Relativistic QRPA calculation of total muon capture rates
Abstract
The relativistic protonneutron quasiparticle random phase approximation (PNRQRPA) is applied in the calculation of total muon capture rates on a large set of nuclei from C to Pu, for which experimental values are available. The microscopic theoretical framework is based on the Relativistic HartreeBogoliubov (RHB) model for the nuclear ground state, and transitions to excited states are calculated using the PNRQRPA. The calculation is fully consistent, i.e., the same interactions are used both in the RHB equations that determine the quasiparticle basis, and in the matrix equations of the PNRQRPA. The calculated capture rates are sensitive to the inmedium quenching of the axialvector coupling constant. By reducing this constant from its freenucleon value by 10% for all multipole transitions, the calculation reproduces the experimental muon capture rates to better than 10% accuracy.
pacs:
21.60.Jz, 23.40.Bw, 24.30.Cz, 25.30.Mrtoday
I Introduction
Semileptonic weak interaction processes in nuclei are very sensitive to detailed properties of nuclear ground states and excitations. In astrophysical applications, in particular, weak interaction rates (decay half lives, neutrinonucleus cross sections, electron capture rates) must be calculated for hundreds of isotopes. Many of those are located far from the valley of stability, and thus not easily accessible in experiments. For a consistent description, reliable predictions and extrapolations of these processes it is, therefore, essential to employ a consistent theoretical framework based on microscopic nuclear structure models.
At present the framework of nuclear energy density functionals (NEDF) provides the most complete description of groundstate properties and collective excitations over the whole nuclide chart. At the level of practical applications the NEDF framework is realized in terms of selfconsistent meanfield (SCMF) models. With a small set of universal parameters adjusted to data, the SCMF approach has achieved a high level of accuracy in the description of structure properties over the whole chart of nuclides, from relatively light systems to superheavy nuclei, and from the valley of stability to the particle driplines BHR.03 (); Vretenar2005a ().
In a series of recent studies we have used a fully consistent microscopic approach based on relativistic energy density functionals to analyze decay halflives of neutronrich nuclei Niksic2005 (); Marketin2006 (), and to model inclusive chargedcurrent neutrinonucleus reactions Paar2008 (). In this framework nuclear ground states are described using the relativistic HartreeBogoliubov (RHB) model Vretenar2005a (), and transitions to excited nuclear states are calculated in the relativistic quasiparticle randomphase approximation (RQRPA) Paar2003 (); Paar2004 (). There are important advantages in using functionals with manifest covariance, the most obvious being the natural inclusion of the nucleon spin degree of freedom. The resulting nuclear spinorbit potential has the correct empirical strength and isospin dependence. This is, of course, especially important in the description of excitations in the spinisospin channel, e.g. semileptonic weak interaction processes. In addition, by employing a single universal effective interaction in modeling both groundstate properties and multipole excitations in various mass regions of the chart of nuclides, the calculation of weakinteraction rates is essentially parameter free, and can be extended to regions of nuclei far from stability, including those on the process path.
To successfully extend a particular microscopic approach to regions of unknown nuclei far from stability, it is necessary to perform extensive tests and compare results with available data. Reliable prediction of weak interaction rates, in particular, require a fully consistent description of the structure of ground states and multipole excitations. For instance, calculated decay halflives are very sensitive to lowenergy GamowTeller transitions, but can only test excitations of lowest multipoles. Higher multipoles are excited in neutrinonucleus reactions in the lowenergy range below 100 MeV, and these reactions could play an important role in many astrophysical processes, including stellar nucleosynthesis. There are, however, only few data on neutrinonucleus reactions, and these are limited to relatively light nuclei. Much more data are available for total muon capture rates. Muon capture on stable nuclei has been studied in details since many years, both experimentally and theoretically Primakoff1959 (); Mukhopadhyay1977 (); Measday2001 (); Walecka75 (). In this process the momentum transfer is of the order of the muon mass and, therefore, the calculation of total muon capture rates presents an excellent test of models that are also used in studies of lowenergy neutrinonucleus reactions.
