Detailed description of exclusive muon capture rates using realistic two-body forces
DDetailed description of exclusive muon capture rates using realistic two-body forces
P.G. Giannaka ∗ and T.S Kosmas † Division of Theoretical Physics, University of Ioannina, GR-45110 Ioannina, Greece (Dated: September 10, 2018)Starting from state-by-state calculations of exclusive rates of the ordinary muon capture (OMC),we evaluated total µ − -capture rates for a set of light- and medium-weight nuclear isotopes. Weemployed a version of the proton-neutron quasi-particle random phase approximation (pn-QRPA,for short) which uses as realistic nuclear forces the Bonn C-D one boson exchange potential. Specialattention was paid on the percentage contribution to the total µ − -capture rate of specific low-spinmultipolarities resulting by summing over the corresponding multipole transitions. The nuclearmethod used offers the possibility of estimating seperately the individual contributions to the totaland partial rates of the polar-vector and axial-vector components of the weak interaction Hamilto-nian for each accessible final state of the daughter nucleus. One of our main goals is to provide areliable description of the charge changing transitions matrix elements entering the description ofother similar semileptonic nuclear processes like the charged-current neutrino-nucleus reactions, theelectron capture on nuclei, the single β ± -decay mode, etc., which play important role in currentlyinteresting laboratory and astrophysical applications like the neutrino-detection through lepton-nucleus interaction probes, and neutrino-nucleosynthesis. Such results can be also be useful invarious ongoing muon-capture experiments at PSI, Fermilab, JPARC and RCNP. PACS numbers: 23.40.-S, 23.20.-Js, 25.30.Mr, 23.40.-Hc, 24.10.-iKeywords: semi-leptonic charged-curent reactions, ordinary muon capture, nuclear matrix elements, Quasi-Particle Random Phase Approximation, neutrino nucleosynthesis
I. INTRODUCTION
In recent years, various sensitive experiments take ad-vantage of the powerful muon beams produced in wellknown muon factories (PSI, Fermilab, JPARC, RCNP inOsaka and others) for standard and non-standard muonphysics probes [1, 2]. Among the standard model probesthose involving muon capture on nuclei specifically thoseemitting X-rays and/or several particles (p, n, α , etc.)after µ − -capture, which are important to understand therates and spectra of these particles, are investigated [1].For example, at PSI researchers are interested in exper-iments based on the emission of charged particles frommuonic atoms of Al, Si and Ti or neutron emission follow-ing muon capture from Fe, Ca ,Si and Al [1]. Also veryrecently, in the highly intense muon facilities MuSIC atRCNP, Osaka, Japan, nuclear muon capture reactions(on Mo, Pb, etc.) are planned in order to study, nuclearweak responses (for neutrino reactions, etc.) [2]. For ex-periments like the above, it is important, before going tothe rates of the emitted particles, to know the first stagemuon capture process.As it is well known, when negative muons, µ − , pro- ∗ Electronic address: [email protected] † Electronic address: [email protected] duced in a meson factory, slow down in matter, it is pos-sible for them to be captured in atomic orbits. After-wards, fast electromagnetic cascades bring these muonsdown to the innermost (1s or 2p) quantum orbits (in thisway muonic atoms are produced) [3–7]. A bound muonin the muonic atom may disappear either by decay knownas muon decay in orbit or by capture by the nucleus themain channel of which is the ordinary muon capture rep-resented by the reaction [8–10] µ − b + ( A, Z ) → ( A, Z − ∗ + ν µ , (1)where ( A, Z ) denotes the initial atomic nucleus with massnumber A and proton number Z while ( A, Z − ∗ standsfor an excited state of the daughter nuclear isotope.The reaction (1) is a well known example of symbio-sis of atomic, nuclear and particle physics. In this work,however, we will concentrate on its nuclear physics as-pects. As muon-capture in nuclei presents many advan-tages for the study of both nuclear structure and the fun-damental electro-weak interactions [11–14], process (1)has been the subject of extensive experimental and the-oretical investigations started early on 50’s using closureapproximation or sum-over partial rates to find the to-tal µ − -capture rate (the measured quantity). [3–6, 16–22]. In the plethora of the relevant papers, the mostimportant motivation was rested on the hope to explainhow nucleons (hadronic current) inside the nucleus cou-ple weakly to the lepton field (leptonic current). The nu- a r X i v : . [ nu c l - t h ] J un clear physics aspects of process (1), however, still possesssome yet unresolved fundamental problems, e.g. thoserelated to the nucleon-nucleon and lepton-nucleus inter-actions, the question whether the individual propertiesof the nucleons change when they are packed togetherin the nucleus or remain essentially unaffected like thecoupling of the nucleon to the leptonic field, etc.The interest of studying µ − -capture has recently beenrevived [23–25] due to its prominent role in testing thenuclear models employed in several physical applicationsin neutrino physics and astrophysics [26–28]. Specifically, µ − -capture is a very useful test for various nuclear meth-ods used to describe semi-leptonic weak charged-currentreactions [8, 16] as the electron capture in stars (criti-cal in the collapse of supernovae) [26, 28, 29], the neu-trino nucleus scattering (important in the detection of as-trophysical neutrinos) [27, 28], and other reactions [12].This is due to the fact that the muon-capture involvesa large momentum transfer and, hence, it can providevaluable information about effects which are not foundin processes like the beta-decay modes (on medium mo-mentum transfer processes however, useful informationcan be also obtained from low-spin forbidden transitionsof beta-decays and charge exchange reactions) [30]. Fur-thermore, there is an intimate relation between the inclu-sive muon capture rate and the cross section for the an-tineutrino induced charged-current reactions, since bothare governed by the same nuclear matrix elements andproceed from the same set of initial to the same final nu-clear states [10, 26–28]. Moreover, from the ground statetransition matrix elements of the µ − -capture process onemay also derive cross sections for the beta decay modes[29]. Calculations on single beta decay which are moredifficult to calculate, need explicit nuclear structure cal-culations [31].The purpose of the present work is to perform detailedstate-by-state calculations [32–41] of exclusive muon cap-ture rates and concentrate on the individual contribu-tion of each basic multipole operator inducing low-lyingexcitations in the daughter nucleus. In contrast, mostof the previous muon capture calculations have beenperformed within the assumptions of closure approxi-mation [17, 18, 44]. Towards this aim, the pn-QRPAprovides a reliable description of the required nucleartransition matrix elements [3, 5, 13–15, 45–50]. Ourextensive channel-by-channel calculations would be car-ried out for the exclusive, partial and total muon cap-ture rates, and the results refer to the nuclear iso-topes Si, S, T i, F e, Zn and Zr , which coverthe light- and medium-weight region of the periodic ta-ble. We also specialize on the individual contributions ofthe polar-vector and axial-vector components of the µ − -capture operators in each of the multipole states and inthe total Ordinary Muon Carture (OMC) rate. Despitethe fact that the semileptonic process (1) is studied for along time [3–7, 11, 16, 19–22], essentially only the totalmuon-capture rates have been measured for a great num-ber of nuclear isotopes [8, 10, 23–25]. On the theoretical side, various nuclear methods using several residual in-teractions allowed the calculation of total capture rateson many nuclei with an accuracy of about 10% comparedto the experimental rates. However, for only few isotopesexclusive capture rates to specific states in the daughternucleus have been determined [8, 10, 24, 25]. As the ex-perimental data for muon capture rates are quite precise,and the theoretical techniques of evaluating the nuclearresponse in the relevant nuclear systems are well devel-oped [8, 34, 35], it is worthwhile to see to what extentthe exclusive capture rates are theoretically understood.Furthermore, we mention that, there appear recently,clear indications that the axial-vector coupling constant g A = 1 .
