Charge-changing transitions and capture strengths of pf-shell nuclei with T z =−2 at proton drip-line
CCharge-changing transitions and capture strengths ofpf-shell nuclei with T z = − at proton drip-line Muneeb-Ur Rahman • Jameel-Un Nabi Abstract
Charge-changing transitions, commonly re-ferred to as Gamow-Teller (GT) transitions, and elec-tron capture/ β + -decay strengths for pf-shell nuclei with T z = − + strength values and electron capture/ β + -decay ratesare needed for the study of the late stages of the stel-lar evolution. The pn-QRPA theory is used for a mi-croscopic calculation of GT strength distribution func-tions and associated stellar electron capture/ β + -decayrates of proton-rich pf-shell nuclei with T z = − ≤ A ≤
56 at proton drip-line. Stan-dard quenching factor of 0.74, usually implemented inthe shell model calculation, has been incorporated forthe comparison with experimental data (wherever avail-able). The calculated GT strength of the two proton-rich nuclei, Ni and Zn are compared with experi-mental data of corresponding mirror nuclei. It has beenfound that the pn-QRPA results are in good agreementwith the experimental data as well as shell model re-sult. It is noted that the total GT strength increaseslinearly with the increase of mass number. The elec-tron capture/ β + -decay rates for proton-rich nuclei arecalculated on a temperature and density scale relevantto presupernova evolution of massive stars. The β + de-cay half-lives are compared with measured and othertheoretical calculations. Muneeb-Ur RahmanDepartment of Physics, Islamia College Peshawar, KP, Pakistanemail: [email protected] NabiFaculty of Engineering Sciences, GIK Institute of EngineeringSciences and Technology, Topi 23640, Swabi, KP, Pakistanemail: [email protected]
Keywords
Gamow-Teller (GT) strength distribution;electron capture/ β + -decay; pn-QRPA; proton drip-line; stellar dynamics; core-collapse. The simulation of supernovae explosion mechanism de-pends on many input parameters to be fed in relevantmega codes. Alone the parameters related to nuclearphysics involve data for a large number of nuclei re-quired to simulate this complex scenario. The moresusceptible nuclei for electron capture and beta decaytend to be small in numbers, however, weak rate timesabundance is the quantity important for necessary ac-tion. Synthesis of the iron group and other heavierelements substantially depend on many input parame-ters. Number of electrons per baryon ( Y e ), energeticsof the shock waves, entropy of the stellar core, massand metalicity of the progenitor, mixing and fallback,and explosion energy are among the few. The hydro-dynamic shock is believed to be formed at the edge ofhomologous core and its energy is related to Y e as: E S (cid:39) ( GM HC /R HC )( Y ef − Y ei ) (1) (cid:39) M / HC ( Y ef − Y ei ) (cid:39) Y / ef ( Y ef − Y ei ) , where M HC , R HC , Y ei , and Y ef are mass of the un-shocked inner core (the homologous core), radius of ho-mologous core, and initial and final lepton fraction, re-spectively (Kar et al. 1994; Nabi et al. 2004; Rahmanet al. 2014). The central part of the massive stars( M ≥ M (cid:12) ) consists of iron core which grows withthe passage of time. When this core exceed the appro-priate Chandrasekhar mass limit, the implosion ensues.This collapse is subsonic and homologous in the innerregion of the core and is supersonic in the outer re-gions. The behavior of the supernovae after collapse is a r X i v : . [ nu c l - t h ] F e b very sensitive to the mass of homologous core, and con-sequently to final lepton fraction Y ef which in turn isdependent on the electron capture (Bethe 1990). Themass of the inner core has important consequences suchas it sets the value of kinetic energy imparted to theshock wave, mass cut for material which the shock hasto plough, and sets the amount of matter and angu-lar momentum that can be dynamically relevant in theastrophysical scenario. For core density greater than10 g/cm the electromagnetic radiation and heat con-duction are transported very slowly as compared to thetime scale of the collapse (Juodagalvis 2010). Thus, forstellar core densities ≤ g/cm , provided the starsare not too massive, neutrinos bleed away from the sur-face to support low entropy condition and keep the nu-cleons bound inside nuclei in stellar interior. This re-duction in entropy in stellar interior favors smaller massof the iron core and consequently facilitate