can be important, in keeping with the high polarizability displayed by these systems as compared with, for example, Pb. The energy of the two–phonon pairing vibrational mode given in Fig. 6 for Li (as well as that associated with Be, see Sect. V below) could, in principle, be strongly modified by such effects, as well as by coupling to states consisting of two (particle-hole-like) phonons (see Fig. 9(c); within this connection see Broglia:71b () as well as App. B).

iv.1 The optical potential

In keeping with the fact that structure and reactions are just but two aspects of the same physics and that in the study of light halo nuclei, continuum states are to be treated on, essentially, equal footing in the calculation of the wavefunctions describing bound states (structure) as well as of the asymptotic distorted waves entering in the calculation of the absolute two-particle transfer differential cross sections (reaction; see Figs. 11 (a) and 11 (b)), the calculation of the optical potentials is essentially within reach (reaction, see Fig. 11 (c) and Fig.12, see also e.g. Fernandez:10a (); Fernandez:10b () and refs. therein).

Because the real and imaginary parts of complex functions are related by simple dispersion relations (see e.g. Mahaux:85 () and refs. therein) it is sufficient to calculate only one of the two (real or imaginary) components of the self-energy function to obtain the full scattering, complex, nuclear dielectric function (optical potentials). Now, absorption is controlled by on-the-energy shell contributions. Within this scenario it is likely that the simplest way to proceed is that of calculating the absorptive potential and then obtain the real part by dispersion (see e.g. Broglia:81b (), Pollarolo:83 (); see also Broglia:05c ()). Of notice that in heavy ion reactions, one is dealing with leptodermous systems. Thus, the real part of the optical potential can, in principle, be obtained by convolution of the nuclear densities and of the surface tension (see e.g. Broglia:05c () and refs therein).

Within the present context, one can mention the ambiguities encountered in trying to properly define a parentage coefficient relating the system of nucleons to the system of nucleons, and thus a spectroscopic amplitude (see e.g. Dickhoff:05 (); Jennings:11 (), see also Mahaux:85 ()). In other words, a prefactor which allows to express the absolute one-particle transfer differential cross section in terms of the elastic cross section.

Making use of NFT diagrams like the one shown in Fig. 12, it is possible to calculate, one at a time, the variety of contributions leading to one- and two-particle transfer processes (in connection with this last one see Fig. 11). Summing up the different contributions, taking also proper care of those arising from four-point vertex, tadpole processes, etc. (see refs. Bes:76a (); Bes:76b (); Bes:76c (); Bes:75 (); Mottelson:76 (); Bortignon:77 (); Broglia:76 () and refs. therein, see also Dickhoff:05 ()), a consistent description of the different channels can be worked out, in which the predicted quantities to be directly compared with observables are absolute differential cross sections, or, more generally, absolute values of strength functions for different scattering angles.

V The pairing vibrational spectrum of Be

Calculations similar to the ones discussed in previous sections have been carried out in connection with the expected shell closure pairing vibrational band of . In Fig. 13 we display the associated pairing vibrational spectrum in the harmonic approximation (see App. A, in particular section of this Appendix). Also given are the absolute two-nucleon transfer differential cross sections associated with the excitation of the one-phonon pair addition and pair subtraction modes excited in the reactions Be(p,t)Be(gs) and Be(p,t)Be(gs) respectively, calculated for a bombarding energy appropriate for planned studies making use of inverse kinematic techniques Kanungo:11 ().

The ((2p-2h)-like) two-phonon pairing vibration state of Be is expected, in this approximation, to lie at 4.8 MeV, equal to the sum of the energies of the pair removal = 0.5 MeV and of the pair addition = 4.3 MeV modes. In keeping with the fact that the lowest known excited state of Be appears at about 6 MeV Alburger:69 (), we have used this excitation energy in the calculation of the value associated with the Be(p,t)Be(pv) cross section. The associated shift in energy from the harmonic value of 4.8 MeV can, arguably, be connected with anharmonicities of the Be pairing vibrational spectrum, as discussed in the case of Li in connection with Figs. 8 and 9. Medium polarization effects (see e.g. Fig. 8(b)) may also lead to conspicuous anharmonicities in the pairing vibrational spectrum.