262 in a nuclear medium is reduced from its freenucleon value [8, 10, 23, 51–54]. The evidences for such arenormalization of the value g A come primarily from theanalysis of beta decay modes between low-lying states ofmedium-heavy nuclei [54] but the use of a quenched g A value is mainly invoked from the second-order core polar-ization caused by the tensor force [55] and the screeningof the Gamow-Teller (GT) operator by the ∆-hole pairs[56]. Thus, it is necessary to scrutinize on the in-mediumquenching of the axial vector coupling constant which isin agreement with various well-known indications that g A is reduced to the value of g A ≈ . µ -capture rates obtained withthe values (i) g A = 1 .
262 and (ii) g A = 1 .
135 with othertheoretical ones obtained with the latter value.The rest of the paper, is organized as follows. InSection 2, we summarize briefly, the main character-istics of the effective charged-current weak interactionHamiltonian and present the main formalism of the or-dinary muon capture rates which is based on our com-pact formalism for the relevant nuclear transition ma-trix elements (relying on the Donnelly-Walecka projec-tion method) and in the expressions for exclusive, partialand total muon capture rates [5, 15, 34]. Special focusis given on the calculation of the nuclear wave functionsderived within the context of the pn-QRPA. In Section3, we concentrate on the determination of the requiredmodel parameters for the nuclear ground state, derivedby solving the BCS (Bardeen Cooper Schrieffer) equa-tions, as well as of the excited states (solution of thepn-QRPA equations). Our results (Sestion 4) refer toexclusive, partial and total muon capture rates, of theabove mentioned nuclear isotopes, which cover the light-and medium-weight region of the periodic table. We alsoinclude the individual contributions of the polar-vectorand axial-vector operators in each of the multipole statesand in the total Ordinary Muon Carture (OMC) rate.Finally, in Section 5, we summarize the main conclusionsextracted from the present work.
II. FORMALISM OF MUON CAPTURE RATES
The ordinary muon-capture process, that takes placein muonic atoms and is represented by the semi-leptonic reaction (1), proceeds via a charged-currentweak-interaction Hamiltonian which is written as a prod-uct of a leptonic, j leptµ , and a hadronic current, ˆ J µ , as[5, 15, 34, 37] ˆ H w = G √ j leptµ ˆ J µ (2)where G = G F cosθ c with G F and θ c being the well knownweak interaction coupling constant and the Cabbibo an-gle, respectively.From the nuclear theory point of view, the main task isto calculate the partial and total capture rates of the re-action (1) which are based on the evaluation of exclusivenuclear transition matrix elements of the form (cid:104) f | (cid:100) H w | i (cid:105) = G √ (cid:96) µ (cid:90) d x e − i qx (cid:104) f | (cid:99) J µ | i (cid:105) . (3)(the integration is performed in the region of the nuclearsystem). In the latter expression | i (cid:105) and | f (cid:105) denote theinitial (ground) and the final nuclear states, respectively.The quantity (cid:96) µ e − i qx stands for the leptonic matrix el-ement written in coordinate space with q being the 3-momentum transfer. The magnitude of −→ q is defined fromthe kinematics of the process and is approximately givenby [57] q ≡ q f = m µ − (cid:15) b + E i − E f (4)where m µ is the muon rest mass, (cid:15) b is the muon-bindingenergy in the muonic atom, E i denotes the energy of theinitial state of the parent nucleus and E f the final energyof the corresponding daughter nucleus.In the unified description of all semi-leptonic electro-weak processes in nuclei developed by Donnelly andWalecka [3, 5, 13–15], the calculation of the required tran-sition strengths of Eq. (3) is based on a multipole decom-position of the hadronic current density which leads to aset of eight independent irreducible tensor multipole op-erators (four of them come from the polar-vector compo-nent and the other four from the axial-vector componentof the nuclear current). In the present work we assumethat the pn-QRPA excitations | J πm (cid:105) have good quantumnumbers of angular momentum (J), parity ( π ) and energywhich is a basic assumption for the Donnelly-Waleckaprojection method to be applicable. In this spirit, thecomputation of each partial transition rate of the muoncapture is written in terms of the eight different nuclearmatrix elements (between the initial | J i (cid:105) and the final | J f (cid:105) states) asΛ i → f = 2 G q f J i + 1 R f (cid:104) (cid:12)(cid:12) (cid:104) J f (cid:107) Φ s ( (cid:99) M J − (cid:98) L J ) (cid:107) J i (cid:105) (cid:12)(cid:12) (5)+ (cid:12)(cid:12) (cid:104) J f (cid:107) Φ s ( (cid:98) T elJ − (cid:98) T magnJ ) (cid:107) J i (cid:105) (cid:12)(cid:12) (cid:105) where Φ s represents the muon wave function in the1s muonic orbit [44]. The operators in Eq. (5) referto as Coulomb (cid:99) M J , longitudinal (cid:98) L J , transverse elec-tric (cid:98) T elJ and transverse magnetic (cid:98) T magnJ multipole op-erators and contain polar-vector and axial-vector parts(see Appendix A). The factor R f in Eq. (5) takesinto consideration the nuclear recoil which is written as R f = (cid:16) q f /M targ (cid:17) − , with M targ being the mass ofthe target nucleus. III. DESCRIPTION OF THE NUCLEARMETHOD
For reliable predictions of partial muon-capture rates,a consistent description of the structure of the groundstate | J i (cid:105) of the parent nucleus as well as of the multipoleexcitations | J f (cid:105) of the daughter nucleus are required. Inthe present work, the state-by-state muon capture ratesare evaluated using Eq. (5) with the transition matrix el-ements between the states | J i (cid:105) and | J f (cid:105) determined withthe use of the BCS and pn-QRPA equations, respectively(the BCS equations determine the ground state and thepn-QRPA equations provide the excited states as it isshown below) [34–41]. To this end, at first we have cho-sen the active model space (the same for proton and neu-tron configurations) for each studied isotope consisted ofthe single particle j-shells shown in Table I. TABLE I: The used active model space with the respectiveharmonic oscilator parameter for all the studied nuclei. Inthe last column the major harmonic oscillator shells N plusthe individual orbits used for each nucleus are listed.Model SpaceNucleus b(h.o) Core ActiveLevels N ( (cid:126) ω ) Si S g / ,0 g / T i g / ,0 g / F e O