the shock’soutward march (Timmes et al. 1996). Lower entropyof the core favors the explosion because less energy isstored in the nuclear excited states in the collapsingcore and presupernova environment and consequentlythe implosion process leads to higher density to pro-duce stronger bounce and an energetic shock (Bethe etal. 1979). Bethe and collaborators (Bethe 1990; Betheet al. 1979) pointed out that the lower central entropymakes it sensitive to the Chandrasekhar mass limit anddue to the overlying matter’s pressure the mass of thefinal collapsing core is small as compare to the Chan-drasekhar mass. The structure of the presupernova core(which determines the extent of the convective shells)and nucleosynthesis in stars are greatly effected by theentropy profile. Neutrino bleeding, entropy profile andelectron to baryon ratio ( Y e ) are dependent on weak de-cay rates in stellar matter. Electron capture reduces thevalue of Y e and electron degeneracy pressure in the coreto accelerate the collapse. Various authors (Rahman etal. 2014; Juodagalvis et al. 2010; Heger et al. 2001;Nabi et al. 1999a; Liu 2013; Rahman et al. 2013) im-plemented different models to calculate the weak decayrates at temperature and density scale relevant to as-trophysical environment. These authors noted that theelectron capture rates are substantially suppressed instellar core as compared to the seminal work of Fuller,Fowler and Newman (FFN)(Fuller et al. 1980, 1982a,1982b, 1985). After recognition of the pivotal role ofGamow-Teller (GT) strength functions in astrophysi-cal environment, FFN used the parameterization basedon the independent particle model to calculate the GTcontributions to the stellar rates. They inserted theexperimental data, available at that time, for the dis-crete transitions and assigned a value of log ft = 5 tounmeasured allowed GT transitions. However, Caurier and collaborators (Caurier et al. 1999) observed thatfor even-even nuclei their GT centroid was at lower ex-citation energy in the daughter nucleus as comparedto the FFN. It has also been noted that for odd-Aand odd-odd nuclei, FFN placed GT centroid at toolow excitation energies than the shell model and exper-imental data (Aufderheide et al. 1996; Langanke et al.1998; Nabi and Rahman 2005). Experimental measure-ments showed that, contrary to the independent parti-cle model, the total GT strength both in GT + and GT − direction is quenched and fragmented over many finalstates in the daughter nucleus (Rapaport et al. 1983;Vetterli et al. 1989; Anderson et al. 1990; Ronnqvistet al. 1993; El-Kateb et al. 1994). This quenching andfragmentation of the GT strength is due to the residualinteraction among the valence nucleons and an accu-rate description of these correlations is essential andplay significant role for the calculation of stellar weakdecay rates. Various models have been proposed andcan be found in literature elsewhere.The pn-QRPA is an efficient way to calculate GTstrength and associated weak decay rates (Nabi et al.1999a; Nabi 2009; Nabi 2012; Nabi et al. 2013).The pn-QRPA model has access to a liberal modelspace of 7 (cid:126) ω to perform the required calculation. Forisospin symmetry, Fermi transitions are simple and theyare only important for β + decay of proton-rich nucleiwith Z > N (Sarriguren 2013). The authors (Cole etal. 2012) used QRPA to calculate and compare GTstrength and electron capture rates with experimentaland other models for various pf-shell nuclei and dis-cussed the pros and cons of their results for astrophys-ical scenario. The following section shows the neces-sary mathematical formulae for the calculation of GTstrength distributions and weak decay rate at densityand temperature scale that are relevant for astrophys-ical environment. The GT strength distributions, cal-culated in the present study, are compared with shellmodel results and measurements in Section 3. The elec-tron capture rates for few selected proton drip-line nu-clei ( Mn, Fe, Co, Ni and Zn) are presented inSection 4. Finally we conclude our study in Section 5.
For the calculation of GT strength distributions andelectron capture/ β + -decay rates on proton rich nucleiin stellar environment, the following main assumptionswere taken into account:1) Only allowed GT and superallowed Fermi transi-tions were calculated in the present study. Forbiddentransitions are relatively negligible for the density andtemperature scales considered here.
