Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 398796, 11 pages http://dx.doi.org/10.1155/2015/398796
Research Article Electron Capture Cross Sections for Stellar Nucleosynthesis
P. G. Giannaka and T. S. Kosmas
Division of Theoretical Physics, University of Ioannina, 45100 Ioannina, Greece
Correspondence should be addressed to P. G. Giannaka; [email protected]
Received 11 July 2014; Accepted 8 October 2014
Academic Editor: Athanasios Hatzikoutelis
Copyright Β© 2015 P. G. Giannaka and T. S. Kosmas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.
In the first stage of this work, we perform detailed calculations for the cross sections of the electron capture on nuclei under laboratory conditions. Towards this aim we exploit the advantages of a refined version of the proton-neutron quasiparticle random- phase approximation (pn-QRPA) and carry out state-by-state evaluations of the rates of exclusive processes that lead to any of the accessible transitions within the chosen model space. In the second stage of our present study, we translate the abovementioned β π -capture cross sections to the stellar environment ones by inserting the temperature dependence through a Maxwell-Boltzmann 66 distribution describing the stellar electron gas. As a concrete nuclear target we use the Zn isotope, which belongs to the iron group nuclei and plays prominent role in stellar nucleosynthesis at core collapse supernovae environment.
1. Introduction is the fact that electron capture on nuclei (specifically on nuclei of the pf shell) plays a key role [14, 15]. Weak interaction processes occurring in the presence of Intheearlystageofcollapse(fordensitieslowerthanafew 10 β3 nuclei under stellar conditions play crucial role in the late 10 gcm ), the electron chemical potential is of the same β stagesoftheevolutionofmassivestarsandinthepresu- order of magnitude as the nuclear π value, and the π -capture pernova stellar collapse [1β6].Asitisknown,thecoreofa cross sections are sensitive to the details of GT strength dis- massive star, at the end of its hydrostatic burning, is stabilized tributions in daughter nuclei. For this reason, some authors by electron degeneracy pressure as long as its mass does not restrict the calculations only to the GT strength and evaluate β exceed an appropriate mass (the Chandrasekhar mass limit, π -captureratesonthebasisoftheGTtransitions(atthese π π Ch)[6β10].Whenthecoremassexceeds Ch, electron densities, electrons are captured mostly on nuclei with mass degeneracy pressure cannot longer stabilize the center of the number π΄β€60)[9β12, 15, 16]. Various methods, used for β star and the collapse starts. In the early stage of collapse calculating π -capture on nuclei during the collapse phase, electrons are captured by nuclei in the iron group region have shown that this process produces neutrinos with rather [6, 10]. low energies in contrast to the inelastic neutrino-nucleus During the presupernova evolution of core collapse reactions occurring in supernova [17β20]. These neutrinos supernova, the Fermi energy (or equivalently the chemical escape the star carrying away energy and entropy from the potential) of the degenerate electron gas is sufficiently large to core which is an effective cooling mechanism of the exploding πΈ πΈ πβ overcome the threshold energy thr ( thr is given by negative massive star [21]. For higher densities and temperatures, π values of the reactions involved in the interior of the stars) capture occurs on heavier nuclei π΄β₯65[8, 10, 13β15]. As [11] and the nuclear matter in the stellar core is neutronized. a consequence, the nuclear composition is shifted to more This high Fermi energy of the degenerate electron gas leads neutron-rich and heavier nuclei (including those with π> β to enormous π -capture on nuclei and reduces the electron 40) which dominate the matter composition for densities 10 β3 to baryon ratio ππ [12, 13]. In this way, the electron pressure is larger than about 10 gcm [1, 15, 21, 22]. reducedandtheenergyaswellastheentropydrop.Oneofthe The first calculations of stellar electron capture rates for important characteristics of the early pre-explosion evolution iron group nuclei have been performed by employing the 2 Advances in High Energy Physics independent particle model (IPM) [2β4]. Recently, simi- on the evaluation of the nuclear transition matrix elements lar studies have been addressed by using continuum RPA between the initial |πβ© and a final |πβ© nuclearstatesoftheform (CRPA) [25], large scale shell model [26, 27], RPA [11], and β σ΅¨ σ΅¨ πΊ σ΅¨ σ΅¨ so forth [28]. In the present work π -capture cross sections σ΅¨Μσ΅¨ π 3 βπqβ x σ΅¨Μσ΅¨ β¨π σ΅¨π»π€σ΅¨ πβ© = β β«π π₯π β¨π σ΅¨Jπσ΅¨ πβ© . (3) are obtained within a refined version of the quasiparticle β2 random phase approximation (QRPA) which is reliable for βππβπqβ x constructing all the accessible final (excited) states of the The quantity stands for the leptonic matrix ele- q daughter