12 2,3,4 Zn O
12 2,3,4 Zr O
16 2,3,4,0 h / ,0 h / ,1 f / ,1 f / As it is well known, in a rather good approximation,the nucleus can be considered as a system of Z protonsand N neutrons moving independently inside the nuclearvolume and attracted by the nuclear center through acentral strong nuclear force. This central attraction iswell described by a mean field which, in our case, is as-sumed to be a Woods-Saxon potential with a Coulombcorrection and a spin-orbit parts [35]. For the latter po-tential we tested two different parametrizations: i) thatof Bohr and Motelson [58], and ii) that of the IOWAgroup [59] and found that both give rather similar re-sults. For the purposes of the present work, however, weadopted the more realistic IOWA parametrization [59].For a reliable nuclear Hamiltonian, in addition to themean field, the two-nucleon correlations, known as resid-ual two-body interaction, are necessary to be included.Towards this aim, we employed the pn-Bonn C-D one-boson exchange potential, but, since the initially eval-uated bare nucleon-nucleon matrix-elements of the lat-ter potential refer to all nuclides with mass number A,for a specific isotope ( A, Z ) studied, a renormalizationof these two-body matrix elements was carried out withthe use of four multiplicative parameters: The first two,known as pairing parameters g p,npair , for protons (p) andneutrons (n), renormalize the monopole (pairing) inter-action which is the part of the correlations involved at theBCS level for the description of the considered indepen-dent quasi-particles. The third, g pp , tunes the particle-particle channel and the fourth, g ph , renormalizes theparticle-hole interaction of the Bonn C-D potential. Webriefly summarize the adjustement of these parametersbelow (subsection III B). A. Determination of the parent nucleus groundstate
The ground state of the parent nucleus, is obtainedwithin the context of the BCS theory where the one-quasi-particle states are deduced by solving (iteratively)the BCS equations. Towards this aim one is definingquasi-particle creation, α † , and annihilation, α , opera-tors related to the particle-creation, c † κ , and particle an-nihilation, c κ , operators through the Bogolyubov-Valatintransformations [60, 61] α † κ = u k c † κ − υ k ˜ c κ , ˜ α κ = u k ˜ c κ + υ k c † κ , (6)where ˜ c κ denotes the time reversed particle annihila-tion operator defined as ˜ c κ = ( − j k + m k c − κ with − κ =( k, − m k ). The probability amplitudes v k and u k for the k single particle level to be occupied or unoccupied, re-spectively, are [60] υ p,n ) k = 12 (cid:20) − (cid:15) p ( n ) k − λ p ( n ) E p ( n ) k (cid:21) , (7)( u k = 1 − υ k ) where (cid:15) k is the single particle energy ofthe j κ -level and λ p ( λ n ) denotes the chemical potentialfor protons (neutrons). Moreover, the solution of the rel-evant BCS equations gives the single quasi-particle ener-gies [46, 60] E p ( n ) k = (cid:113) ( (cid:15) p ( n ) k − λ p ( n ) ) + ∆ k (8)with ∆ κ being the theoretical energy gaps (∆ k = − (cid:80) k (cid:48) > ¯ υ k ¯ kk (cid:48) ¯ k (cid:48) u k (cid:48) υ k (cid:48) ) [60]. From the solution of the gapequation [45, 46]∆ kp ( n ) = g p ( n ) pair j k ] (cid:88) k (cid:48) [ j k (cid:48) ] ∆ k (cid:48) E p ( n ) k (cid:104) ( kk )0 |G| ( k (cid:48) k (cid:48) )0 (cid:105) (9) (here the notation is, [ j ] = √ j + 1) one obtains thepairing gaps for protons ∆ kp and neutrons ∆ kn throughthe renormalization of the proton and neutron pairingmatrix elements (cid:104) ( kk )0 |G| ( k (cid:48) k (cid:48) )0 (cid:105) of the residual interac-tion, using the parameters g ppair and g npair . The lowestquasi-particle energy, obtained from the gap equation, isdetermined, through the pairing parameters g p ( n ) pair enter-ing the theoretical gaps of Eq. (9) so as to reproduce theexperimental (empirical) energy gaps ∆ expp,n given fromthe three point formula [46]∆ expp ( n ) = − (cid:104) S p ( n ) [( A − , Z − Z ))] − S p ( n ) [( A, Z )]+ S p ( n ) [( A + 1 , Z + 1( Z ))] (cid:105) . (10)In the latter equation S p and S n are the experimen-tal separation energies for protons and neutrons, respec-tively, of the target nucleus (A,Z) and of the neighboringnuclei ( A ± , Z ±
1) and ( A ± , Z ). Here, we used themethod of Ref. [46] to obtain the g p,npair values for thestudied nuclei and tabulate them in Table II. We notethat, in order to achieve the reproducibility of the exper-imental energy spectrum in similar QRPA calculationssome authors modify slightly the Woods-Saxon protonand neutron single particle energies in the vicinity of thenuclear Fermi surfaces [26, 36]. In this work, we pay spe-cial attention on the reproducibility of the energy spec-trum of the daughter nucleus as is discussed in detail inthe next section. TABLE II: Parameters for the renormalization of the inter-action of proton pairs, g ppair , and neutron pairs, g npair . Theyhave been fixed in such a way that the corresponding experi-mental gaps, ∆ expp and ∆ expn , are quite accurately reproduced.Nucleus g npair g ppair ∆ expn (MeV) ∆ theorn (MeV) ∆ expp (MeV) ∆ theorp (MeV) Si S T i F e Zn Zr B. The pn-QRPA excitation spectrum of thedaughter nucleus
For the purposes of the present study, transitions be-tween the | + (cid:105) ground state of a rather spherical even-even parent-nucleus and the excited states of the re-sulting daughter nucleus are the basic ingredients. Forseveral charged-current reactions, the pn-QRPA methodprovides a reliable description of the nuclear excitedstates of the resulting odd-odd nuclear system in Eq. (1)[46]. Here, we exploit this advantage in order to derivethe excitation spectrum of the daughter nucleus producedin the µ -capture process. In this context, we first definethe two quasi-fermion operators A † and ˜ A (which obeyboson commutation relations in a correlated RPA groundstate) as [12, 32–41, 60, 61] A † mi ( JM ) = [ a † j m a † j i ] JM (11)= (cid:88) m m ( m i ) (cid:104) j m j i m m m i | JM (cid:105) α † j m m m α † j i m i , ˜ A mi ( JM ) = ( − J − M A mi ( J − M ) . (12)Afterwards, we write down the pn-QRPA phonon opera-tors Q ν † J π M = (cid:88) m ≤ i [ X νmi A † mi ( JM ) + Y νmi ˜ A mi ( JM )] , (13) ν enumerates the multipole states of the multipolarity J π , that creates the excitation | ν (cid:105) ≡ | J πν (cid:105) by acting onthe QRPA vacuum | ˜0 (cid:105) QRP A as [12, 33, 35, 36, 38–41] | J πν (cid:105) = Q ν † J π M | ˜0 (cid:105) QRP A . (14)The X (forward) and Y (backward) scattering ampli-tudes entering Eq. (13) are obtained by solving the pn-QRPA equations (pn-QRPA eigenvalue problem) whichin matrix form is written as [60] (cid:18) A B−B −A (cid:19) (cid:18) X ν Y ν (cid:19) = Ω νJ π (cid:18) X ν Y ν (cid:19) , (15)Ω νJ π denotes the excitation energy of the QRPA state | J πν (cid:105) . Thus, the X and Y amplitudes are calculated seper-ately for each multipole set of states (multipolarity).The reliability of the QRPA excitations Ω νJ π and ofthe corresponding many-body nuclear wave functions ischecked through the reproducibility of the energy spec-trum of the final odd-odd nucleus. The values of particle-particle ( g pp ) and particle-hole ( g ph ) parameters in theset of isotopes chosen (determined separately for eachmultipolarity) [35, 39, 41] lie in the region 0.65 - 1.20(with the exception of the 1 + and 2 − multipolarities insome isotopes, for which the values are rather small, 0.2- 0.6) [62]. Such small values of the strength param-eters come out in studies of charged current reactions( e − -capture, single- and double-beta decays) when fit-ting simultaneously the QRPA parameter, g pp , and theaxial vector coupling constant, g A [47, 51, 63–65]. Westress that in our QRPA method the strength parame-ters are determined through the reproduction of the en-ergy spectrum of the daughter nucleus but we have alsomade an effort to test them through the GT energy po-sition and the total GT strength [25, 28]. Even thoughour GT-type operator contributes differently (due to the presence of the Bessel function), we found that, the to-tal GT strength differs significantly (more than a factorof 2.5) from the experimental one, although, the energyposition is well reproduced. In our muon capture (and e − -capture) rates the simultaneous variation of g A and g pp parameters has not been checked extensively (see Ref.[31].We furthermore note that, in order to achieve the re-producibility of the experimental energy spectrum of thedaughter nucleus and for measuring the excitation en-ergies of the daughter nucleus from the ground state ofthe initial (even-even) nucleus, some authors shift theentire set of QRPA spectrum by about λ p − λ n in themuon capture process [24]. In our present study we alsoadopt the latter treatment, so, the calculated pn-QRPAenergy spectrum of each individual multipolarity J π isshifted in such a way that the first calculated value ofeach multipole state (i.e. 1 +1 , +1 ...etc), to approach asclose as possible to the corresponding lowest experimen-tal energy of the daughter nucleus. Such a shifting isnecessary whenever in the pn-QRPA a BCS ground stateis used, a treatment adopted by other groups too [47, 48].Table III shows the shifting applied to the QRPA spec-trum for each multipolarity of the studied nuclei. Wenote that, a similar treatment is required in QRPA cal-culations for double-beta decay studies where the excita-tions derived for the intermediate odd-odd nucleus (in-termediate states) through p-n or n-p reactions from theneighboring nuclei do not match each other [47]. The re-sulting low-energy spectrum (up to 3.0 MeV) using ourpn-QRPA method, agrees well with the experimental oneas can be seen from Fig. 1.Before proceeding to our results, it is worthwhile tobriefly summarize the advantages of the calculationalprocedure followed in performing the present detailed cal-culations of partial and total muon capture rates as com-pared to the methods used by other groups [8, 16, 23, 24].In the earlier pioneering work of Foldy and Walecka [16],the authors related the dipole capture rate to the exper-imental photoabsorption cross section and used symme-try arguments to compare polar-vector and axial-vectormatrix elements. The afore mentioned authors derived µ − -capture sum rules based on the GDR strength ex-cited after µ − -capture. The required GDR amplitudesare obtained (for light and medium nuclei) from the cor-responding photo-absorption cross sections. Later, onthe calculations of Eramzhyan et al. [24] employed atruncated model space with ground state correlations andadopted the standard free nucleon coupling constants. Inthe work of Kolbe et al. [8], for the calculations of muoncapture rates, use of the continuum RPA method wasmade with the free nucleon form factors, while recently,Zinner et al. [23] proposed the use of a quenched value forthe axial-vector coupling constant g A in order to reliablyevaluate the true Gamow-Teller transitions.It is worth mentioning that, recent studies of singleand double beta-decays as well as of neutrino-nucleus re-actions under stellar conditions, have demonstrated an TABLE III: The shift of the spectrum seperately of each state in MeVPositive Parity States Negative Parity States J + 28 Si S T i F e Zn Zr J − Si S T i F e Zn Zr + − + − + − + − + − + − Mn and Y nuclei (for the other spectra see Ref. [29, 62]).The agreement is quite good at least for low excitation energies. important role of the quenched value of the coupling con-stant g A [29, 51, 52]. In the present calculations we alsouse a quenched value of g A same for all multipole transi-tions (see the following section). IV. RESULTS AND DISCUSSION
In the ordinary muon capture on complex ( A ≥ m µ minus the binding energy (cid:15) b of the muon in the muonicatom restricted from below by the mass difference of theinitial and final nuclei and from above by the muon mass[see Eq. (4)]. The phase space and the nuclear responsefavor lower nuclear excitations, namely the nuclear statesin the giant resonance region (GDR and GT resonance)are expected to dominate [8].In our calculational procedure we followed three steps.(i) In the first step we performed realistic state-by-state calculations on exclusive OMC rates in the isotopes Si, S, T i, F e, Zn and Zr , a set which coversa rather wide range of the periodic Table from light- tomedium-weight nuclei. These calculations have been per-formed twice: Once with the use of the free nucleon cou-pling constants g A = 1 .