2) It was assumed that the gas in the stellar mediumis completely ionized and electrons are no longer boundto the nucleus and obey Fermi-Dirac distribution.3) Neutrino and anti-neutrino captures escape freelyfrom the interior and surface of the star for densityscales considered in this project.4) The effect of particle emission from the excitedstates were taken into account.5) All excited states having energy less than S p ( S n )(separation energy of protons (neutrons)) were assumedto decay directly to the ground state through γ transi-tions.The Hamiltonian, in the present study, takes theformH QRPA = H sp + V pair + V phGT + V ppGT , (2)where H sp , V pair , V phGT , V ppGT are the single-particleHamiltonian, the pairing force (pairing was treated inthe BCS approximation), the particle-hole (ph) GTforce, and the particle-particle (pp) GT force, re-spectively. Single particle energies and wave func-tion were calculated in the Nilsson model, which takesinto account nuclear deformations. The proton-neutronresidual interactions occurred as particle-particle andparticle-hole interaction. The interactions were givenseparable form and were characterized by two inter-action constants κ and χ , respectively. The detailsand fine tuning of these GT strength parameters canbe found in literature (Staudt et al. 1990; Muto etal. 1992; Hirsch et al. 1993). Other parameters suchas Nilsson potential parameters (taken from (Nilsson1955)), the deformations, pairing gaps, and Q-valueof the reactions were used in the calculation of weakdecay rates. Nilsson oscillator constant was taken as (cid:126) ω = 41 A − / (MeV) for both neutrons and protons.The traditional choice of ∆ p = ∆ n = 12 / √ A ( M eV )was used for the pairing gaps. Deformation of the nu-clei was calculated as δ = 125( Q )1 . Z )( A ) / , (3)where Z and A are the atomic and mass numbers, re-spectively and Q is the electric quadrupole momenttaken from Ref. (M¨oller et al. 1981). Q-value of thereaction was taken from the mass compilation of Audiand Wapstra (Audi et al. 2003).The electron capture (ec) and positron decay (pd)rates of a transition from the i th state of the parent tothe j th state of the daughter nucleus are given by λ ec ( pd ) ij = (cid:20) ln 2 D (cid:21) (cid:104) B ( F ) ij + (cid:0) g A / g V (cid:1) B ( GT ) ij (cid:105)(cid:104) f ec ( pd ) ij ( T, ρY e , E f ) (cid:105) . (4) The value of D was taken to be 6146 ± B ( F ) ij and B ( GT ) ij are the reduced transition probabilities dueto Fermi and GT transitions: B ( F ) ij = 12 J i + 1 | < j (cid:107) (cid:88) k t k ± (cid:107) i > | . (5) B ( F ) ij = [ T ( T + 1) − T zi T zf ] . (6) B ( GT ) ij = 12 J i + 1 | < j (cid:107) (cid:88) k t k ± (cid:126)σ k (cid:107) i > | . (7)Here (cid:126)σ k is the spin operator and t k ± stands for theisospin raising and lowering operator and ( g A / g V ) eff isthe effective ratio of the axial-vector ( g A ) to the vector( g V ) coupling constants that takes into account the ob-served quenching of the GT strength (Osterfeld 1992).In the present work ( g A / g V ) eff was taken as( g A / g V ) eff = 0 .
74 ( g A / g V ) bare (8)with ( g A / g V ) bare taken as -1.257. In the present workthe dominant Gamow-Teller transition strength aretaken into account for the electron capture/ β + -decayrate calculation. In order to calculate the Fermi transi-tions, the Coulomb displacement energy was calculatedas (cid:52) E c = (1 .
444 ´ Z/ ( A ) / ) − . M eV. (9)Where ´ Z is average charge of the pair of the respectivenucleus and A is the mass number. The energy positionof the isobaric analog state (IAS) was computed usingthe following equation: E IAS = (cid:52) E c + (cid:52) − ( m n − m p ) (10)where (cid:52) is the beta decay energy (to the ground state ofthe daughter nucleus), m n and m p are mass of neutronand proton, respectively.The f ec ( pd ) ij are the phase space integrals and arefunctions of stellar temperature ( T ), electron density( ρY e ) and Fermi energy ( E f ) of the electrons. They areexplicitly given by f ecij = (cid:90) ∞ w l w (cid:112) w − w m + w ) F (+ Z, w ) G − dw. (11)and by f pdij = (cid:90) w m w (cid:112) w − w m − w ) F ( − Z, w )(1 − G + ) dw, (12)In Eqs. (11) and (12), w is the total energy of theelectron including its rest mass. w m is the total β -decayenergy, w m = 1 m e c ( m p − m d + E i − E j ) , (13)where m p and E i are masses and excitation energiesof the parent nucleus, and m d and E j of the daughternucleus, respectively. F( ± Z,w) are the Fermi func-tions and were calculated according to the procedureadopted by Gove and Martin (Gove et al. 1971). G ± are the Fermi-Dirac distribution functions for positrons(electrons). G + = (cid:20) exp (cid:18) E + 2 + E f kT (cid:19) + 1 (cid:21) − , (14) G − = (cid:20) exp (cid:18) E − E f kT (cid:19) + 1 (cid:21) − , (15)here E is the kinetic energy of the electrons and k isthe Boltzmann constant.The neutrino blocking of the phase space was nottaken into account for reasons mentioned earlier. Thetotal capture/ β + -decay rate per unit time per nucleusis finally given by λ ec ( pd ) = (cid:88) ij P i λ ec ( pd ) ij . (16)The