The two-nucleon spectroscopic amplitudes corresponding to the reaction Be(p,t)Be(gs) and displayed in Table III were obtained solving the RPA coupled equations (determinant) associated with the Be(gs) pair-removal mode, making use, as explained in Sect. b of App. A, of two pairing coupling constants , to properly deal with the difference in matrix elements (overlaps) between core-core, core-halo and halo-halo two-particle configurations. In other words with a “selfconsistent” treatment of the halo particle states (), in particular of the halo state. The absolute differential cross sections displayed in the figure were calculated making use of the optical parameters of refs. An:06 (); Fortune:94 ().

The two–nucleon spectroscopic amplitudes associated with the reaction correspond to the numerical coefficients appearing in Eq. (12) below, and associated with the wavefunction describing the neutron component of the Be ground state (cf. ref. Gori:04 ()):

(10)

with

(11)

and

(12)

the states , , being the corresponding lowest states of Be, calculated with the help of a multipole separable interaction in the RPA (see e.g. Table IV). It is of notice that a rather similar absolute differential cross section to the one displayed in Fig. 13 for the BeBe(gs) reaction is obtained making use of the spectroscopic amplitudes provided by the RPA wavefunction describing the Be pair addition mode (see Table III and App. A). This can be seen from the results displayed in Fig. 14.

To assess the correctness of the structure description of Be(gs) provided by the wavefunction (10-12) and of the second order DWBA-reaction mechanism (successive, simultaneous plus non-orthogonality) employed to calculate the absolute value of the BeBe(gs) differential cross section, we compare in Fig. 15 the predictions of the model for the reaction Be(p,t)Be(gs) at 17 MeV triton bombarding energy with the experimental data. Theory provides an overall account of observation within experimental errors.

It is of notice that the components proportional to and of the state (10) can lead, in a Be reaction, to the direct excitation of the and states of Be. Such results will add to the evidence obtained in the reaction H (Li(gs),Li(;2.69 MeV))H Tanihata:08 () of phonon mediated pairing Potel:10 (). The role of these components is assessed by the fact that (wrongly) normalizing the state (12) to 1, one obtaines a value of mb (), a factor 2 larger than the experimental value Fortune:94 () (see Fig. 15).

Let us now return to Fig. 13. 4 The ratio of the integrated absolute cross section at = 7 MeV in the range 10 appropriate for planned experimental studies making use of inverse kinematic techniques Kanungo:11 () is,

(13)

a result which testifies to the clear distinction between occupied and empty states taking place at , and thus of the bona fide nature of this magic number for halo, drip line nuclei. The ratio (13) reflects the fact that the pairing Zero Point Fluctuations (ZPF in gauge space) displayed by the Be as embodied in the pair addition and pair removal modes, and quantified by the absolute values of the associated two-nucleon transfer cross sections, are of the same order of magnitude. This is an intrinsic property of the vibrational modes, in the same way in which e.g. the width (lifetime) of a nuclear state is an intrinsic (nuclear structure) property of such a state. An experiment displaying an energy resolution better than the intrinsic width of the states under study will provide structure information. Otherwise, eventually an upper limit. Within this scenario and in keeping with the fact that the successive transfer induced by the single-particle potential is the intrinsically (structure) dominant contribution to the absolute two-particle transfer cross section, Q-value (kinematic) effects can strongly distort the picture. In particular in the case in which single-particle transfer channels are closed at the studied bombarding energies.

Let us elaborate on these arguments in the case in which the reaction Be(p,t)Be populates the Be ground state, where the two correlated nucleons participating in the process are the two valence neutrons of Be. As the first neutron leaves Be, it leads to Be where labels , and . In Fig. 16 we report, for concreteness, the value scenario where , although the calculations of the absolute cross sections displayed in Figs. 13 and 14 were carried out making use of the full wavefunction, and thus of all possible (, and ) intermediate states in the Be channel. Of notice that, because of ground state correlations, situations in which and are also possible (as hole states). The corresponding contributions are not negligible (see Table 3), although the most important ones arise from those mentioned above.