262 and the other with the useof the value g A = 1 . µ -capture rate. We also estimated the percentage (por-tion) of their contribution in the total rate for the mostimportant multipolarities. (iii) In the last step, we eval-uated total muon-capture rates for the above set of iso-topes. For all the above calculations, the required wavefunctions (for the initial (ground) state and for all acces-sible final states) were constructed by solving the BCSand QRPA equations, respectively, as described before(see Sections III A and III B). A. State by State calculations of exclusivetransition Rates in µ -capture At first, we evaluated the exclusive µ − -capture ratesΛ i → f of Eq (5) for all multipolarities with J π ≤ ± .In Eq. (5) transitions between the ground state | i (cid:105) ≡| + gs (cid:105) of a spherical target nucleus and an excited state | J πf (cid:105) ≡ | f (cid:105) of the resulting odd-odd nucleus are consid-ered. In most of the previous studies a mean value of themuon wave function, Φ µ ( −→ r ), with −→ r being the spheri-cal coordinate, has been utilized (see Appendix B). Anaccurate description of the reaction (1) (and of any re-action having the same initial state with it, i.e. a muonorbiting around an atomic nucleus ( A, Z )), however, re-quires the exact muon wave function derived by solvingthe Schr¨odinger equation (or the Dirac equations) thatobeys a bound muon within the extended Coulomb fieldof the nucleus in such muonic atoms [23].Assuming, that the muon wave function in the regionof the nuclear target is nearly constant, the integrals en-tering Eq. (5) can be performed by taking out of them anaverage value (cid:104) Φ s (cid:105) . Hence, the exclusive muon capturerates Λ J πf can be rewritten as:Λ gs → J πf ≡ Λ J πf = 2 G (cid:104) Φ s (cid:105) R f q f · (cid:104) (cid:12)(cid:12) (cid:104) J πf (cid:107) ( (cid:99) M J − (cid:98) L J ) (cid:107) + gs (cid:105) (cid:12)(cid:12) + (cid:12)(cid:12) (cid:104) J πf (cid:107) ( (cid:98) T elJ − (cid:98) T magnJ ) (cid:107) + gs (cid:105) (cid:12)(cid:12) (cid:105) (16)On the basis of the latter expression, we initially, per-formed state-by-state calculations, for the above men-tioned set of nuclear isotopes, by using the free nucleoncoupling constant g A for the axial-vector form factor.Then, we repeated these calculations (with the excep-tion of Si and S isotopes) by taking into accountthe quenching effect of the axial-vector coupling constant g A = 1 . − , 1 + and 2 − low-lying multipole excitations,of the particle bound spectrum and of the giant dipole,spin and spin-dipole resonances.As mentioned before, our code initially gives resultsfor exclusive muon capture rates, Λ J πf , seperately foreach multipolarity (in ascending order with respect tothe pn-QRPA excitation energy Ω νJ π ). In order to studythe dependence of the rates on the excitation energy ω throughout the entire pn-QRPA spectrum of the daugh-ter isotopes, a rearrangement of all possible excitationsin ascending order with respect to ω and with the cor-responding rates, is required. This was performed byusing a special code (appropriate for matrices). Totally,for J π ≤ ± in the model space chosen for each isotope,we have 286 states for the Si isotope, 440 states foreach of the S and T i isotopes, 488 states for eachof the F e and Zn , and 912 states for Zr isotopein the corresponding daughter nucleus. The variation ofthe exclusive rates throughout the entire excitation spec-trum of the daughter nucleus in the case of the abovetarget isotopes are demonstrated in Figs. 2, 3 and 4.For all reactions, the rates present some characteristicclearly pronounced peaks at various excitation energies ω and specifically for transitions J π = 1 + , 1 − but alsofor J π = 0 + , 0 − and 2 − transitions.More specifically, in the daughter Al isotope themaximum peak corresponds to the 1 +7 QRPA transitionat ω = 7 . M eV (see Fig. 2). Other two characteristicpeaks are at ω = 18 . M eV and at ω = 18 . M eV which correspond to the 0 − and 1 − transitions respec-tively. In the case of P isotope the maximum peakcorresponds to the 1 +5 transition at ω = 4 . M eV . An-other characteristic peak is at ω = 15 . M eV whichcorresponds to the 1 − transition as shown in Fig. 2 (left).For the Sc isotope, the pronounced peaks correspondto the first excited 0 + state (0 +1 ) (at ω = 4 . M eV ),the 2 − (at ω = 9 . M eV ), the 1 +13 ( ω = 10 . M eV )and the 1 − transitions ( ω = 18 . M eV ). From Fig.3 (right pannel), for the daughter isotope M n , we seethat the maximum peak appears at ω = 8 . M eV andcorresponds to the 1 +10 transition. Another importanttransition is that of 1 − at ω = 18 . M eV . As shownin Fig 4, in the case of the daughter isotope Cu , themaximum peak appears at ω = 6 . M eV and cor-responds to 1 +10 state and a pronounced peak for the1 − at ω = 14 . M eV . Finally, for the Y isotope,the maximum peak appears for the 1 − transition at FIG. 2: Individual contribution of the Polar-Vector Λ V (pannel(a)) and Axial-Vector Λ A (pannel (b)) to the total muon-capturerate (pannel(c)) as a function of the excitation energy ω for the Si and S nuclei. ω = 18 . M eV and for the 1 +36 at ω = 9 . M eV .From the above results, we conclude that in general,a great part of the OMC rate comes from the excitationenergy region where the centroid of the GT strength islocated for each daughter nucleus. As it is known fromclosure approximation studies [17, 18], the mean excita-tion energy in muon capture (about 15 MeV) is nearlyequal to the energy of the giant