summation over the initial and final states was car-ried out until satisfactory convergence was achieved isstellar weak rates calculation. Here P i is the probabil-ity of occupation of parent excited states and followsthe normal Boltzmann distribution. Due to the advancement of Radioactive Ion Beam(RIB) facilities worldwide (e.g. projectile- fragmenta-tion facilities at GANIL with powerful LISE3 separatorsfor in-flight isotope separation), the study of mediummass proton drip-line nuclei are now in experimentalreach (Pougheon et al. 1987; Borrel et al. 1992). As onemoves on the proton-rich side of the nuclear landscapethe Q energy window in the GT + (electron capture) di-rection increases and thereby allows access to large GTstrength. The pn-QRPA theory was used to calculatethe GT strength in the isospin raising direction, T > , for pf-shell nuclei at proton drip-line. The proton-richnuclei Mn, Fe, Co, Ni and Zn were studied inthis project and are also shown on the nuclear chart inFig. 1. These nuclei have an excess of four protons andrepresent the bound nuclei (Pougheon et al. 1987).The representative GT strength distributions in theelectron capture direction (GT + ) for Mn and Coare shown in Fig. 2 and Fig. 3, respectively. The en-ergy scale refers to excitation energy in daughter nu-clei. The energy position of the isobaric analogue state(IAS) is shown by dashed-line in Fig. 2 and Fig. 3. The E IAS for Mn, Fe, Co, Ni, and Zn is 25.06MeV, 19.38 MeV, 25.75 MeV, 19.99 MeV, 22.09 MeV,respectively. The ground state Q ec values and mass ex-cess of these nuclei with T z = − Q EC value is represented by an arrow and itis evident that most of the GT strength structure ap-pears within the Q window of the reaction. A sharpdecline in the GT strength can be seen at high excita-tion energy beyond the Q window in the daughter Cr(Fig.2). In this work a total unquenched and quenchedstrength of 12.45 and 6.82, respectively, was calculatedas compared to the corresponding shell model values of12.89 and 6.98, respectively (Caurier et al. 1998).For even-even Fe, the pn-QRPA extracted an un-quenched and quenched total GT strength of 12.62 and6.91, respectively. This quenched value is close to thequenched total strength of 7.18 reported by (Caurier etal. 1998). The mass-excess of the Fe ground state,taken from (Audi et al. 2003), is taken to be (-18160 ±
70) keV.For the odd-odd nucleus Co the GT strength(shown in Fig. 3) is fragmented over many states in thedaughter Fe due to the correlations effect among thenucleons. The over all morphology of the strength is ingood agreement with calculated GT strength of (Cau-rier et al. 1998). They used the KB3 interaction (Wanget al. 1988) to calculate the GT strength and theseinteractions are modified version of Kuo-Brown inter-actions. The pn-QRPA extracted more GT strengththan calculated by shell model (Caurier et al. 1998).This decrease in their calculated GT strength could bedue to the truncated model space used in their cal-culation. It is noted that bulk of the GT strength isdistributed below or very close to the Q window. Thetotal GT strength is very crucial in the high densityregions in stellar core where the electrons are degener-ate and their Fermi energy grows faster than the corre-sponding Q value. The pn-QRPA and shell model total unquenched and quenched GT strengths for the protondrip-line nuclei are given in Table 1. These strengthswere computed with the additional assumption that theproton that has been converted to a neutron lie withinthe same major shell. It is noted that the pn-QRPAcalculated GT strength is in reasonable agreement withcorresponding shell model results. It is further notedthat for heavier nuclei the reported strengths are muchbigger than shell model results (for reasons mentionedabove). The pn-QRPA calculated GT strength is com-pared with experimental strength wherever available.Due to lack of measurements of the ground state β − transition, one can use the data for transitions of themirror nucleus given by (p,n) charge-exchange reac-tions. It is well known that the GT strength is expectedto be same in both isospin direction and the symmetryequation B ( GT + ) = B ( GT − ) can be used for compar-ison. The total GT strengths observed in the (p, n)charge-exchange reaction (Wang et al. 1988; Rapaportet al. 1983) for the mirror nuclei Cr and Fe with T z = +2 are used for comparison in Table 1. It is seenthat the pn-QRPA result is in good agreement with theupper limit of the total strength observed in the (p, n)charge-exchange reaction for the case of A = 56. At thispoint, we reiterate the remarks in (Caurier et al. 1998)about the reliability of the experimental data of theseproton rich nuclei. The strong splitting of the strengthmakes the observation more difficult and thus imposesstrict limits for the background in the GT spectrum.The improvement in production rate of these protonrich nuclei can assist in the possible observation of theshape of the main components of the GT distributionby beta decay study.The increase of GT + strength with mass number isnatural as the number of protons increases with theincrease of mass number for the selected proton drip-line nuclei. The total strength for these nuclei (with T z = − ≤ A ≤