The case in which one populates the excited pairing vibrational state MeV) is illustrated in Fig. 17. The fermions involved correspond to the correlated (valence) holes characterizing the ground state of Be, the two correlated (valence) neutrons building the Be ground state acting, in the harmonic approximation, only as spectators. Within this scenario, the most important contributions to the Be intermediate states are hole states moving in the and orbitals. As seen from the figure, at MeV, the channels (e.g. Be) are barely open, thus quenching in a major way the excitation of the MeV) state. The closing or opening of single-particle transfer channels due to Q-value effects, may constitute a unique opportunity to learn about the mechanism which is at the basis of two-nucleon transfer reaction processes (see Figs. 18-20). By properly adjusting the bombarding conditions (excitation function), one may tune on situations in which two-particle transfer switches from successive- to simultaneous-dominated transfer regimes.

Simple mechanical interpretations of the above mentioned Q-value effects can be given within the framework of the semiclassical approximation Broglia:05c (). The transfer formfactors associated with the excitations of pairing correlated modes are, as a rule, smoothly varying functions along the trajectories of relative motion, displaying none or few nodes, in the neighborhood of the distance of closest approach. To obtain the variety of contributions to the two-nucleon transfer amplitudes the different formfactors are to be weighted with (imaginary) exponentials (i.e. periodic functions), resulting from the mismatch in momentum (recoil effects, Galilean-like transformations), and in stationary state phases (i.e. exp) associated with the different relative orbitals. Large transfer amplitudes and eventually two-nucleon transfer cross sections are obtained when the arguments of the phases appearing in the imaginary exponential are small along the relative motion trajectory (little mismatch). The situation is reversed when the large (stationary) energy differences are not compensated by recoil effects and viceversa. In this case, the harmonic functions change rapidly sign along the relative motion trajectories, thus canceling the (smooth behavior) transfer formfactor contributions (large mismatch).

The extension (of the same arguments) to situations of long wavelength of relative motion (fully quantal scenario) can be made in terms of the WKB approximation (e.g. in terms of Stokes lines), as well as in terms of the Feynman path integral method. In this case all possible trajectories are to be considered. The associated formfactors are weighted with action phases. The different amplitudes thus result from the interweaving of these two types of functions. Depending on their mismatch different bombarding energies, angles, etc. will be privileged over others, as in the simpler (semiclassical) scenario discussed above.

Vi Conclusions

It is an open question whether (mainly) phonon mediated neutron halo pairing can lead to a well developed (quasi harmonic) pairing vibrational multiphonon band. Inverse kinematics two–particle transfer processes, as well as standard two–particle transfer reactions can test the validity of the harmonic spectrum discussed above, and thus the actual nature of the shell closure, as well as shed light on the role medium polarization effects play in these fragile, highly polarizable systems. In particular concerning the induced pairing interaction. The question of how large the coupling between particle–hole phonons and the () pairing modes is, could also be tested through the above mentioned two–particle transfer experiments. Last, but not least, the NFT description of structure and reactions provides a natural framework for the calculation of the optical potential, a possibility which would bring one step further the concrete realization of the fact that structure and reactions are but two aspects of the same physics, in particular in the case of halo nuclei.

Discussions with I. Tanihata and R. Kanungo are gratefully acknowledged.

Financial support from the Ministry of Science and Innovation of Spain Grant No. FPA2009-07653 is acknowledged by G. P.

F.B. acknowledges financial support from the Ministry of Science and Innovation of Spain grants FPA2009-07653 and ACI2009-1056.

Figure 1: Diagramatic representation of: a) inelastic scattering process (transfer quantum number ) of a collective particle–hole like (quadrupole surface vibration) induced by the nonhomogeneous time dependent field created by a nucleon (projectile, curved arrowed line) passing by the target nucleus, b) two–particle transfer process in which a dineutron is stripped from the incoming triton exciting the pair addition mode (transfer quantum number ) of e.g. a closed shell system (double lines labeled with a conventionally empty arrow corresponds to two neutrons bound in the projectile (triton); when labeled by a solid arrow, it corresponds to a pair addition mode, i.e. to two nucleons bound to the core nucleus). The solid dot indicates the particle vibration coupling vertex measured by the strength , where the subindex indicate the angular momentum and eventually, if the mode is fragmented, whether the state is the lowest, next to lowest, etc. in energy.