dipole resonance (GDR)which is slowly decreasing with A or Z [23]. On the otherhand, the GT-like operators (in which the full sphericalBessel functions is taken into account) contribute verylittle in heavier nuclei where most of the active neutronsand protons are in different oscillator shells. In lighter nu-clei, however, i.e. for nuclei having N and Z smaller than40, the GT strength is significant and it is concentratedat the low energy region. Regarding the giant spin reso-nance ( J π = 1 + ) for all nuclei the peak of the exclusive µ -capture rate is located between 5-11 MeV. It should be stressed that concerning the pronounced contributionto the 1 − states, it may contain a small portion of thespurious center of mass motion part (up to about 17%in our QRPA method) [35]. This is due to the isoscalarmovement of the nucleons in the mean field (dipole os-cillation of the whole nucleus). As it is known this isusually removed by using specific methods [35].As it becomes clear from Figs. 2, 3 and 4, for the stud-ied nuclei the muon capture response presents maximumpeak in the very important giant dipole resonance region(GDR), which is located in the energy region of 18-19MeV for Si , T i , F e and Zr isotopes, and in theregion of 15-16 MeV for S and Zn isotopes. These re-sults can be compared with the empirical expression, formedium-weight and heavy isotopes, which gives the en-ergy location of the giant dipole resonance, E IV D , basedon the Jensen-Steinwedel and Goldhaber-Teller models
FIG. 3: The same as Fig. 2 but for the nuclei T i and F e . (a hydrodynamical view of the giant resonance) as [66] E IV D = 31 . A − / + 20 . A − / (17)(A is the atomic mass of the nucleus). Even though thisformula refers to pp- and nn-reactions, it can however beused to our results referred to pn-reactions ( µ − -capture),on the basis of the well known Foldy-Walecka theoremaccording to which the giant dipole resonance in µ − -capture rates, are calculated starting from the experi-mental photo-absorption cross sections [16]. Accordingto Eq. (17) for T i the maximum 1 − peak is locatedat 18.668 MeV, for F e at 18.716 MeV, for Zn at17.945 MeV and for Zr at 16.684 MeV which are ina good agreement with our results (the worst case oc-curs for Zn where the empirical peak is at about 15MeV). Moreover, our results are in good agreement withthe conclusions of Ref. [24] where authors mention thatfor the stable Ni isotopes ( . , N i ) the peak appearsin the range of 18-19 MeV. We note that similar conclu- sion is extracted from the study of the charged currentreaction F e ( ν e , e − ) Co by Kolbe and Langanke [27],where the peak of the giant dipole resonance appears atabout 17 MeV (see Fig 1 of Ref. [27]).As can be seen from Figs. 2, 3 and 4, the main con-tributions coming from the polar-vector operator are the1 − and 0 + states while, the most important transitionsdue to the axial-vector operator are the 0 − , 1 + and 2 − excitations, namely the lowest spin states.We note that the figures of this section, have been de-signed by using the ROOT program of Cern with binningwidth 0.112, 0.105, 0.105, 0.15, 0.14, 0.11, respectively,for Si , S , T i , F e , Zn , Zr nuclei. B. Contribution of Multipole Transitions
The second step of our study includes calculations ofthe partial µ − -capture rates for various low-spin multi-0 FIG. 4: The same as Fig. 2 but for the nuclei Zn and Zr . polarities, Λ J π (for J π ≤ ± ), in the chosen set of nuclei.These partial rates have been found by summing over thecontibutions of all the individual multipole states of thestudied multipolarity asΛ J π = (cid:88) f Λ gs → J πf = 2 G (cid:104) Φ s (cid:105) · (18) (cid:104) (cid:88) f q f R f (cid:12)(cid:12) (cid:104) J πf (cid:107) ( (cid:99) M J − (cid:98) L J ) (cid:107) + gs (cid:105) (cid:12)(cid:12) + (cid:88) f q f R f (cid:12)(cid:12) (cid:104) J πf (cid:107) ( (cid:98) T elJ − (cid:98) T magnJ ) (cid:107) + gs (cid:105) (cid:12)(cid:12) (cid:105) where f runs over all states of the multipolarity | J π (cid:105) .As mentioned before, these calculations have been per-formed first by using the free nucleon axial-vector cou-pling constant g A = 1 . g A = 1 . Si [67, 68] and S isotopes, these cal- culations were performed only for the free nucleon cou-pling constant g A = 1 . µ − -capture rates of these isotopes are illustrated in Fig. 5,from which one can see that, as it is expected, the mostimportant multipole transitions are the J π = 1 + and 1 − .More specifically, for Si isotope, the contributions of all J π = 1 − transitions exhaust the 36% of the total muon-capture rate and the J π = 1 + about 30%. Significantcontribution, about 14%, comes from the J π = 0 − mul-tipolarity and about 13% from the J π = 2 − . A similarpicture is found in S isotope, where the dominant con-tributions to the total muon-capture rate are the J π = 1 − (38%) and the J π = 1 + (30%). From the rest of themultipolarities rather significant portions come from theabnormal parity transitions 0 − and 2 − about 13% and14% respectively.Because, as mentioned in the Indroduction, for elec-tromagnetic and weak charged-current nuclear processes,1 FIG. 5: Partial muon capture rates Λ J π of different multipole transitions in Si and S isotopes. In both isotopes thepronounced contributions are the J π = 1 − and J π = 1 + multipolarity. the free nucleon coupling constant g A must be modifiedfor medium-weight and heavy nuclei [23], in µ − -captureon T i, F e, Zn, Zr isotopes we repeated the state-by-state calculations by using g A = 1 .