56) is plotted as afunction of mass number in Fig. 4. It is noted that totalstrength increases roughly linearly with the increase ofproton number. This is attributed to the correlationeffects between the nucleons in-built within the QRPAmodel. The number of protons increases as one movesfrom Mn to Zn along the proton drip-line. This re-sults in the enhancement of Coulombic repulsion amongthe protons and consequently the Fermi surface of theproton’s orbital is pushed up and is close to the contin-uum. β + -Decay Rates forProton Drip-Line Nuclei Bethe and coworkers (Bethe et al. 1979) pointed out theimportance of GT transitions for electron capture on heavy nuclei in the presupernovae and collapse phasesof stellar core. The entropy per baryon in stellar coreis very important as it determines the free proton frac-tion. The proton-rich pf-shell nuclei may be used inthe studies of nucleosynthesis and energy generation inX-ray bursts and other rp-process sites (Pruet et al.2003; Wallace et al. 1981). The authors in (Pruet etal. 2003) argued the need of weak rates on proton-richnuclei up to mass number 110 for nucleosynthesis in therp-process.Electron capture rates based on independent particlemodel led to incorrect conclusions due to the Pauli-blocking of the GT transitions. This Pauli-blockingof the GT transitions is overcome by the correlationseffect among the nucleons (Caurier et al. 1999), andtemperature effects (Fuller et al. 1980; Cooperstein etal. 1984). This correlation effect is taken into accountfor calculation of GT transitions in this project. Theelectron capture and β + decay rates for proton-rich pf-shell nuclei with T z = − ≤ A ≤
56, at various various densities and temperatures, aregiven in Tables 2-6. It has been noted that the β + de-cay rates dominate the electron capture rates at lowtemperature and low densities in the stellar interior.However, for densities ≥ gcm − , the electron cap-ture process dominates even at low temperatures. Stel-lar electron capture rates for proton-rich pf-shell nuclei Mn, Fe, Co, Ni and Zn are shown in Figs. 5- 9,respectively. These proton rich nuclei are characterizedby their large decay-values of 17.10 MeV, 11.16 MeV,17.28 MeV, 11.27 MeV, and 12.87 MeV, respectively.On the other hand nuclei, closer to the valley of sta-bility, have usually smaller decay-values. The electroncapture rates for all five cases (Figs. 5- 9) follow a moreor less similar behavior which we explain below. It isevident from discussion on GT strength in previous sec-tion that a fair proportion of the GT strength lies atthe upper end of the Q-value window. This impedesthe electron capture rates in the low temperature anddensity regions of the stellar core. As the temperatureof the stellar matter increases in low density regions theparticipant nucleon will get a fair chance of occupyinghigher energy levels and in turn assists the electron cap-tures on the nucleon. For densities ρ ≤ g − cm − ,the stellar weak rates are dominated by Fermi (appli-cable to beta decay only) and GT transitions. Thereare two quantities that drive the electron capture rates:the chemical potential of the electrons and the nuclearQ-value. Chemical potential grows like ρ / (Nabi andRahman 2005) and this growth of the chemical poten-tial is much faster than the Q-value of the nuclei in thestellar core. Thus, the impedance posed by large Q-values of the reactions for these nuclei is overcome by the fast growth of the chemical potential of the electronsand consequently led to the enhancement of the elec-tron capture rates. At low densities, where the chemi-cal potential approximately equals nuclear Q-value, thecapture rates are very sensitive to the available phasespace and detailed description of the GT strength dis-tribution is then desirable. It is noted that in the lowdensity region the beta decays compete with the elec-tron capture rates. These beta decays during and af-ter silicon shell burning increase the value of Y e in thestellar core and cool the core efficiently as against com-peting electron capture rates. The sensitivity to thephase space and details of the GT strength distributionis less important as a result of fast growth in chemicalpotential at high densities. The capture rates are nomore dependent on the details of GT strength distribu-tion. They rather depend on the total GT strength. Inthis scenario crude nuclear models might also be ableto give an estimate of the stellar capture rates.The experimentally measured and calculated half-lives for proton-rich pf-shell nuclei with T z = − ≤ A ≤
56 are mutually compared in Ta-ble 7. The pn-QRPA results are also compared with theresults of the gross theory of beta decay of (Tachibanaet al. 1988) and (M¨oller et al. 1997). In addition ourcalculated half-lives are also compared with the shellmodel calculation (Caurier et al. 1998). The results ofthe pn-QRPA, for cases of Co and Zn, are in goodagreement with the shell model results and deviationsare found for the rest of the cases.