nuclei in the iron group region of the periodic ment written in coordinate space with being the 3- table [29β38]. For the description of the required correlated momentum transfer. For the calculation of these transition nuclear ground states we determine single-particle occupa- matrix elements one may take advantage of the Donnelly- tion numbers calculated within the BCS theory [29, 31, 32]. Walecka multipole decomposition which leads to a set of Our nuclear method is tested through the reproducibility eight independent irreducible tensor multipole operators of experimental muon capture rates relying on detailed containing polar-vector and axial-vector components [44] calculations of exclusive, partial, and total muon capture (see Appendix A). rates [23, 24, 39β42]. The agreement with experimental data In the present work, in (3) the ground state of the parent |πβ© provided us with high confidence level of our method and nucleus is computed by solving the relevant BCS equations we continued with the calculations of electron capture cross which give us the quasiparticle energies and the amplitudes π π sections in supernova conditions (where the densities and and that determine the probability for each single temperatures are high) using the pn-QRPA method. In this particle level to be occupied or unoccupied, respectively [31]. 66 paper, we performed calculations for Zn isotope (it belongs Towards this aim, at first, we consider a Coulomb corrected totheirongroupnuclei)thatplaysprominentroleincore Woods-Saxon potential with a spin orbit part as a mean field collapse supernovae stellar nucleosynthesis [18, 19, 43]. for the description of the strong nuclear field [45, 46]. For Our strategy in this work is, at first, to perform extensive the latter potential we adopt the parameterization of IOWA calculations of the transition rates for all the abovementioned group [47]. Then, we use as pairing interaction the monopole part of the Bonn C-D one meson exchange potential. The nuclear processes, assuming laboratory conditions, and then 66 to translate these rates to the corresponding quantities within renormalization of this interaction, to fit in the Zn isotope, ππ,π π(π) stellar environment through the use of an appropriate con- is achieved through the two pairing parameters pair, for volution procedure [9, 14, 15, 21, 26]. To this purpose, we proton(neutron)pairs,andthevaluesofwhicharetabulated assume that leptons under such conditions follow Maxwell- in Table 1. ππ,π Boltzmann energy distribution [9, 26]. As it is well known, the pairing parameters, pair,are determined through the reproduction of the energy gaps, Ξexp 2. Construction of Nuclear Ground and π,π, from neighboring nuclei as follows (3-point formula): 1 Excited States Ξexp =β [π (π΄β1,π) β2π (π΄, π) +π (π΄+1,π)], π 4 π π π Electrons of energy πΈπ arecapturedbynucleiinteracting πβ 1 weakly with them via boson exchange as follows: Ξexp =β [π (π΄β1,πβ1) (4) π 4 π β β (π΄, π) +π σ³¨β (π΄, π) β1 + ]π. (1) β2ππ (π΄, π) +ππ (π΄+1,π+1)],
The outgoing ]π neutrino carries energy πΈ] while the daughter where ππ and ππ are the experimental separation energies nucleus (π΄, π β 1) absorbs a part of the incident electron for protons and neutrons, respectively, of the target nucleus energy given by the difference between the initial πΈπ and the (π΄, π) and the neighboring nuclei (π΄Β±1,πΒ±1)and (π΄Β±1, π). final πΈπ nuclear energies as πΈ] =πΈπ βπΈπ. For the readers convenience in Table 2 we show the values of 66 The nuclear calculations for the cross sections of reaction experimental separation energies for the target 30Zn and the 65 67 65 67 (1) start by writing down the weak interaction Hamiltonian neighboring nuclei 29Cu, 31Ga, 30Zn, and 30Zn. Μ lept Hπ€ which is given as a product of the leptonic, ππ , Subsequently, the excited states |πβ© of the studied daugh- π 66 and the hadronic, JΜ , currents (current-current interaction ter nucleus Cu are constructed by solving the pn-QRPA Hamiltonian) as follows: equations [29β38], which are written as follows in matrix form [29]: πΊ HΜ = πleptJΜπ, AB π] π] π€ π (2) ( )( )=Ξ©] ( ), β2 βB βA π] π½π π] (5)
] π where πΊ=πΊπΉ cos ππ with πΊπΉ and ππ being the well-known Ξ©π½π denotes the excitation energy of the QRPA state |π½] β© with weak interaction coupling constant and the Cabibbo angle, spin π½ and parity π. respectively [31, 32, 44]. The solution of (5) is an eigenvalue problem which From the nuclear theory point of view, the main task is provides the amplitudes for forward and backward scattering ] to calculate the cross sections of reaction (1)whicharebased π and π, respectively, and the QRPA excitation energies Ξ©π½π Advances in High Energy Physics 3
ππ ππ Table 1: Parameters for the renormalization of the interaction of proton pairs, pair, and neutron pairs, pair. They have been fixed in such a exp exp 66 way that the corresponding experimental gaps, Ξ π and Ξ π of Zn isotope, are quite accurately reproduced.