135 (a value smallerby about 10 −
12% compared to the g A = 1 . g A , cameout of the following studies: (i) In the analysis of mea-surements on the nuclear beta-decays that lead to low-lying excitations [54], and (ii) in the interpretation of themissing Gamow-Teller strength revealed in forward angle(p,n) and (n,p) charge-exchange reactions [53]. We notethat, in (n,p) reactions many authors use quenched valuesof g A lying in the region of 0 . < g A < . < A <
64 [65, 69, 70]. In β − -decay and(p,n) reactions the quenching is mainly related to the ne-glect of configurations outside the model space used andthe non-consideration of the meson-exchange currents.A quenched value of g A was recently suggested to beused in other weak interaction processes such as the neu-trino induced nuclear reactions. As has been found [25],the consideration of a quenched factor instead of the freenucleon axial-vector coupling constant, leads to betteragreement of the theoretical results with the experimen-tal muon capture rates. Since the axial-vector form factor F A ( q ) multiplies all four operators (see Eqs. (A1)-(A4)),a quenched value of g A must enter the multipole opera-tors generating the pronounced excitations 0 − , ± ... etc.In Ref. [23], a quenched value of g A is used only for thetrue Gamow-Teller transitions. In our study, we find thatfor the reproducibility of the experimental data, as themass number A of the nucleus increases the quenchingbecomes more signifficant and can not be ignored as wehave done in the case of the Si and S isotope.For the medium-weight nuclei T i, F e, Zn and Zr , we used the moderate quenched value g A = 1 . g A for the contributions of the different multipole transi-tions in the isotopes F e, Zn and Zr , we found thatthe most important peaks correspond to the J π = 1 + and 1 − . For the T i isotope, however, we found that agreat part of the total rate comes from the J π = 1 − and2 − , as is shown in Fig. 6. In more detail, in the caseof T i isotope the 1 − multipolarity contributes about44%, the 2 − about 17%, the 1 + about 16% and the 0 − about 11%. Significant contribution (about 7%) origi-nates also from the 0 + multipolarity. For F e isotopethe most important contribution about 42% comes fromthe 1 − multipolarity. Other multipolarities with signif-icant contributions are the 1 + (22%) , − (13%) , − (10%)and 0 + (8%). A similar picture appears in the other twoisotopes, Zn and Zr , where the major contribution isderived from the 1 − multipolarity, about 44% and 42%respectevelly. The 1 + multipolarity contributes about21% in Zn and about 20% in Zr isotope. Corre-spondingly, the 2 − contributes about 13% for Zn andabout 14% for Zr , the 0 + about 8% and 9% respectiv-elly and finally the 0 − multipolarity offers about 8% for Zn and about 7% for Zr .In Table IV we present the partial muon-capture ratesobtained for the low-spin multipole transitions up to J π = 4 ± evaluated with our pn-QRPA code. Corre-spondingly, in Table V we tabulate the individual por-tions to the total OMC rate, for the low-spin nultipoletransitions up to J π = 4 ± . As can be seen, for all nu-clei the contribution of 1 − multipole transitions is themost important multipolarity, exhausting more than 39%of the total muon-capture rate. Ordinary muon captureproceeds mainly through spin-multipole transitions, themost important of which are the Gamow-Teller transi-tions ( j ( kr ) σt + operator), and the spin-dipole transi-tions ( j ( kr )[ Y ⊗ σ ] J t + operator) where j and j arethe spherical Bessel functions of zero and first order, re-spectively [24]. Such important contribution is found in O and in Ca isotopes studied in Ref. [10].There are no similar results for the isotopes Si , S , T i , F e and Zn to compare with our portions. Forthe Zr , however, Kolbe, Langanke and Vogel [10] foundabout 28% (for 1 − ), 25% (for 1 + ) and about 13% (for2 − ) multipolarities which, with the exception of 1 − con-2 FIG. 6: Contribution of multipole transition rates Λ J π (up to J π = 4 ± ) with the total muon capture rate in T i , F e , Zn and Zr isotopes with (filled histograms) and without (double dashed histogramms) quenching effect. The dominanceof J π = 1 − and 1 + multipolarities is obvious in all nuclei.TABLE IV: Muon capture rates Λ J π (in 10 s − ) of eachmultipolarity evaluated with our pn-QRPA code. Si S T i F e Zn Zr − + − + − + − + − + . − . − . − . − . − . − tribution, are in good agreement with our results listedin Table V. The difference in 1 − multipolarity is mostlydue to the fact that Zr is a double closed shell nucleusand the QRPA convergence is treated as in Ref. [33, 71]. TABLE V: The percentage of each multipolarity into thetotal muon-capture rate evaluated with our pn-QRPA code. Si S T i F e Zn Zr − + − + − + − + − + C. Total Muon-Capture-Rates
In the last stage of our present work, we computed thetotal rates of muon-capture on the chosen set of nuclei.3These rates are obtained by summing over all partial mul-tipole transition rates in two steps. At first, we sum upthe contribution of each final state of a specific multipo-larity, and then, we sum over the multipole responses (upto J π = 4 ± ) asΛ tot = (cid:88) J π Λ J π = (cid:88) J π (cid:88) f Λ J πf (19)Such calculations have been carried out twice: one with g A = 1 .
262 (free nucleon axial-vector coupling constant)and the other with the quenched value g A = 1 .
135 [23].The results are tabulated in Table VI, where for the sakeof comparisson we also include the experimental totalrates as well as the theoretical ones of Ref. [23]. More-over, in Table VI we show the individual contribution inthe total muon capture rate of the polar-vector (Λ
Vtot ),the axial-vector (Λ
Atot ), and the overlap (Λ
V Atot ) parts.
TABLE VI: Individual contribution of Polar-vector, Axial-vector and Overlap part to the total muon-capture rate. Com-parison between the total muon capture rates obtained by us-ing the pn-QRPA with the quenched value of g A = 1 .
135 formedium-weight nucleus ( T i, F e, Zn and Zr ) and thefree nucleon coupling constant g A = 1 .
262 for the light nu-cleus Si and S , with the available experimental data andwith the theoretical rates of Ref [23].Total Muon-capture rates Λ tot ( × ) s − pn-QRPA Calculations Experiment RPA DeanNucleus Λ Vtot Λ Atot Λ V Atot Λ tot Λ exptot Λ theortot [23] Si S T i F e Zn Zr As can be seen, our results obtained with the quenched g A are in very good agreement with the experimental to-tal muon-capture rates. For all studied nuclei the de-viations from the corresponding experimental rates aresmaller than 7% when using the quenched g A (the devi-ation is much bigger when using the g A = 1 . λ = ω calc ω exp (20)for the results obtained with the above two values of g A (with and without quenching). The filled circles repre-sent the results for the free g A and the X symbols theresults for the quenched g A . It is evident the betteragreement of our calculations with quenched value of g A .We furthermore, compare our results with the available calculated rates Zinner [23] obtained by using differentapproach and the comparison is good. FIG. 7: Ratio of the calculated and experimental total muoncapture rates as a function of Z. Circles and X symbols corre-spond to rates calculated with free nucleon g A and quenchedvalue of g A respectively. Finally, it is worth noticing that, in medium-weightnuclei the contribution comes mainly from transitionsfor which the angular momentum transfer is L=0,1 and2 but, in heavy nuclei, some contributions from highermultipolarities become noticeable.