Figure 2: In the case of Li the bare interaction (e.g. Argonne potential; horizontal dotted line) acting to infinite order between the halo neutrons (see (a)) does not lead to a bound Cooper pair (double solid arrowed line), but merely shifts downwards (lowers the energy) of the resonant configurations and by about 100 keV individually, without leading to any appreciable mixing [8]. The exchange of a dipole pigmy resonance of the halo field and of a quadrupole vibration of the Li core (wavy lines, see (b)) provides essentially all of the glue for the Cooper pair to become bound ( keV, theory [8] as compared to the experimental data of 3785 keV [32]; 369.150.65 keV [33]). In (c) a schematic representation of the process (see [34, 26]) is displayed (see also Figs. 3 and 4).

Figure 3: (a) NFT–Feynman diagram associated with the process H (Li(gs),Li(;2.69 MeV)H), which treats on equal footing the nuclear structure () and the reaction mechanism (). The neutron correlated Cooper pair (pair addition mode), has a somewhat different structure when bound to Li to make Li, than to the proton to make the triton (H) (depicted with a solid and with an open arrow respectively). This is the reason why there is a finite overlap, called between the corresponding relative motion wavefunctions displaying zero,one, etc. nodes (cf. e.g. [4] and references therein). In keeping with this fact one can posit that although () reactions are quite specific to probe pairing correlations in nuclei, in particular in the case of the Li neutron Cooper pair, they display limitations. In particular the specific probe in the case under discussion is the Li(Li,Li)Li reaction (), although obviously much more demanding experimentally. 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 (Li) and final (Li(; 2.69 MeV)) nuclear states are given, respectively. The ordinate to the right indicates time. Following NFT of both structure [10, 11, 12, 13, 14, 15, 16] and reaction [17] processes, all possible time orderings are to be considered in the calculations.It is of notice that curved arrowed lines indicate continuum scattering states, while standard arrowed lines correspond to bound states.

Figure 4: Gedanken (two–particle transfer)–( decay) coincidence experiments aimed at better individuating the couplings involved in the neutron halo Cooper pair correlations in Li and of the member of Li excited in the H(Li,Li)H reaction [34, 26].

Figure 5: Schematic representation of the phonon pairing () and density vibrational () spectrum based on Li (harmonic approximation), constructed making an analogy with studies of two–particle transfer reactions carried out around Pb (cf. e.g. [1] p. 109 Fig. 5.5 and refs. therein; see also [8]). The Fermi representation of occupied proton () and neutron () states is schematically shown in terms of grey areas, where holes are shown as empty circles, and particles are indicated in terms of crosses. A single–arrowed line pointing upwards (downwards) represent a particle (hole) neutron or proton state. A double arrow indicates the correlated neutron pair addition (subtraction) modes. The predicted two–particle transfer cross sections and excitation energy associated with the two–phonon (pair–addition), (pair–subtraction) and ( like excitation) pairing vibrational (pv) state expected in Li are, in the harmonic approximation: MeV, mb/sr [34], and mb/sr [35], (see also Fig. 6).

Figure 6: Pairing vibrations around Li and absolute cross sections associated with removal [35] (see also [36]) and addition [34] modes. The theoretical absolute differential cross sections for (addition: a) is reported in [26]. The theoretical absolute differential cross section associated with the reaction (removal: r) was carried out making use of the wavefunction associated with the RPA solution of the pairing Hamiltonian (see [4], [18] and App. A as well as Table 1) , adjusting the coupling constant to reproduce the correlation energy of the two neutron holes in the core of Li (i.e. in the ground state of Li). The optical potential parameters used were taken from ref. [35, 37]. Of notice that throughout in this paper, in particular in connection with this figure, we report absolute differential cross sections (see also Table 2).

Figure 7: Absolute differential cross section of the pair addition and pair removal modes of Li in comparison with the experimental findings. (a) The NFT results of the calculations of the absolute values of associated with the reaction (pair addition mode) reported in Fig. 6 are compared with those labeled Tamm-Dancoff approximation (TDA), in which the interweaving of single-particle and particle-hole like vibrational modes are neglected while the components of the two-neutron wavefunction are normalized to one (see text). (b) The absolute value of the differential cross section associated with the reaction (pair removal mode) and calculated making use of the RPA two-nucleon transfer spectroscopic amplitudes (-values, Table 1) also reported in Fig. 6, is compared with that obtained neglecting ground state correlations and labeled TDA (see text).