ππ ππ Ξexp Ξtheor Ξexp Ξtheor Nucleus pair pair π (MeV) π (MeV) π (MeV) π (MeV) 66 Zn 1.0059 0.9271 1.7715 1.7716 1.2815 1.2814
Table 2: The experimental separation energies in MeV for protons and neutrons of the targetπ΄ ( , π) and neighboring (π΄ Β± 1, π Β± 1) and (π΄ Β± 1, π) nuclei.
Nucleus ππ(π΄ β 1, π)π π (π΄, π)π π (π΄ + 1, π)π π (π΄ β 1, π β 1)π π (π΄, π)π π (π΄ + 1, π + 1) 66 Zn 7.979 11.059 7.052 7.454 8.924 5.269
[31β34]. In our method the solution of the QRPA equations is 2.5 π 2+ carried out separately for each multipole set of states |π½ β©. + For the renormalization of the residual 2-body inter- 2 1β 3+ action (Bonn C-D potential), the strength parameters, for 1+ + + β (π ) (π ) 1+ 4 2 the particle-particle pp and particle-hole ph interaction 3 2.0 3+ entering the QRPA matrices A and B, are determined β 2β β 5 1+ (separately for each multipolarity) from the reproducibility 4 1β 1β 1+ of the low-lying experimental energy spectrum of the final β β 5 + 5 + β 2 nucleus. The values of these parameters in the case of the 1β β 66 2 2 + 1 spectrum of Cu are listed in Table 3. 1+ 2 1.5 + At this point, it is worth mentioning that, for measuring 2+ + 66 2 + 1 3 + the excitation energies of the daughter nucleus Cu from 1+ 1 66 β β the ground state of the initial one Zn, a shifting of the 4 + 4
(MeV) 2 entire set of QRPA eigenvalues is necessary. Such a shifting π + is required whenever in the pn-QRPA a BCS ground state is + 1 1.0 3 3+ used, a treatment adopted by other groups previously [9, 40, 66Cu 29 48, 49]. The shifting, for the spectrum of the daughter nucleus 2+ 66 + Cu, is done in such a way that the first calculated value of 3+ 3 66 + + + + each multipole state of Cu (i.e. 11 ,21 ,...,etc.)approaches 4 4 0.5 + as close as possible the corresponding lowest experimental 2+ 2 multipole excitation. Table 4 shows the shifting applied to our 1+ 1+ QRPA spectrum for each multipolarity of the parent nucleus 3+ 3+ 66 + Zn. We note that a similar treatment is required in pn- 2+ 2 QRPA calculations performed for double-beta decay studies where the excitations derived for the intermediate odd-odd 0.0 nucleus (intermediate states) through π-π and π-π reactions |gsβ© β‘1|Jπβ©= | +β© from the neighboring nuclei, left or right nuclear isotope, do QRPA exp. not match to each other [48, 49]. The resulting low-energy spectrum, after using the parameters of Tables 1 and 3 and the Figure 1: Comparison of the theoretical excitation spectrum (result- ing from the solution of the QRPA eigenvalue problem) with the shifting shown in Table 4, agrees well with the experimental 66 one (see Figure 1). low-lying (up to about 3 MeV) experimental one for Cu nucleus. We must also mention that, usually, in nuclear structure Asitcanbeseen,theagreementisverygoodbelow1MeVbutfor calculations we test a nuclear method in two phases: first higher excitation energies it becomes moderate. through the construction of the excitation spectrum as dis- cussedbeforeandsecondthroughthecalculationsofelectron thepn-QRPAmethod.Therequirednuclearmatrixelements scattering cross sections or muon capture rates. Following between the initial |π½πβ© and the final |π½πβ© states are determined the above steps, we test the reproducibility of the relevant by solving the BCS equations for the ground state [29, 31, 32] experimental data for many nuclear models employed in and the pn-QRPA equations for the excited states [31β34](see nuclear