V. SUMMARY AND CONCLUSION
In the present work, relying on an advantageous nu-merical approach constructed by our group recently, weperformed detailed calculations for all multipole transi-tion matrix elements entering the exclusive muon-capturerates. The required nuclear wave functions were obtainedwithin the context of the pn-QRPA using realistic two-body forces (Bonn C-D potential). Results for the exclu-sive rates through extensive state-by-state calculationsand subsequently for the total muon capture rates onthe set of isotopes Si , S , T i , F e , Zn and Zr were computed.Because the capture rates are rather sensitive to thequenching of the axial-vector coupling constant, we ex-amined the known as in-medium effect of the nucleon,by reducing this constant from its free nucleon value g A = 1 .
262 to the effective value g A = 1 .
135 for all multi-pole transitions, and found that the experimental muoncapture rates are well reproduced with an accuracy bet-ter than 10%. Detailed study of this effect, however,required for experiments at RCNP [74, 75] is under wayand results are expected to be obtained soon.The muon-capture studies on these nuclei demonstratethat the used pn-QRPA method may provide an accuratedescription of the charged current semileptonic weak in-teraction processes in the Z-range of the isotopes chosen.4As the inclusive muon capture rates and the cross sectionof the antineutrino-induced charged current reactions areclosely related (both of them are governed by the samenuclear matrix elements and proceed via the same initialand final states), we have adopted this method to studyother types of charge-changing weak interaction processesas, electron-capture, beta-decay modes, etc. [29, 31] incurrently interesting nuclei from a nuclear astrophysicspoint of view.
Acknowledgments
This research has been co-financed by the EuropeanUnion (European Social Fund-ESF) and Greek nationalfunds through the Operational Program “Education andLifelong Learning” of the National Strategic ReferenceFramework (NSRF) - Research Funding Program: Her-acleitus II. Investing in knowledge society through theEuropean Social Fund.
Appendix A: Nuclear Matrix Elements
The eight different tensor multipole operators enter-ing Eq. (5) refer to as Coulomb (cid:99) M J , longitudinal (cid:98) L J ,transverse electric (cid:98) T elJ and transverse magnetic (cid:98) T magnJ ,contain polar-vector as well as axial-vector parts and arewritten as: (cid:99) M JM ( qr ) = (cid:99) M coulJM + (cid:99) M coul JM (A1)= F V M JM ( qr ) − i qM N [ F A Ω JM ( qr )+ 12 ( F A + q F p )Σ (cid:48)(cid:48) JM ( qr )] (cid:98) L JM ( qr ) = (cid:98) L JM + (cid:98) L JM (A2)= q q F V M JM ( qr ) + iF A Σ (cid:48)(cid:48) JM ( qr ) (cid:98) T elJM ( qr ) = (cid:98) T elJM + (cid:98) T el JM (A3)= qM N [ F V ∆ (cid:48) JM ( qr ) + 12 µ V Σ JM ( qr )]+ iF A Σ (cid:48) JM ( qr ) (cid:98) T magnJM ( qr ) = (cid:98) T magnJM + (cid:98) T magn JM = − qM N [ F V ∆ JM ( qr ) − µ V Σ (cid:48) JM ( qr )] + iF A Σ JM ( qr ) (A4)where the form factors F X , X=1,A,P and µ V are func-tions of the 4-momentum transfer q µ .These multipole operators, due to the Conserved Vec-tor Current (CVC) theory, are reduced to seven new ba-sic operators expressed in terms of spherical Bessel func-tions, spherical harmonics and vector spherical harmon-ics (see Refs. [5, 34]). The single particle reduced matrix elements of the form (cid:104) j (cid:107) T Ji (cid:107) j (cid:105) , where T Ji representsany of the seven basic multipole operators ( M JM , Ω JM ,Σ JM , Σ (cid:48) JM , Σ (cid:48)(cid:48) JM , ∆ JM , ∆ (cid:48) JM ) of Eq. (A1)-(A4), have beenwritten in closed compact formulae as [34, 35] (cid:104) ( n l ) j (cid:107) T J (cid:107) ( n l ) j (cid:105) = e − y y β/ n max (cid:88) µ =0 P Jµ y µ (A5)where the coefficients P Jµ are given in Ref. [34]. In thelatter summation the upper index n max represents themaximun harmonic oscillator quanta included in the ac-tive model space chosen as n max = ( N + N − β ) / N i = 2 n i + l i , i=1,2, and β is related to the rankof the above operators [34].In the context of the pn-QRPA, the required reducednuclear matrix element between the initial | + gs (cid:105) and anyfinal | f (cid:105) state entering the rates of Eq. (16) are given by (cid:104) f (cid:107) (cid:98) T J (cid:107) + gs (cid:105) = (cid:88) j ≥ j (cid:104) j (cid:107) (cid:98) T J (cid:107) j (cid:105) [ J ] · (cid:2) X j j u pj υ nj + Y j j υ pj u nj (cid:3) (A6)where u j and υ j are the probability amplitudes for the j -level to be unoccupied or occupied, respectively (see thetext) [32].These matrix elements enter the description of varioussemi-leptonic weak interaction processes in the presenceof nuclei [3, 5, 13–15, 34–43]. Appendix B: Muon wave function in the muonicatom
The calculation of the exact muon wave function,Φ s ( r ), entering Eq. (5) needs the use of a specific nu-merical method. This however, can be avoided by usingeither its value at r (cid:39)
0, namely the Φ s ( r (cid:39) (cid:104) Φ s (cid:105) , which is givenin terms of the effective nuclear charge Z eff that sees thebound muon as (cid:104) Φ s (cid:105) = 1 π α m µ Z eff Z (B1)( α denotes the fine structure constant). The quantity Z eff is approximated by Z eff = πα (cid:104) ρ (cid:105) , where α is themuon Bohr radius and (cid:104) ρ (cid:105) is the mean charge densityof the parent nucleus [72]. For light nuclei Z eff (cid:39) Z but for heavier ones Z eff (cid:28) Z . In recent studies theexact wave functions for the bound muon are obtainedby solving the Schroedinger and Dirac equations by usingneural network techniques or genetic algorithms [73]. Inthe work of Zinner, Langanke and Vogel [23], for thedescription of the exact bound muon wave functions (w-fs), the muon density beyond the site of the nucleus isconsidered for solving the Dirac equation. These authorsuse exact muon wave functions, for other muonic orbits,Φ p , etc, which are considered to have rather signifficantcontributions [23].5 Appendix C: pn-QRPA equations
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12 (Ω / m P m + Ω − / m R m ) Y m = (cid:114)
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