Recent advancements in the accelerator driven technol-ogy has led to noticeable improvement in nuclear in-puts for core-collapse supernova models. It has beennoted that the collapse phase is dominated by electroncapture rates on nuclei rather than on free protons.The GT + strength and associated electron capture/ β + -decay rates were calculated within the domain of thepn-QRPA model. These weak interaction mediatedrates are key nuclear physics input to simulation codesand a reliable and microscopic calculation of these ratesfrom ground-state and excited states is desirable. Thetransitions in GT + direction and associated electroncapture/ β + -decay rates for pf-shell nuclei are impor-tant from astrophysical point of view because these nu-clei are key input in the modeling of the explosion dy-namics of massive stars. Our calculation shows that β + -decay rates dominate electron capture rates at lowstellar temperatures and densities.The pn-QRPA theory with improved model param-eters was used to calculate weak-interaction mediated rates for the proton drip-line nuclei. These calculationswere carried out in a luxurious model space of 7 (cid:126) ω .The pn-QRPA calculated half-lives of Co and Znare in reasonable agreement with shell model calcula-tion but differ in other cases. The GT strength distri-bution is important for the description of the electroncapture/ β + rates related to pre-supernova and super-nova conditions. The pn-QRPA’s total unquenched andquenched GT strength for proton drip-line nuclei arecompared with shell model values and with the avail-able experimental data. The pn-QRPA calculated GTstrengths are in reasonable agreement with the shellmodel and experimental data. This may affect the evo-lution timescale and dynamics of collapsing supermas-sive stars as the capture rates are dependent on thetotal GT strength in the high temperature and den-sity regions in stellar interior. It has been found thattotal GT strength increases linearly with increase ofproton numbers as expected. The electron capture/ β + -decay rates for these proton drip-line nuclei, for densityand temperature scale relevant to astrophysical sce-nario, may be requested from the authors as ASCIIfiles. Core-collapse simulators are encouraged to em-ploy these rates in simulation codes to check for possibleinteresting outcomes. References
Anderson, B.D., Lebo, C., Baldwin, A.R., Chittrakarn, T.,Mandey, R., Watson, J.W. 1990, Phys. Rev. C, , 1474Aufderheide, M.B., Bloom, S.D., Mathews, G.J., Resler,D.A. 1996, Phys. Rev. C, , 3139-3142Audi, G., Wapstra, A.H., Thibault, C. 2003, Nucl. Phys. A, , 337Audi, G., Bersillon, O., Blachot, J., Wapstra, A.H. 2003,Nucl. Phys. A, , 3-128Bethe, H.A., Brown, G.E., Applegate, J., Lattimer, J. 1979,Nucl. Phys. A, , 487Bethe, H.A. 1990, Rev. Mod. Phys., , No. 4, 801-866Borrel, V., Anne, R., Bazin, D., Borcea, C. Chubarian, G.G., Del Moral, R., Dtraz, C., Dogny, S., Dufour, J.P.,Faux, L., Fleury, A., Fifield, L.K., Guillemaud-Mueller,D., Hubert, F., Kashy, E., Lewitowicz, M., Marchand,C., Mueller, A.C., Pougheon, F., Pravikoff, M.S., Saint-Laurent, M.G., Sorlin, O. 1992, Z. Phys. A - Hadrons andNuclei, , 135Cole, A.L., Anderson, T.S., Zegers, R.G.T., Austin, SamM., Brown, B.A., Valdez, L., Gupta, S., Hitt, G.W.,Fawwaz, O. 2012, Phys. Rev. C, , 015809Cooperstein, J., Wambach, J. 1984, Nucl. Phys. A, ,591-620Caurier, E., Didierjean, F., Nowacki, F., Walter, G. 1998,Phys. Rev. C, , 2316Caurier, E., Langanke, K., Mart´ınez-Pinedo, G., Nowacki,F. 1999, Nucl. Phys. A, , 439-452Dossat, C., Blank, B. et al. 2007, Nucl. Phys. A, , 18El-Kateb, S., Jackson, K.P., Alford, W.P., Abegg, R.,Azuma, R.E., Brown, B.A., Celler, A., Frekers, D.,Husser, O., Helmer, R., Henderson, R.S., Hicks, K.H.,Jeppesen, R., King, J.D., Shute, G.G., Spicer, B.M.,Trudel, A., Raywood, K., Vetterli, M., Yen, S. 