Nuclear Field Theory predictions for Li and Be: shedding light on the origin of pairing in nuclei
(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.
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
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.
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 region
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),
The excitation energy of a state of the vibrational band (again in the harmonic approximation), can be written as
The absolute value of the different transitions between these states can be expressed in terms of the basic cross–sections
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).
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 ()),
with and , and
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 ()):
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.
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.
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.
|(Li(gs) Li (gs)) b)|
|(Li(gs) Li (gs)) b)|
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 . 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  and refs. therein, see also ). 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
(, , , …) 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. ), 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
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).
Calculation of energy and wavefunctions of pairing vibrations
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
creates a pair of particles coupled to angular momentum zero in levels with energy (), while
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)
where (note that is simplified in what follows into ),
while measure the pair degeneracy of the single particle orbital .
The RPA amplitudes appearing in (17) are defined as
The particle-vibration coupling strengths are determined from the normalization conditions
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. ),
being the radial single-particle wavefunction describing initial () and final () configurations. Making use of the approximation (constant value of inside the nucleus of radius ),