applications (nuclear structure and nuclear reactions) Section 2). For the calculations of the original cross sections, andinnuclearastrophysics[1, 9]. a quenched value of ππ΄ (see Appendix B) is considered which subsequently modifies all relevant multipole matrix elements 3. Results and Discussion [23, 24, 50, 51]. At this point of the present work and in order to increase In this work we perform detailed cross section calculations the confidence level of our method, we perform total muon 66 for the electron capture on Zn isotope on the basis of capture rates calculations [23, 24, 39β42]. The comparison 4 Advances in High Energy Physics
(π ) (π ) Table 3: Strength parameters for the particle-particle pp and particle-hole ph interaction for various multipolarities (for the rest of π Β± 66 multipolarities, π½ β€5 , the bare 2-body interaction has been used), in the case of the spectrum of Cu nucleus.
Positive parity states Negative parity states π½π 0+ 1+ 2+ 3+ 4+ 1β 2β 3β 4β π pp 0.827 0.547 0.686 0.854 1.300 0.994 0.200 0.486 0.622 π ph 0.336 0.200 1.079 0.235 0.200 1.200 0.200 1.200 1.200
Table 4: The shift (in MeV) applied on the spectrum (seperately of where Ξ¦1π representsthemuonwavefunctioninthe1π 66 each multipole set of states) of Cu isotope, daughter nucleus of the 66 muonic orbit. The operators in (6) are refered to as Coulomb electron capture on Zn. Μ Μ Μel Mπ½,longitudinalLπ½, transverse electric Tπ½ ,andtransverse Μmag Positive parity states Negative parity states magnetic Tπ½ multipole operators (see Appendix A). The + β π 0 0.90 0 5.00 factor π in (6) takes into consideration the nuclear recoil β1 + β π =(1+π/π ) π 1 2.50 1 6.80 which is written as π π targ ,with targ being + β 2 2.55 2 3.85 the mass of the target (parent) nucleus. + β Duetothefactthattherearenoavailabledatainthe 3 2.50 3 2.60 literature for exclusive muon capture rates, the test of our 4+ 4β 1.75 3.55 method is realized by comparing partial and total muon cap- + β 5 0.55 5 3.00 ture rates with experimental data and other theoretical results [23, 24]. Towards this purpose, our second step includes β calculations of the partial π -capture rates for various low- π Β± with experimental and other theoretical results is shown spin multipolarities, Ξ π½π (for π½ β€4), in the studied in Section 3.1.Afterwards,westudyindetailtheelectron nucleus. These partial rates are found by summing over the capture process as follows. (i) Initially we consider laboratory contributions of all the individual multipole states of the conditions; that is, the initial (parent) nucleus is considered in studied multipolarity as follows: the ground state and no temperature dependence is assumed (see Section 3.2.1). (ii) Second, we consider stellar conditions; Ξ π½π = βΞ ππ π βπ½ π thatis,theparentnucleusisassumedtobeinanyinitial π β excited state and due to the π -capture process it goes to any final excited state of the daughter nucleus. At these σ΅¨ σ΅© σ΅© σ΅¨2 =2πΊ2β¨Ξ¦ β©2 [βπ2 π σ΅¨β¨π½π σ΅©(MΜ β LΜ )σ΅© 0+ β©σ΅¨ conditions it is necessary to take into account the temperature 1π π πσ΅¨ π σ΅© π½ π½ σ΅© ππ σ΅¨ dependence of the cross sections (see Section 3.2.2)[10]. [ π