1994, Phys.Rev. C, , 3129Faux, L., et al., 1994, Phys. Rev. C , 2440Faux, L., et al., 1996, Nucl. Phys. A , 167Fuller, G.M., Fowler, W.A., Newman, M.J. 1980, Astro-phys. J. Suppl. Ser., , 447 (1980).Fuller, G.M., Fowler, W.A., Newman, M.J. 1982a, Astro-phys. J. Suppl. Ser., , 715Fuller, G.M., Fowler, W.A., Newman, M.J. 1985, Astro-phys. J., , 1-16Gove, N.B., Martin, M.J. 1971, Nucl. Data Tables, , 205Hirsch, M., Staudt, A., Muto, K., Klapdor-Kleingrothaus,H.V. 1993, At. Data Nucl. Data Tables, , 165Heger, A., Woosley, S.E., Mart´ınez-Pinedo, G., Langanke,K. 2001, The Astrophysical Journal, , 307-325Jokinen, A., Nieminen, A., yst, J., Borcea, R., Caurier, E.,Dendooven, P., Gierlik, M., Grska, M., Grawe, H., Hell-strm, M., Karny, M., Janas, Z., Kirchner, R., La Com-mara, M., Mart´ınez-Pinedo, G., Mayet, P., Penttil, H.,Plochocki, A., Rejmund, M., Roeckl, E., Sawicka, M.,Schlegel, C., Schmidt, K., Schwengner, R. 2002, Euro.Phys. J. Direct A , 1Juodagalvis, A., Langanke, K., Hix, W.R., Mart´ınez-Pinedo, G., Sampaio, J.M. 2010, Nucl. Phys. A, ,454-478 Jin-Jing Liu 2013, Mon. Not. R. Astron. Soc., , 1108-1113Kar, K., Ray, A., Sarkar, S. 1994, The Astrophysical Jour-nal, , 662-683Langanke, K., Mart´ınez-Pinedo, G. 1998, Phys. Lett. B, , 19Langanke, K., Mart´ınez-Pinedo, G. 1999, Phys. Lett. B, , 187M¨oller P., Nix J.R. 1981, At. Data Nucl. Data Tables, ,165M¨oller, P., Nix, J.R., Kratz, K.-L. 1997, At. Nucl. DataTables, , 131Muto, K., Bender, E., Oda, T., Klapdor-Kleingrothaus, H.V. 1992, Z. Phys. A, , 407Mart´ınez-Pinedo, G., Poves, A., Caurier, E., Zuker, A.P.1996, Phys. Rev. C, , 2602-2605Nilsson, S.G. 1955, Mat. Fys. Medd. Dan. Vid. Selsk, ,16Nabi J.-Un, Klapdor-Kleingrothaus H.V. 1999a, Eur. Phys.J. A, , 337Nabi J.-Un, Klapdor-Kleingrothaus H.V. 2004, At. DataNucl. Data Tables, , 237Nabi, J.-U., Rahman, M.-U. 2005, Phys. Lett. B, , 190-196Nabi J.-Un. 2009, Eur. Phys. J. A, , 223Nabi J.-Un. 2012, Astrophys. and Space Sci., , 305-315Nabi J.-Un. and Johnson C.W. 2013, J. Phys. G, , 065202Osterfeld, F. 1992, Rev. Mod. Phys. , 491Pougheon, F., Jacmart, J.C., Quiniou, E., Anne, R., Bazin,D., Borrel, V., Galin, J., Guerreau, D., Guillemaud-MueUer, D., Mneller, A.C., Roeckl, E., Saint-Laurent,M.G., Dtraz, C. 1987, Z. Phys. A - Atomic Nuclei, ,17Pruet, J., Fuller, G.M. 2003, Astrophys. J. Suppl. , 189Rapaport, J., Taddeucci, T., Welch, T.P., Gaarde, C.,Larsen, J., Horen, D.J., Sugarbaker, E., Koncz, P., Fos-ter, C.C., Goodman, C.D., Goulding, C.A., Masterson,T. 1983, Nucl. Phys. A, , 371Ronnquist, T., Cond´e, H., Olsson, N., Remstr (cid:12) o m, E.,Zorro, R., Blomgreen, J., H ˙ a kansson, A., Ringbom, A.,Tibell, G., Jonsson, O., Nilsson, L., Renberg, P.U., Vander Werf, S.Y., Unkelbach, W., Brady, F.P. 1993, Nucl.Phys. A, , 225Rahman, M.-U, Nabi J.-U. 2013, Astrophys. and Space Sci., , 427-435Rahman, M.-U, Nabi, J.-U. 2014, Astrophys. Space Sci., , 235-242, DOI:10.1007/s10509-014-1831-0.Sarriguren, P., 2013, Phys. Rev. C, , 045801Staudt, A., Hirsch, M., Muto, K., Klapdor-Kleingrothaus,H. V. 1990, Phys. Rev. Lett., , 1543Tachibana, T., Yamada, M., Nakata, K., Report of Sci. andRes. Lab., Waseda University, No. 88-3 (1988)Timmes, F.X., Woosley, S.E., Weaver, T.A. 1996, Astro-phys. J., , 834Vetterli, M.C., H (cid:12) a usser, O., Abegg, R., Alford, W.P.,Celler, A., Frekers, D., Helmer, R., Henderson, R., Hicks,k.H., Jackson, K.p., Jappesen, R.G., Miller, C.A, Ray-wood, K., Yen, S. 1989, Phys. Rev. C, , 559Wallace, R.K., Woosley, S.E. 1981, Astrophys. J. Suppl., , 389-420Wang, D., Rapaport, J., Horen, D.J., Brown, B.A., Gaarde,C., Goodman, C.D., Sugarbaker, E., Taddeucci, T.N.1988, Nucl. Phys. A, 480, 285 This manuscript was prepared with the AAS L A TEX macros v5.2.