Figure 8: Pairing modes phonon–phonon interaction arizing from: (a) Pauli principle processes between pairing modes, and (b) between single–particle and ph- phonon mediated induced pairing interaction (so called CO diagrams, cf. e.g. [38]. See also Fig. 10 (inset)).

Figure 9: (a),(b) Examples of pair addition and pair removal modes interactions. (c) Interaction between the two-phonon pairing vibration state and the two-phonon particle-hole state.

Figure 10: Single-particle states for isotones around Be associated with parity inversion. The thin horizontal lines represent the single-particle state, while the thick ones the orbital. In the case of Li one reports the centroid of the virtual and of the resonant states. stands for excitation energy and is the neutron separation energy. In the case of Li e.g. , while MeV and MeV. In the inset the correlation (CO) and polarization (PO) (virtual) contribution to the single-particle self energy are shown. An arrowed line pointing upwards represent a particle moving in a level with energy , a downwards pointing line represent a hole state , while a wavy line stands for a ph-like vibrational state. Their contribution to the real (single-particle “legs” propagating to times) processes dressing the and neutron states of Li and Be are (a) and (b) respectively. In the first case the phonon corresponds essentially only to the vibration of the corresponding core (Li and Be respectively), and pushes the orbital upwards (Pauli principle, Lamb-shift-like process) making the dressed orbital more strongly unbound than what it was originally in the Saxon–Woods potential (see [31] Eqs. (2-180)–(2-182) pp. 238 and 239). In the case of the orbital, it is mainly the process (b) which dresses the state making it almost bound (virtual state) as compared with the Saxon–Woods state. Within this context, it is of notice that in the binding of the two halo neutrons of Li to the Li core, it is essentially the pigmy resonance of Li which provides the largest contribution, the coupling to the vibration of the core Li giving a small shift in energy (nonetheless, it is this weak component of the self energy which is responsible for the excitation, in the Li(p,t)Li reaction [34], of the MeV state [26]). In the case of Be the (p-h) vibrations are the , and of the core Be, in keeping also with the fact that Be does not display a pigmy resonance, not at least based on the ground state. It is of notice that graphs (a) and (b) give rise to an effective mass known as the -mass. Associated with it are the occupation factors (discontinuity at the Fermi energy; for details see ref. [38] and refs. therein).

Figure 11: NFT diagrams summarizing the physics which is at the basis of the structure of Li [8] and of the analysis of the reaction [26]. At variance with similar diagrams shown in Fig. 2 (c) and Fig. 3, in the present figure emphasis is set on intermediate (like e.g. Li, see (a) and (b)) and elastic (see (c), see also Fig. 12) channels.

Figure 12: NFT diagrams and summary of the expression (see e.g. [38] and refs. therein) entering in the calculation of one of the contributions (that associated with one-particle transfer and, arguably, the dominant one) to the Li elastic channel. The self-energy function is denoted , while the real and imaginary parts are denoted and respectively, the subindex indicating the incoming proton. These quantities are, in principle, a function of frequency and momentum.

Figure 13: Pairing vibrational spectrum of Be (see also Fig. 21 below) and associated absolute two-nucleon transfer differential cross section calculated as explained in the text.

Figure 14: Absolute differential cross section associated with the reaction at = 7 MeV, calculated making use of : (a) the wavefunction (10) (already shown in Fig. 13) and (b), the RPA wavefunction describing the Be pair addition mode (see Table III).

Figure 15: Absolute differential cross section measured [48] in the reaction at 17 MeV triton bombarding energy (solid dots). The theoretical calculations (continuous solid curve) were obtained making use of the spectroscopic amplitudes associated with the wavefunction in Eqs. (10)-(12), and the optical parameters of refs. [37] and [48] taking into account successive, simultaneous and non-orthogonality processes.

Figure 16: Q-values associated with the , also for the successive transfer (see text).

Figure 17: As for Fig. 16 but for the reaction .

Figure 18: Absolute differential cross section associated with the reaction at two bombarding energies (a) has already been displayed in Fig. 13). Also given are the integrated values of the cross section in the angular range , as well as the different contributions to the total differential cross section.