In this work we test the fully consistent RHB plus protonneutron RQRPA model in the calculation of total muon capture rates on a large set of nuclei from C to Pu, for which experimental values are available Suzuki1987 (). Previous calculation of muon capture rates on selected nuclei using the RPA approach include the consistent HartreeFock (HF) RPA model Auerbach1984 (); Auerbach1997 (), in which the HF mean field and the particlehole interaction result from the same Skyrme effective force, and a series of studies Kolbe1994 (); Kolbe2000 (); Zinner2003 (); Zinner2006 () in which both the continuum and standard RPA were used, and the effect of quenching of axialvector coupling was analyzed. The present analysis parallels the recent study by Zinner, Langanke and Vogel Zinner2006 (), where the nonrelativistic RPA was used to systematically calculate muon capture rates for nuclei with . There are, however, significant differences between the two approaches. The model employed in Refs. Kolbe1994 (); Kolbe2000 (); Zinner2003 (); Zinner2006 () uses a phenomenological WoodsSaxon potential to generate the basis of singlenucleon states. The strength of the potential is adjusted to experimental proton and neutron separation energies in individual nuclei. In a second step the RPA with a phenomenological LandauMigdal residual interaction is used to calculate nuclear excitations. The present approach, as already emphasized above, is fully consistent: both the basis of singlenucleon states and multipole excitations of nuclei are calculated from the same energy density functional or nuclear effective interaction. Results will be compared with data and discussed in relation to those reported in Ref. Zinner2006 (). In particular, we will consider the important issue of quenching of the axialvector strength.
Ii Theoretical framework
The capture of a negative muon from the atomic orbit on a nucleus
(1) 
presents a simple semileptonic reaction that proceeds via the charged current of the weak interaction. Detailed expressions for the reaction rates and the transition matrix elements can be found in Refs. O'Connell1972 (); Walecka75 (); Walecka2004 (). The capture rate reads
(2) 
where denotes the quantization volume and is the muon neutrino energy. The Hamiltonian of the weak interaction is expressed in the standard currentcurrent form, i.e. in terms of the nucleon and lepton currents
(3) 
and the transition matrix elements read
(4) 
is the muon wave function, the fourmomentum transfer is , and the multipole expansion of the leptonic matrix element determines the operator structure for the nuclear transition matrix elements O'Connell1972 (); Walecka75 (); Walecka2004 (). The expression for the muon capture rate is given by
(5) 
where is the weak coupling constant, the phasespace factor accounts for the nuclear recoil, and is the mass of the target nucleus. The nuclear transition matrix elements between the initial state and final state , correspond to the charge , longitudinal , transverse electric , and transverse magnetic multipole operators:

the Coulomb operator
(6) 
the longitudinal operator
(7) 
the transverse electric operator
(8) 
and the transverse magnetic operator
(9)
where all the form factors are functions of , and . These multipole operators contain seven basic operators expressed in terms of spherical Bessel functions, spherical harmonics, and vector spherical harmonics O'Connell1972 (). By assuming conserved vector current (CVC), the standard set of form factors reads Kuramoto1990 ():
(10) 
(11) 
(12) 
(13) 
The muon capture rates are evaluated using Eq. (5), with the transition matrix elements between the initial and final states determined in a fully microscopic theoretical framework based on the Relativistic HartreeBogoliubov (RHB) model for the nuclear ground state, and excited states are calculated using the relativistic quasiparticle random phase approximation (RQRPA). The RQRPA has been formulated in the canonical singlenucleon basis of the relativistic HartreeBogoliubov (RHB) model in Ref. Paar2003 (), and extended to the description of chargeexchange excitations (protonneutron RQRPA) in Ref. Paar2004 (). In addition to configurations built from twoquasiparticle states of positive energy, the relativistic QRPA configuration space must also include pairconfigurations formed from the fully or partially occupied states of positive energy and empty negativeenergy states from the Dirac sea. The RHB+RQRPA model is fully consistent: in the particlehole () channel effective Lagrangians with densitydependent mesonnucleon couplings are employed, and pairing () correlations are described by the pairing part of the finite range Gogny interaction. Both in the and channels, the same interactions are used in the RHB equations that determine the canonical quasiparticle basis, and in the matrix equations of the RQRPA. In this work we use one of the most accurate mesonexchange densitydependent relativistic meanfield effective interactions – DDME2 Lalazissis2005 () in the channel, and the finite range Gogny interaction D1S Berger1991 () in the channel.
