GAMOW-TELLER STRENGTH DISTRIBUTIONS for -Decaying Nuclei Within Continuum-Qrpa
A version of the pn-continuum-QRPA is outlined and applied to describe the Gamow-Teller strength distributions for -decaying open-shell nuclei. The calculation results obtained for the pairs of nuclei Cd-Sn and Te-Xe are compared with available experimental data.
PACS numbers: 25.40.Kv, 23.40.-s, 24.30.Gd, 27.50.+e
Description of weak interaction in nuclei is often a challenge for models of nuclear structure. Numerous calculations of the nuclear -decay amplitudes well illustrate this statement (see, e.g. Refs.  and references therein). Uncertainties in theoretical calculations of the Gamow-Teller (GT) -decay amplitude have stimulated experimental studies of the GT-strengths of the states virtually excited in the decay process (see, e.g. Refs. [2, 3]).
As a double charge-exchange process, -decay is enhanced by nucleon pairing which is due to the singlet part of the particle-particle (p-p) interaction. The discrete quasiboson version of the quasiparticle RPA (pn-dQRPA) which accounts for the nucleon pairing is usually applied to calculate the -decay amplitudes in open-shell nuclei . In spite of differences in model parameterizations of the nuclear mean field and residual interaction in the particle-hole (p-h) and p-p channels, all pn-dQRPA calculations reveal marked sensitivity of the amplitude to the ratio of the triplet to singlet strength of the p-p interaction. Physical reasons for such a general feature of all calculations were analyzed in Ref. , where they were attributed to violation of the spin-isospin SU(4) symmetry in nuclei. An identity transformation of the amplitude into sum of two terms was used in Ref. . One term, which is due to the p-p interaction only, depends linearly on and vanishes at when the SU(4)-symmetry is restored in the p-p sector of a model Hamiltonian. The second term is a smoother function of at , but exhibits a quadratic dependence on the strength of the mean-field spin-orbit term, which is the main source of violation of the spin-isospin SU(4)-symmetry in nuclei.
Understanding of general properties of the amplitude helps to improve reliability of evaluation of -decay amplitudes. For a quantitative analysis, we use here an isospin-selfconsistent pn-continuum-QRPA (pn-cQRPA) approach of Ref. , where this approach was applied to evaluate the GT strength distributions in single-open-shell nuclei. In the reference the full basis of the single-particle (s-p) states was used in the p-h channel along with the Landau-Migdal forces, while the nucleon pairing was described within the simplest version of the BCS-model based on discrete basis of s-p states. A rather old version of the phenomenological isoscalar nuclear mean field (including the spin-orbit term) was used in Refs. [5, 4], as well.
The first application of the pn-cQRPA approach of Ref.  to description of the -decay observables in several nuclei has been given recently in Ref. . Realistic (zero-range) forces have been used in the p-p channel to describe the nucleon pairing within the BCS model realized on a rather large discrete+quasidiscrete s-p basis.
The pn-cQRPA approach of Refs. [5, 7] is further extended here by using a modern version of the phenomenological isoscalar mean field (including the spin-orbit term) deduced in Ref.  from the isospin-selfconsistent analysis of experimental single-quasiparticle spectra in double-closed-shell nuclei. In the present contribution we give a brief overview of the approach and its applications to description of different GT strength functions for the pairs of nuclei Cd-Sn and Te-Xe.
2 Coordinate representation of the pn-cQRPA equations and GT strength functions
In formulation of a version of the pn-cQRPA we follow Ref. , where the pn-dQRPA equations originally written in terms of the forward and backward amplitudes are transformed in equivalent equations for the 4-component radial transition density . The latter is defined in terms of the , amplitudes by Eqs. (39), (40) of Ref.  and related to the GT excitations having the wave functions and energies . The expression for the transition density related to GT excitations follows from that for by the substitution . Hereafter, we often use the notations of Ref.  (except for the energies) and refer to some equations from this reference. Limiting ourselves in this contribution to description of the GT transitions only, we omit the quantum numbers indices , for spin-monopole excitations. The spin-angular variables in all expressions are separated out as well.
The pn-dQRPA solutions are related to the excitation energies measured from the ground-state energy of the corresponding daughter nuclei as:
Here, is the chemical potential for the proton (neutron) subsystem found from the known BCS equations, are the total binding-energy differences, are the excitation energies measured from ( is the nucleon mass). The energies are usually described by a model Hamiltonian.