Table 1
Comparison of the pn-QRPA calculated GT + total strength with shell model calculation (Caurier et al. 1998)and measured GT strength for the pf-shell nuclei with T z = − ≤ A ≤ Nucleus Unquenched Unquenched Quenched Quenched Σ B ( GT ± ) experimentalΣ B ( GT + ) pn − QRP A Σ B ( GT + ) SM Σ B ( GT + ) pn − QRP A Σ B ( GT + ) SM (mirror nuclei) Mn 12.45 12.89 6.82 6.98 − Fe 12.62 13.26 6.91 7.18 − Co 16.73 14.68 9.16 7.95 − Ni 17.51 15.33 9.59 8.30 5.9 ± Zn 21.91 16.69 11.99 9.04 9.9 ± Table 2 β + -decay rates and electron capture rates on Mn for different selected densities and temperatures in stellarmatter. T represents the temperature in 10 K and ρY e denotes the stellar density in units of g/cm . Rates are given inlog to base 10 scale and have units of s − . ρY e T λ β + λ ec ρY e T λ β + λ ec
10 0.01 0.852 -5.000 10
10 1.011 0.68110 30 0.977 2.536 10
30 0.977 2.54010
10 1.009 0.588 10
10 1.019 2.37610
30 0.977 2.536 10
30 1.012 2.84910
10 1.009 0.589 10
10 1.019 5.15610
30 0.977 2.536 10
30 1.048 5.216
Table 3
Same as Table 2 but for Fe. ρY e T λ β + λ ec ρY e T λ β + λ ec
10 0.01 -0.020 -5.695 10
10 0.715 0.32510 30 1.637 2.687 10
30 1.638 2.69110
10 0.714 0.232 10
10 0.720 2.06510
30 1.637 2.688 10
30 1.661 2.99710
10 0.714 0.233 10
10 0.720 4.96110
30 1.637 2.688 10
30 1.685 5.302
Table 4
Same as Table 2 but for Co. ρY e T λ β + λ ec ρY e T λ β + λ ec
10 0.01 1.510 -4.473 10
10 1.602 1.11310 30 1.550 2.842 10
30 1.550 2.84610
10 1.600 1.020 10
10 1.608 2.76910
30 1.550 2.842 10
30 1.583 3.15110
10 1.600 1.021 10
10 1.608 5.39610
30 1.550 2.842 10
30 1.617 5.435 Table 5
Same as Table 2 but for Ni. ρY e T λ β + λ ec ρY e T λ β + λ ec
10 0.01 0.810 -5.162 10
10 0.973 0.60110 30 1.549 2.866 10
30 1.549 2.87010
10 0.972 0.508 10
10 0.978 2.32710
30 1.549 1.549 10
30 1.574 3.17810
10 0.972 0.508 10
10 0.978 5.17010
30 1.549 2.866 10
30 1.598 5.517
Table 6
Same as Table 2 but for Zn. ρY e T λ β + λ ec ρY e T λ β + λ ec
10 0.01 1.399 -4.429 10
10 1.392 1.08210 30 1.331 2.822 10
30 1.332 2.82610
10 1.390 0.990 10
10 1.399 2.74510
30 1.331 2.822 10
30 1.365 3.13210
10 1.390 0.991 10
10 1.399 5.38110
30 1.331 2.822 10
30 1.399 5.417 Table 7
Comparison of half-lives of the pn-QRPA model with experimental: (a) Faux et al. 1994, (b) Borrel et al. 1992,(c) Faux et al. 1996, (d) Dossat et al. 2007, (e) Audi et al. 2003b and other theoretical calculations : (f) Caurier et al. 1998,(g) Tachibana et. al 1988 and (h) M¨oller et.al 1997 for the pf-shell nuclei with T z = − ≤ A ≤ s − . Nucleus [T / ] Exp. [T / ] ( e ) [T / ] pn − QRP A [T / ] ( f ) [T / ] ( g ) [T / ] ( h )46 Mn 41 +7( a ) −
37 97.4 29 53 15 Fe 44 ± ( b )
44 125.8 53 48 60 Co 44 ± ( b )
44 21.4 27 36 47 Ni 38 ± ( c )
38 107.3 50 35 7740 . ± ( d )56 Zn −
36 27.6 24 24 83
Fig. 1 (Color online) The studied proton-rich pf-shell nuclei with T z = − Fig. 2 (Color online) Calculated GT strength distribution for Mn. E j refers to the excitation energy (MeV) in daughter Cr. The isobaric analog state (IAS) is shown with a dashed line while the Q-value for the electron capture reaction isshown by solid arrow. Fig. 3 (Color online) Calculated GT strength distribution for Co. The energy scale refers to the excitation energy(MeV) in daughter Fe. The isobaric analog state (IAS) is shown with a dashed line while the Q-value for the electroncapture reaction is shown by solid arrow. Fig. 4 (Color online) Calculated total GT strength as a function of mass number for the T z = − ≤ A ≤ Fig. 5 (Color online) Stellar electron capture rates for Mn as a function of temperature. Electron capture (EC) ratesare given in log (to base 10) scale in units of s − . T represents temperature in units of 10 K. Densities in inset are givenin units of g-cm − . Fig. 6 (Color online) Same as Fig. 5 but for Fe. Fig. 7 (Color online) Same as Fig. 5 but for Co. Fig. 8 (Color online) Same as Fig. 5 but for Ni. Fig. 9 (Color online) Same as Fig. 5 but for56