66ππ σ΅¨ σ΅© σ΅© σ΅¨2 3.1. Calculations of Muon Capture Rates for . Despite the +βπ2 π σ΅¨β¨π½π σ΅©(TΜel β TΜmagn)σ΅© 0+ β©σ΅¨ ] π πσ΅¨ π σ΅© π½ π½ σ΅© ππ σ΅¨ fact that the muon capture on nuclei does not play a crucial π role in stellar-nucleosynthesis, it is, however, important to ] start our study from this process since the nuclear matrix (7) elements required for an accurate description of the π-capture π (π runs over all states of the multipolarity |π½ β©). We also esti- are the same for all semileptonic charge-changing weak mate the percentage (portion) of their contribution into the interaction processes. In addition, the excitation spectrum of total π-capture rate for the most important multipolarities. the daughter (π΄, π β 1) nucleus, as we saw before, is in good In Table 5 we tabulate the individual portions of the low-spin agreement with the experimental data. π Β± multipole transitions (π½ =4)intothetotalmuoncapture The calculations of the muon capture rates are performed β rate. As it can be seen, the contribution of the 1 multipole in three steps. In the first step we carry out realistic state- transitions is the most important multipolarity exhausting by-state calculations of exclusive ordinary muon capture 66 about 44% of the total muon capture rate. Such an important (OMC) rates in Zn isotope for all multipolarities with 16 48 π½π β€5Β± contribution was found in Oand Ca isotopes studied in (higher multipolarities contribute negligibly). The [41]. appropriate expression for the exclusive muon capture rates In the last step of testing our method, we evaluate total is written as follows: 66 muon capture rates for the Zn isotope. These rates are Ξ ππ π βπ½ β‘Ξπ½π obtained by summing over all partial multipole transition π π π Β± rates (up to π½ =4 ) as follows: 2 2 2 =2πΊβ¨Ξ¦1π β© π πππ Ξ = βΞ π½π = ββΞ π½π . σ΅¨ σ΅© σ΅© σ΅¨2 (6) tot π (8) Γ[σ΅¨β¨π½π σ΅©(MΜ β LΜ )σ΅© 0+ β©σ΅¨ π½π π½π π σ΅¨ π σ΅© π½ π½ σ΅© ππ σ΅¨
σ΅¨ π σ΅© σ΅© + σ΅¨2 π +σ΅¨β¨π½ σ΅©(TΜel β TΜmagn)σ΅© 0 β©σ΅¨ ], For the sake of comparison, the abovementioned - σ΅¨ π σ΅© π½ π½ σ΅© ππ σ΅¨ capturecalculationshavebeencarriedoutusingthe Advances in High Energy Physics 5
Table 5: The percentage of each multipolarity into the total muon + βW (πΈπ,πΈ]) capture rate evaluated with our pn-QRPA method. π½β₯0
σ΅¨ σ΅© σ΅© σ΅¨2 Positive parity transitions Negative parity transitions σ΅¨ σ΅©Μ σ΅© σ΅¨ π π Γ{(1+πΌcos Ξ¦) σ΅¨β¨π½π σ΅©Mπ½σ΅© π½πβ©σ΅¨ π½ Portions (%)π½Portions (%) σ΅¨ σ΅© σ΅© σ΅¨ 0+ 0β 8.22 7. 9 4 2 σ΅¨ σ΅© σ΅© σ΅¨2 + β +(1+πΌ Ξ¦β2π Ξ¦) σ΅¨β¨π½ σ΅©LΜ σ΅© π½ β©σ΅¨ 1 21.29 1 44.21 cos sin σ΅¨ π σ΅© π½σ΅© π σ΅¨ 2+ 2β 2.85 13.32 π σ΅© σ΅© + β β[ (1+πΌ Ξ¦) +π]2 β¨π½ σ΅©LΜ σ΅© π½ β© 3 1.58 3 0.34 π cos Re π σ΅© π½σ΅© π + β 4 0.01 4 0.23 σ΅© σ΅© β σ΅©Μ σ΅© Γβ¨π½π σ΅©Mπ½σ΅© π½πβ© }} , quenched value ππ΄ = 1.135 [23, 24]. The results are listed in Table 6, where we also include the experimental total rates (9) and the theoretical ones of [23, 24]. Moreover, in Table 6 we where πΉ(π,π πΈ ) isthewell-knownFermifunction[18]. The show the individual contribution into the total muon capture π(πΈ ,πΈ )=πΈ2/(1 + πΈ /π ) Ξπ Ξπ΄ factor π ] ] ] π accounts for the nuclear rate of the polar-vector ( tot), the axial-vector ( tot), and the π Ξππ΄ recoil [8], π is the mass of the target nucleus, and param- overlap ( tot) parts. As it can be seen, our results obtained eters πΌ, π,andπ are given, for example, in [31]. The nuclear with the quenched ππ΄ coupling constant are in very good transition matrix elements between the initial state |π½πβ© and agreement with the experimental total muon capture rates Μ the final state |π½πβ© correspond to the Coulomb Mπ½π,longitu- (the deviations from the corresponding