The system of equations for (Eq.(41) of Ref. ) contains the explicit expression for the matrix of the free two-quasiparticle propagator (Eq.(43) of Ref. ; ). These propagators are the main quantities in description of charge-exchange excitations within the pn-QRPA. In particular, in terms of one can formulate a Bethe-Salpeter-type equation for the effective propagator . The spectral expansion of in terms of allows one to express the pn-QRPA strength functions in terms of the effective propagator, or, equivalently, in terms of the 4-component effective fields . Some relevant formulas are shown below.
The GT strength functions, corresponding to the external fields (probing operators) , , are defined as follows:
with GT strengths . The strength function can be expressed in terms of the corresponding effective field , which is different from the external one due to the residual interaction :
The residual interaction here is supposed to be of zero-range type with intensities : , . For the p-h and p-p channels the corresponding strengths are: and , respectively. Dimensionless values , , () are the well-known Landau-Migdal p-h strength parameters. The same parameterization we use for the p-p interaction strengths: , . For calculation of one can use Eqs. (3), (4) with substitution . An alternative way is based on the symmetry properties of : . As a result, we get the expression for in terms of and . This expression is obtained from Eqs. (3), (4) with the substitution .
The nuclear GT amplitude for -decay into the ground state of the product nucleus is given by the expression:
where . To calculate within the pn-QRPA, the vacua and should be identified. As a result of such identification, one has , in accordance with Eq. (1).
The amplitude (5) can be expressed in terms of a “non-diagonal” GT strength function :
where is defined as follows:
The corresponding pn-QRPA expression for is:
An alternative expression for is obtained in terms of the “non-diagonal” static polarizibility :
Decomposition of the amplitude (5) into two terms 
where is the energy of GT giant resonance (GTR), allows us to clarify the sensitive dependence of -decay amplitude as a function of (for details, see Ref. ). The “non-diagonal” energy-weighted sum rule is straightforwardly expressed in terms of the strength function of Eq. (6):
again supposing the QRPA vacuum is identified with that of .
3 Calculation of strength function within the pn-cQRPA
Starting from the coordinate representation of the pn-dQRPA equations outlined above, we are able to take exactly into account the s-p continuum in the p-h channel and, therefore, to formulate a version of the pn-cQRPA. The pairing problem is solved on a rather large basis of bound+quasibound proton and neutron s-p states within the present version of the model. To take the s-p continuum into account, the following transformations of the expression for  are done: (i) the Bogolyubov coefficients , and the quasi-particle energies are approximated by their non-pairing values , , and for those s-p states (), which lie far above the chemical potential (i.e. ), (ii) the Green function of the s-p radial Schrödinger equation , which is calculated via the regular and irregular solutions of this equation, is used to perform explicitly the sum over the s-p states in the continuum. As a result, the properly transformed free two-quasiparticle propagator is obtained, upon which a corresponding version of the pn-cQRPA is based.
The solution of the pairing problem is simplified by using the “diagonal” approximation for the p-p interaction for the neutral channel. In this approximation the nucleon-pair operators are assumed to be formed only from the pair of nucleons occupying the same s-p level . The nucleon pairing is described with the use of the Bogolybov transformation with the gap parameter dependent on . The same number of bound+quasibound states forming the basis of the BCS problem is used for both the neutron and proton subsystems. These numbers are shown in Table 1 for nuclei in question. In evaluation of total binding energies within the model (that is necessary to evaluate the pairing energies ) the blocking effect for odd nuclei is taken into account. In description of the nucleon pairing, different values of the p-p interaction strength parameters and for the neutron and proton subsystems are used. These values are found from comparison of the calculated and experimental pairing energies for nuclei under consideration (Table 1).
|Pair of nuclei||, MeV||, MeVfm||, fm|
The mean field consists of the phenomenological isoscalar part (including the spin-orbit term) along with the isovector and Coulomb part (Eq. (1) of Ref. ). The parameterization of the Woods-Saxon-type isoscalar part contains two strength (, ) and two geometrical (, ) parameters . The mean field isovector part (the symmetry potential) is calculated in an isospin-selfconsistent way (Eqs. (7), (35) of Ref. ) via the neutron-excess density and Landau-Migdal strength parameter . The mean Coulomb field is also calculated selfconsistently via the proton density. All densities are calculated with taking into account the nucleon pairing. Five above-listed model parameters found in Ref.  for a number of double-closed-shell nuclei are properly interpolated for nuclei under consideration (see Table 1; fm is taken for all nuclei).