The spinisospindependent interaction terms are generated by the  and meson exchange. Although the direct onepion contribution to the nuclear ground state vanishes at the meanfield level because of parity conservation, the pion must be included in the calculation of spinisospin excitations. The particlehole residual interaction of the PNRQRPA is derived from the Lagrangian density:
(14) 
where vectors in isospin space are denoted by arrows. For the densitydependent coupling strength of the meson to the nucleon we choose the value that is used in the DDME2 effective interaction Lalazissis2005 (), and the standard value for the pseudovector pionnucleon coupling is , and MeV. The derivative type of the pionnucleon coupling necessitates the inclusion of the zerorange LandauMigdal term, which accounts for the contact part of the nucleonnucleon interaction
(15) 
with the parameter adjusted in such a way that the PNRQRPA reproduces experimental values of GamowTeller resonance (GTR) excitation energies Paar2004 (). The precise value depends on the choice of the nuclear symmetry energy at saturation, and for the DDME2 effective interaction =0.52 has been adjusted to the position of the GTR in Pb. This value is kept constant for all nuclides calculated in this work.
In the evaluation of muon capture rates (Eq. (5)), for each transition operator the matrix elements between the ground state of the eveneven target nucleus and the final state are expressed in terms of singleparticle matrix elements between quasiparticle canonical states, the corresponding occupation probabilities and RQRPA amplitudes:
(16) 
Transitions between the ground state of a spherical eveneven target nucleus and excited states in the corresponding oddodd nucleus are considered. The total muon capture rate is calculated from the expression:
(17)  
with the neutrino energy determined by the energy conservation relation
(18) 
where is the binding energy of the muonic atom.
For each nucleus the muon wave function and binding energy are calculated as solutions of the Dirac equation with the Coulomb potential determined by the selfconsistent groundstate charge density. However, while the RHB singlenucleon equations are solved by expanding nucleon spinors and meson fields in terms of eigenfunctions of a spherically symmetric harmonic oscillator potential, the same method could not be used for the muon wave functions. The reason, of course, is that the muon wave functions extend far beyond the surface of the nucleus and, even using a large number of oscillator shells, solutions expressed in terms of harmonic oscillator basis functions do not converge. The Dirac equation for the muon is therefore solved in coordinate space using the method of finite elements with Bspline shape functions McNeil1989 (); deBoor (). As an illustration, in Fig. 1 we plot the square of the muon wave functions for O, Ca, Sn and Pb. The solutions that correspond to selfconsistent groundstate charge densities are compared with eigenfunctions of the Coulomb potential for the corresponding pointcharge . For light nuclei the radial dependence of the muon wave function is not very different from that of the pointcharge Coulomb potential. With the increase of Z the muon is pulled into the nuclear Coulomb potential, and thus the magnitude of the density inside the nucleus is reduced with respect to the pointcharge value. To test our calculation of muon orbitals in the nuclear Coulomb potential, in Tables 1 and 2 the muon transition energies in Sn isotopes and in Pb, respectively, are compared with available data Piller1990 (); Bergem1988 (). The calculated transition energies are in good agreement with experimental values.
The effect of the finite distribution of groundstate charge densities on the calculated muon capture rates is illustrated in Fig. 2. For a large set of nuclei from C to Pu, we plot the ratio between calculated and experimental muon capture rates. This ratio is for all nuclei when the muon wave functions are determined by selfconsistent groundstate charge densities, whereas for pointcharge Coulomb potentials one notes a distinct increase with , and for the heaviest systems.
Iii Results for muon capture rates
The muon capture rates shown in Fig. 2 are calculated with the standard set of free nucleon weak form factors Eqs. (10) – (13) Kuramoto1990 (), i.e. the calculation does not include any inmedium quenching of the corresponding strength functions. Even with muon wave functions determined selfconsistently by finitecharge densities, the resulting capture rates are larger than the corresponding experimental values by a factor . This is in contrast to the results of Ref. Zinner2006 (), where the experimental values have been reproduced to better than 15% accuracy, using the freenucleon weak form factors and residual interactions with a mild dependency. In fact, it was shown that the calculated rates for the same residual interactions would be significantly below the data if the inmedium quenching of the axialvector coupling constant is employed to other than the true GamowTeller (GT) amplitudes. Consequently, the calculations reported in Ref. Zinner2006 () were performed with quenching only the GT part of the transition strength by a common factor . It was concluded, however, that there is actually no need to apply any quenching to operators that contribute to the muon capture process, especially those involving singlenucleon transitions between major oscillator shells.