experimental rates LΜ TΜel are smaller than 7%). This agreement provides us with high dinal π½π, transverse electric π½π, and transverse magnetic Μmag confidence level for our method. Tπ½π multipole operators (discussed in Appendix A). From the energy conservation in the reaction (1), the πΈ 3.2. Electron Capture Cross Section. After acquiring a high energy of the outgoing neutrino ] is written as follows: confidence level for our nuclear method, we proceed with πΈ =πΈ βπ+πΈ βπΈ , themaingoalofthepresentstudywhichconcernsthecalcu- ] π π π (10) lations of the electron capture cross sections. As mentioned which includes the difference between the initial πΈπ and before, this includes original (see Section 3.2.1)andstellar the final πΈπ nuclear states. The π value of the process is electron capture investigations (Section 3.2.2). determined from the experimental masses of the parent (ππ) and the daughter (ππ) nuclei as π=ππ βππ [9]. 66 3.2.1. Original Electron Capture Cross Section on ππ Isotope. It is worth mentioning that for low momentum transfer, The original cross sections for the electron capture process in various authors use the approximation πβ0for all multi- 66 the Zn isotope are obtained by using the pn-QRPA method pole operators of (9). Then, the transitions of the Gamow- + + considering all the accessible transitions of the final nucleus Teller operator (GT =βπ ππ ππ)providethedominant 66 Cu. In the Donnelly-Walecka formalism the expression for contribution to the total cross section [9]. the differential cross section in electron capture by nuclei While performing detailed calculations for the original 66 reads [10] electron capture cross sections on Zn isotope we assumed 66 that (i) the initial state of the parent nucleus Zn is the ππ πΊ2 2π πΉ (π, πΈ ) |0+β© ππ = πΉcos π π ground state and (ii) the nuclear system is under πΞ© 2π (2π½π +1) laboratory conditions (no temperature dependence of the cross sections is needed). The cross sections as a function of the incident electron energy πΈπ are calculated with the Γ{βW (πΈ ,πΈ ) π ] use of realistic two-body interactions as mentioned before. π½β₯1 The obtained total original electron capture cross sections for 66 Zn target nucleus are illustrated in Figure 2 where the indi- 2 π Β± Γ{[1βπΌcos Ξ¦+πsin Ξ¦] vidual contributions of various multipole channels (π½ β€5 ) arealsoshown.TheelectroncapturecrosssectionsinFigure 2 σ΅¨ σ΅© σ΅© σ΅¨2 σ΅¨ σ΅© σ΅© σ΅¨2 exhibit a sharp increase by several orders of magnitude σ΅¨ σ΅©Μmagσ΅© σ΅¨ σ΅¨ σ΅©Μelσ΅© σ΅¨ Γ[σ΅¨β¨π½π σ΅©Tπ½ σ΅© π½πβ©σ΅¨ + σ΅¨β¨π½π σ΅©Tπ½ σ΅© π½πβ©σ΅¨ ] within the first few MeV above energy-threshold, and this σ΅¨ σ΅© σ΅© σ΅¨ σ΅¨ σ΅© σ΅© σ΅¨ + reflects the GT strength distribution. For electron energy (π +π ) πΈπ β₯10MeV the calculated cross sections show a moderate β[ π π (1βπΌ Ξ¦) βπ] π cos increase. From experimental and astrophysical point of view, the important range of the incident electron energy πΈπ is + up to 30 MeV. At these energies the 1 multipolarity has σ΅© σ΅© σ΅© σ΅© β Γ2 β¨π½ σ΅©TΜmagσ΅© π½ β©β¨π½ σ΅©TΜelσ΅© π½ β© } the largest contribution to the total electron capture cross Re π σ΅© π½ σ΅© π π σ΅© π½ σ΅© π sections [9, 10]. In the present work we have extended the 6 Advances in High Energy Physics