The values of the Landau-Migdal strength listed in Table 1 are obtained by fitting the experimental GTR energy in calculations of the GT strength function. The p-p interaction strength (or its relative value ) is considered as a free parameter. It can be adjusted to reproduce the experimental value (the corresponding values are listed in the last column of Table 1).
Considering the pair Cd-Sn, the GT strength distribution calculated within the pn-dQRPA for the transition SnSb is shown in Fig. 1a (a small imaginary part is added to the s-p potential). To compare the calculation results with the Sn(He,t) experimental data of Ref. , five centroids of the energy, , and their strength relative to the one of the GTR are evaluated (Fig. 1b). The value allows to reproduce the experimental GTR energy in the calculation. The GT strength distribution is almost insensitive to the value ( is taken in the calculation). The GT strength distribution for the transition SnIn is found more sensitive to . Only one state with corresponding to the transition into the In ground state, is found in the calculation within the interval . This weak transition is allowed due to the neutron pairing in Sn. In the Sn(d,He) experiments four states in In were found within the interval with total strength . Population of the states in In has also been studied in the Cd(p,n)-reaction . The result for excitation of the In ground state is only available now. Within the interval the calculated GT strength distribution in In exhibits one state, corresponding to the back-spin-flip transition into the In ground state with the value 1.05 . (for Sn-Sb this transition is Pauli blocked). The -decay amplitude for the decay CdSn can barely be evaluated within the pn-QRPA because the proton shell is closed in Sn.
Coming to the pair Te-Xe, the value is found in the calculation by fitting the experimental GTR energy in I . Then the amplitude (6) (or (9)) and its decomposition (10) are calculated, as a function of (Fig. 2). The corresponding experimental value MeV  can be reproduced in the calculation at . The -decay strength function is calculated for this value of (Fig. 3a) along with the corresponding running sum (Fig. 3b). Figs. 2 and 3 illustrate how the value for the decay TeXe is formed. In particular, as one sees in Fig.3, the experimental studies of are not always sufficient for understanding partial contributions to . The reason is that the intermediate states having a relatively large excitation energy and very small value (like the GTR) can nonetheless play essential role in formation of the -decay amplitude.
In conclusion, an isospin-selfconsistent version of the pn-cQRPA has been outlined and some its applications to description of charge-exchange excitations in open-shell spherical nuclei are presented. Although only general features of the low-energy strength distributions can be described within the approach, it seems applicable to analysis of -decay observables.
This work is supported in part by RFBR (grant 06-02-016883-a) (S.I. and M.U.), NWO (M.U.), DFG (grant FA67/28-2) and EU ILIAS project (contract RII3-CT-2004-506222).
-  A. Faessler and F. Šimkovic, J. Phys. G 24, 2139 (1998); J. Suhonen and O. Civitarese, Phys. Rept. 300, 123 (1998); S.R. Elliott and P. Vogel, Annu. Rev. Nucl. Part. Sci. 52, 115 (2002); S. R. Elliott and J. Engel, J. Phys. G 30, R183 (2004); F. Avignone, S. R. Elliott, J. Engel, arXiv:0708.1033 [nucl-ex].
-  M. Sasano et al., Nucl. Phys. A 788, 76c (2007).
-  S. Rakers et al., Phys. Rev. C 71, 054313 (2005).
-  V.A. Rodin, M.H. Urin, and A. Faessler, Nucl Phys. A 747, 295 (2005).
-  V.A. Rodin and M.H. Urin, Phys. At. Nuclei 66, 2128 (2003), nucl-th/0201065.
-  S.Yu. Igashov, M.H. Urin, Bull. Rus. Acad. Sci. Phys. 70, 212 (2006)
-  V.A. Rodin and A. Faessler, arXiv:0707.2742 [nucl-th].
-  K. Pham et al., Phys. Rev. C 51, 526 (1995).
-  R. Madey et al., Phys. Rev. C 40, 540 (1989).
-  A.S. Barabash, nucl-ex/0602009.