Figure 19: Absolute differential cross section associated with the reaction at two bombarding energies and integrated values of the cross section in the angular range . Also displayed are the various contributions to the total differential cross section.

Figure 20: Absolute differential cross section associated with the reaction at three bombarding energies. Also displayed are the various contributions to the total differential cross section.
[MeV] [MeV]
0.058 1.049 0.244 0.211
Table 1: RPA wavefunction of the pair removal mode of Li. Single–particle energies were deduced from experimental binding energy differences, while MeV, MeV. The results obtained making use of a single or of two pairing coupling constants to take care of the difference of overlaps between core-core, core-halo and halo-halo single-particle wavefunctions are, in the present case, essentially the same (see App. A, in particular Sect. of this appendix).
(Li(gs) Li (gs)) b)
Theory Experiment
6.1 5.7 0.9
(Li(gs) Li (gs)) b)
14.3 14.7 4.4
Table 2: Integrated two-neutron differential transfer cross sections, in the center of mass angular range 20–154.5 for the reaction LiLi and in the center of mass angular range 10–109 for the reaction LiLi in which the measurements have been made, in comparison with the data ([34, 35]).

[MeV] [MeV] 1.28
0.128 1.076 0.232 0.214 0.272
0.080 0.402 0.727 0.588 0.543
Table 3: RPA wavefunctions of pair removal and addition modes of Be, that is, of the ground state of Be and Be. The single–particle energies were deduced from experimental binding and excitation energies, and making use of the coupling constants MeV and MeV (see App. A, in particular Sect. ).
[MeV] [MeV] [MeV]     X     Y
n 5.0
p 7.6
p 12.2
n 25.1
p 28.4
n 17.5
n 28.4
Table 4: Wavefunction of the lowest vibrational state (phonon) of Be (obtained from a QRPA calculation, making use of a quadrupole separable interaction and a value of the proton pairing gap of MeV while setting ). The calculated energy and the transition strength of the low lying are 2.5 MeV and 49.6 fm respectively. These results are to be compared with the experimental values of 3.3 MeV and 52 fm. The quantities and indicate the energy of the hole and of the particle states respectively for either protons (p) or neutrons (n). denotes the associated two-quasiparticle energies, while and are the QRPA amplitudes of the mode.

Appendix A Pairing and surface vibrations for pedestrians

A pairing vibration is a harmonic mode which changes the number of particles in or , and can be observed, around closed shell nuclei, as strong transitions in two-particle transfer processes. For example around Pb, where the monopole, pair addition mode (a) is the ground state of Pb and the pair removal mode (r) is the ground state of Pb [4]. The two-phonon pairing vibration of Pb (Pb( MeV)) is thus a two-particle two-hole state, product of these two ground states, and thus expected, in the harmonic approximation, at an energy of 4.9 MeV above the Pb ground state (see [1] and refs. therein, see also [18]). Consequently, and within the harmonic approximation, in the reaction PbPb() one excites the pair addition mode, with quite similar ( Q-value, angular distribution, cross section, etc. ) observables to those associated with the PbPb reaction. Of notice that the pair addition mode can be viewed as a correlated two-particle state, linear combination of configurations, denoting valence orbitals (, , , …) lying above the Fermi energy. Now, because of ground state correlations, the (RPA) wavefunction of the pair addition mode contains also components of the type , i.e. corresponding to the correlation of two-hole-states in the occupied valence orbitals lying below the Fermi energy (, , , …). These correlations add coherently to the previous ones in binding the two neutrons (Cooper pair partners) of the pair addition mode to the Pb core. Similar arguments apply to the removal mode, but where the role of holes and particles are exchanged. For an example see Table III, where the two-nucleon transfer spectroscopic amplitudes (see e.g. App. 2, ref. [4]), proportional to the RPA wavefunction amplitudes of Be, are collected. In the case of Be one expects the two-phonon pairing vibrational 2p–2h state at an energy of 4.8 MeV above the ground state 5, an energy which emerges directly from two differences in binding energies, namely MeV (see Fig. 21; see also Fig 13 where the same spectrum is displayed, but subtracting to a linear term in , i.e. essentially eliminating the A-dependent term of Weiszäker mass formula, so as to be able to compare the excitation energies of and with respect to .