Table 6: Individual contribution of polar-vector, axial-vector, and overlap parts into the total muon-capture rate. The total muon capture 66 rates, obtained by using the pn-QRPA with the quenched value of ππ΄ = 1.135 for the medium-weight nucleus Zn, are compared with the available experimental data and with the theoretical rates of [23, 24]. Ξ Γ 6 π β1 Total muon capture rates tot( 10 ) Present pn-QRPA calculations Experiment Other theoretical methods Ξπ Ξπ΄ Ξππ΄ Ξ Ξexp Ξtheor Ξtheor Nucleus tot tot tot tot tot tot [23] tot [24] 66 Zn 1.651 4.487 β0.204 5.934 5.809 4.976 5.809
103 process is carried out on heavier and more neutron rich nuclei 66Zn Total π<40 πβ₯40 2 with and [8, 10, 13β15]. 10 In an independent particle picture, the Gamow-Teller 101 transitions (which is the most important in the electron
0 capture cross section calculations) are forbidden for these
) 10
2 nuclei [2β4]. However, as it has been demonstrated in several β1 cm 10 studies, GT transitions in these nuclei are unblocked by finite
β42 πβ β2 temperature excitations [21, 22]. At high temperatures, 10 10 ( 1.5 MeV, GT transitions are thermally unblocked as a result e
π π 10β3 of the excitation of neutrons from the pf-shell into the 9/2 orbital. β4 10 For astrophysical environment, where the finite tem- 10β5 perature and the matter density effects cannot be ignored (the initial nucleus is at finite temperature), in general, the 10β6 0 10 20 30 40 50 initial nuclear state needs to be a weighted sum over an E (MeV) appropriate energy distribution. Then, assuming Maxwell- e Boltzmann distribution of the initial state |πβ© in (9)[9, 26], β 0β 2β 4β the total π -capture cross section is given by the expression + + + 0 2 4 [10] 1β 3β 5β + + + 1 3 5 2 2 βπΈπ/(ππ) πΊπΉcos ππ (2π½π +1)π π(πΈπ,π)= βπΉ (π,π πΈ ) Figure 2: Original total cross sections of electron capture on 2π πΊ (π, π΄,) π 66 π the Zn (parent) nucleus calculated with pn-QRPA method as πΈ σ΅¨ σ΅¨ σ΅¨ σ΅¨2 (11) a function of the incident electron energy π. The individual σ΅¨ σ΅¨Μ σ΅¨ σ΅¨ π Β± 2 σ΅¨β¨π σ΅¨ππ½σ΅¨ πβ©σ΅¨ contributions of various multipole channels (for π½ β€5)arealso Γ β(πΈ βπ+πΈ βπΈ ) σ΅¨ σ΅¨ σ΅¨ σ΅¨ . π π π (2π½ +1) demonstrated. π,π½ π
The sum over initial states in the latter equation denotes a thermal average of levels, with the corresponding partition range of πΈπ up to 50 MeV since at higher energies (around function πΊ(π, π΄, π) [10]. The finite temperature induces the β 40 MeV) the contribution of other multipolarities like 1 , thermal population of excited states in the parent nucleus. In + β 0 ,and0 becomes noticeable and cannot be omitted (see the present work we assume that these excited states in the Figure 2). parent nucleus are all the possible states up to about 2.5MeV. From the study of the original electron capture cross Calculations involving in addition other states lying at higher sections we conclude that the total cross sections can be well energies show that they have no sizeable contribution to the approximated with the Gamow-Teller transitions only in the total electron capture cross sections. As mentioned before, for region of low energies [9β12, 15, 16].Forhigherincidentelec- the evaluation of the total electron capture cross sections, the tron energies the inclusion of the contributions originating use of a quenched value of ππ΄ is necessary [23, 24, 50, 51]. 