As already emphasized in the Introduction, although both calculations are based on the RPA framework, there are important differences between the model of Ref. Zinner2006 (), and the RHB+RQRPA approach employed in the present study. The main difference is probably the fact that the present calculation is fully consistent: for all nuclei both the basis of singlenucleon states and the multipole response are calculated using the same effective interaction, whereas in Ref. Zinner2006 () the phenomenological WoodsSaxon potential was adjusted to individual nuclei and the strength of the residual LandauMigdal force had a mild dependence.
In Fig. 3 we compare the ratios of the theoretical and experimental total muon capture rates for two sets of weak form factors. First, the rates calculated with the free nucleon weak form factors Eqs. (10) – (13) Kuramoto1990 () (circles), and already shown in Fig. 2. The lower rates, denoted by diamonds, are calculated by applying the same quenching to all axial operators, i.e. is reduced by 10% in all multipole channels. In the latter case the level of agreement is very good, with the mean deviation between theoretical and experimental values of only 6%. The factor 0.9 with which the freenucleon is multiplied is chosen in such a way to minimize the deviation from experimental values for spherical, closedshell mediumheavy and heavy nuclei. On the average the results are slightly better than those obtained in Ref. Zinner2006 () (cf. Fig. 2 of Zinner2006 ()). Note, however, that in the calculation of Zinner, Langanke and Vogel Zinner2006 () only the true GamowTeller transition strength was quenched, rather than the total strength in the channel. In the present study considerably better results are obtained when the quenched value of the axialvector coupling constants is used for all multipole operators. The reason to consider quenching the strength in all multipole channels, rather than just for the GT is, of course, that the axial form factor appears in all four operators Eqs. (6) – (9) that induce transitions between the initial and final states, irrespective of their multipolarity. Even more importantly, only a relatively small contribution to the total capture rates actually comes from the GT channel . This is illustrated in Fig. 4, where we display the relative contributions of different multipole transitions to the RHB plus RQRPA muon capture rates in O, Ca, Sn and Pb. For the two lighter nuclei the dominant multipole transitions are and (spindipole). For the two heavier nuclei there are also significant contributions of the and , especially for Pb and for other heavy nuclei. Note that in heavy nuclei the multipole represents transitions, rather than the GamowTeller transitions.
Returning to Fig. 3, we notice that with a 10% quenching of the freenucleon axialvector coupling constant , for mediumheavy and heavy nuclei the calculated capture rates are still slightly larger than the corresponding experimental values, with the ratio typically around , whereas for several lighter nuclei considered here this ratio is actually less than 1 (cf. also Table 3). Overall the best results, with , are obtained near closed shells. The characteristic arches between closed shells can probably be attributed to deformation effects, not taken into account in our RHB+RQRPA model. In addition to the DDME2 interaction, we have also carried out a full calculation of total capture rates from C to Pu, using the density and momentumdependent relativistic effective interaction D3C*. In the study of decay halflives of Ref. Marketin2006 (), this interaction was constructed with the aim to enhance the effective (Landau) nucleon mass, and thus improve the RQRPA description of decay rates. When D3C* is used to calculate muon capture rates, some improvement is obtained only locally, for certain regions of , whereas in other regions ( and ) the results are not as good as those obtained with DDME2. The overall quality of the agreement between theoretical and experimental capture rates is slightly better with DDME2.
The calculated total muon capture rates for natural elements and individual isotopes are also collected in Table 3, and compared with available data Suzuki1987 (). In particular, the calculation nicely reproduces the empirical isotopic dependence of the capture rates Primakoff1959 (), i.e. for a given proton number the rates decrease with increasing neutron number, because of the gradual blocking of available neutron levels. The isotopic trend is also illustrated in Fig. 5, where we plot the experimental and theoretical total muon capture rates on Ca, Cr and Ni nuclei. The latter correspond to the quenching for all multipole operators.