In the reaction PbPb one would also excite, for example, the particle-hole quadrupole vibrational state of Pb, that is PbPb, which decays electromagnetically to the ground state of Pb with a B(E2) value corresponding to approximately 5 Weisskopf single-particle units. On one hand this state, which is very interesting by itself (in particular when exchanged between the two outer neutrons in the Pb ground state, it contributes to the pairing correlations of the pair addition mode), has nothing to do with the pairing vibrational spectrum, in the same way in which the state of Be is not related to the pairing spectrum around magic number . On the other hand, the residual interaction correlating the particle and the hole (linear combination of particle-hole excitations of the quadrupole vibration) pulls the hole close to the particle. This is reason why collective surface vibrations display both enhanced electromagnetic transition probabilities, i.e. B(E) values, and enhanced two-nucleon transfer cross section. The main difference between surface ((p-h)-like) vibrational modes and pairing vibrational modes lies, as far as two-nucleon transfer processes are concerned, in the role played by the ground state correlations. While those associated with the pair addition and pair subtraction modes enhance the two-nucleon transfer cross sections, those associated with particle-hole excitations, while increasing the electromagnetic decay probabilities, decrease the two-nucleon transfer cross section (see e.g. [49, 50]). This competition between ((pp)-(hh))- and (ph)- ground state correlations in nuclei, is at the basis of the studies of pairing in nuclei in terms of the competition between deformed and spherical shapes (see e.g. [14, 51, 52, 53] and Fig. 22).

Figure 21: In the harmonic approximation, the two-phonon pairing vibration state of Be is the state BeBe, its energy being given by the sum of the relative binding energies (BE) of Be and Be as indicated (see also Fig. 13).

Figure 22: (a) Zero Point Fluctuations (ZPF) associated with a pair addition mode and a surface ((p-h)-like) vibration. (b) Pauli principle correction of (pp)-like ZPF in presence of (ph)-like ZPF.

Calculation of energy and wavefunctions of pairing vibrations

The two-phonon, of pairing vibrational state, of the closed shell system can be written, in the harmonic approximation as (cf. [1] Ch. 5, see also [4, 18] and refs. therein)

(14)

where

(15)

and

(16)

are the RPA ground state correlated states of the and the system respectively. The pair addition and pair removal creation operator phonons are written as

(17)

where

(18)

creates a pair of particles coupled to angular momentum zero in levels with energy (), while

(19)

creates a pair of holes in the occupied orbitals, i.e. orbitals with energy (). The index (=1, 2, …) labels the lowest, the first excited, etc. states. In what follows we concentrate on the (ground) modes, otherwise explicitly mentioned.

Assuming and display boson commutation relations, one can linearize the pairing Hamiltonian obtaining the dispersion relation for the lowest () modes (see Fig. 23 for the case of the pairing vibrational spectrum around Be)

(20)

where (note that is simplified in what follows into ),

(21a)
with
(21b)

and

(22)

while measure the pair degeneracy of the single particle orbital .

The RPA amplitudes appearing in (17) are defined as

(23a)
and
(23b)

The particle-vibration coupling strengths are determined from the normalization conditions

(24)

and

(25)

respectively.

Figure 23: Dispersion relation and parameters characterizing the pairing vibrational spectrum of Be. The quantity MeV while MeV, with MeV, being the experimental neutron separation energy in Be. Thus is the energy of the first unoccupied state of Be, calculated respect to the Fermi energy MeV (for details see App. B). Similar expressions are valid for the pair removal mode, namely MeV, MeV, where MeV is the energy of the first occupied state respect to the Fermi energy. The inverse strengths intersect the dispersion relation at values of leading to .

.

Extension of the harmonic approximation to the case of more than one pairing coupling constant

The relation between the pairing coupling constant and the matrix elements between pure two-particle configurations coupled to angular momentum , of a force of strength can be written as (see e.g. Eq. (2-24) p.41 ref. [1]),

(26)

where

(27)

being the radial single-particle wavefunction describing initial () and final () configurations. Making use of the approximation (constant value of inside the nucleus of radius ),

(28)

where