In conclusion, we have tested the RHB plus protonneutron RQRPA model in the calculation of total muon capture rates on a large set of nuclei from C to Pu. The calculation is fully consistent, the same universal effective interactions are used both in the RHB equations that determine the quasiparticle basis, and in the matrix equations of the RQRPA. The calculated capture rates are sensitive to the inmedium quenching of the axialvector coupling constant. By reducing this constant from its freenucleon value to the effective value for all multipole transitions, i.e. with a quenching of approximately 10%, the experimental muon capture rates are reproduced with an accuracy better than . This result can be compared to recent RPAbased calculations Kolbe2000 (); Zinner2003 (); Zinner2006 (), that reproduce the experimental values to better than 15%, using phenomenological potentials adjusted to individual nuclei and dependent residual interactions, but without applying any quenching to the operators responsible for the capture process. The test has demonstrated that the RHB plus QRPA model provides a consistent and accurate description of semileptonic weak interaction processes at finite momentum transfer in mediumheavy and heavy nuclei over a large Zrange. The fully consistent microscopic approach, based on modern relativistic nuclear energy density functionals, can be extended to other types of weak interaction processes (electron capture, neutrinonucleus chargeexchange and neutralcurrent reactions), and to regions of shortlived nuclei far from stability.
ACKNOWLEDGMENTS
This work was supported by MZOS  project 11910051010 and Unity through Knowledge Fund (UKF Grant No. 17/08).
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exp.  calc.  exp.  calc.  

Sn  3432  3439  3478  3485 
Sn  3426  3432  3471  3478 
Sn  3420  3427  3465  3472 
Sn  3421  3466  
Sn  3408  3415  3454  3460 
Sn  3409  3454  
Sn  3400  3404  3445  3450 
Pb  exp.  calc. 

5963  5956  
5778  5773  
2642  2633  
2501  2493  
2458  2450 
Nucleus  Exp.  Calc.  Nucleus  Exp.  Calc.  Nucleus  Exp.  Calc.  Nucleus  Exp.  Calc. 

C  0.039  0.032  Se  6.644  Sn  9.645  Dy  13.540  
O  0.103  0.065  Se  5.796  Sn  8.837  Dy  12.29  14.194  
O  0.088  0.057  Se  4.935  Sn  10.44  10.923  Er  16.129  
Ne  0.204  0.237  Se  5.681  5.950  Te  10.652  Er  14.949  
Mg  0.484  0.506  Sr  8.885  Te  9.830  Er  13.912  
Si  0.871  0.789  Sr  7.393  Te  9.068  Er  13.04  15.270  
S  1.352  1.485  Sr  7.020  7.553  Te  9.270  9.706  Hf  16.434  
Ar  1.355  1.368  Zr  9.874  Xe  9.4  10.631  Hf  15.276  
Ca  2.557  2.340  Zr  9.694  Xe  8.6  8.625  Hf  13.03  15.783  
Ca  1.793  1.851  Zr  8.792  Ba  11.461  W  17.259  
Ca  1.214  1.163  Zr  8.660  9.619  Ba  10.127  W  15.938  
Ti  2.590  2.544  Mo  12.374  Ba  9.940  10.259  W  14.807  
Cr  3.825  4.001  Mo  12.001  Ce  11.888  W  12.36  15.971  
Cr  3.452  3.419  Mo  10.933  Ce  12.142  Hg  17.369  
Cr  3.057  3.065  Mo  9.804  Ce  11.60  11.917  Hg  16.227  
Cr  3.472  3.483  Mo  9.614  10.995  Nd  14.043  Hg  15.205  
Fe  4.411  4.723  Pd  13.182  Nd  14.288  Hg  13.993  
Ni  6.110  6.556  Pd  11.912  Nd  12.981  Hg  12.74  15.733  
Ni  5.560  5.610  Pd  10.795  Nd  12.50  13.861  Pb  15.717  
Ni  4.720  4.701  Pd  9.821  Sm  15.425  Pb  13.718  
Ni  5.932  6.234  Pd  10.00  11.391  Sm  14.132  Pb  13.45  14.348  
Zn  6.862  Cd  12.960  Sm  13.451  Th  12.56  13.092  
Zn  5.809  Cd  11.800  Sm  12.563  U  13.79  14.231  
Zn  4.935  Cd  10.746  Sm  12.22  13.554  U  13.09  13.490  
Zn  5.834  6.174  Cd  9.829  Gd  14.785  U  12.57  12.872  
Ge  6.923  Cd  10.61  11.381  Gd  13.573  Pu  12.90  13.554  
Ge  5.970  Sn  12.395  Gd  12.460  Pu  12.40  12.887  
Ge  5.519  Sn  11.369  Gd  11.82  13.580  
Ge  5.569  6.011  Sn  10.486  Dy  14.917 
From Ref. Fynbo2003 ().
From Ref. Mamedov2000.
From Ref. Haenscheid1990 ().
From Ref. David1988 ().