2 from other multipolarities leads to better agreement [10]. Since the form factor πΉπ΄(π ) multiplies the four components of the axial-vector operator (see (A.2)β(A.5)), a quenched value of ππ΄ must enter the multipole operators generating 66 β Β± 3.2.2. Stellar Electron Capture on ππ Isotope. As it is well the pronounced excitations 0 ,1 ,..., and so forth. For this known, electron capture process plays a crucial role in late reason, in our QRPA calculations we multiplied the free stages of evolution of a massive star, in presupernova and in nucleon coupling constant ππ΄ = 1.262 by the factor 0.8 supernova phases [1β6]. In presupernova collapse, that is, at 10 β3 [23, 24, 50, 51]. densities πβ€10 gcm and temperatures 300 keV β€πβ€ The results coming out of the study of electron capture 800 keV, electrons are captured by nuclei with π΄β€60[9β cross sections under stellar conditions are shown in Figure 3, 12, 15, 16]. During the collapse phase, at higher densities πβ₯ where the same picture as in the original cross section 10 β3 10 gcm and temperatures π β 1.0 MeV, electron capture calculations, but now with larger contribution, is observed. Advances in High Energy Physics 7
3 35 10 + 66Zn Total 30 0 Polar vector 2 MeV 10 25 Eeβ =25.0 /MeV) 2 20 66Zn ( β, ) 66Cu 1 15 e e 10 cm MeV dπ/dπ + T = 0.5 10 2 0 β42 ) 10 5 2 10
( 0 cm 10β1 0 2 4 6 8 101214
β42 β2 π (MeV)
10 10 (
e MeV 20 π β3 T = 0.5 10 1+ Axial vector β4 15 10 + /MeV) 1 + 2 1 66Zn ( β, ) 66Cu β5 10 e e
10 cm + MeV
dπ/dπ 1 T = 0.5 + + + β6 β42 5 1 3 10 1 1+ 0β 10
0 10 20 30 40 50 ( (MeV) 0 Ee 0 2 4 6 8 101214 (MeV) 0β 2β 4β π 0+ 2+ 4+ 35 + 1β 3β 5β 30 0 Total + + + 25 1 3 5 /MeV) 2 20 + 66Zn (eβ, ) 66Cu 15 + 1 e 66 cm 1 + MeV dπ/dπ + 1 T = 0.5 Figure 3: Electron capture cross sections for the Zn parent nucleus 10 1 2β at high temperature (π = 0.5 MeV) in stellar environment obtained β42 5 10
( 0 assuming Maxwell-Boltzmann statistics for the incident electrons. 02468101214 The total cross section and the dominant individual multipole π Β± π (MeV) channels (π½ β€5) are demonstrated as functions of the incident πΈ electron energy π. Figure 4: Individual contributions of the polar-vector, (Ξ π), and axial-vector, (Ξ π΄), components and total electron capture rate as 66 functions of the excitation energy π ( Zn is the parent nucleus). + As discussed before, the dominant multipolarity is the 1 , which contributes by more than 40%tothetotalcrosssection. β cross sections, is required. This was performed by using a In the region of low energies (up to 30 MeV), the total π - special code appropriate for matrices [33]. In the model space 66 π Β± capture cross section can be described by taking into account chosen for Zn isotope, for all multipolarities up to π½ =5 , only the GT transitions, but at higher incident energies the we have a number of 447 final states. The differential electron contributions of other multipolarities become significant and capture cross sections illustrated in Figure 4 present some cannot be omitted. characteristic clearly pronounced peaks at various excitation + + + The percentage contributions of various multipolarities energies π. These peaks correspond mainly to 0 ,1 ,and2 π Β± β 66 (with π½ β€5)intothetotalπ -capture cross section at transitions. More specifically, in the Cu daughter nucleus π = 0.5 πΈ =25 + MeV and for incident electron energy π MeV the maximum peak corresponds to the 01 QRPA transition at are tabulated in Table 7. In addition, in this table we list the π = 2.538 β MeV and other characteristic peaks correspond to values of the individual π -capture cross sections of each 1+ 1+ 1+ π = 3.194 π Β± 7 , 8 ,and 10 transitions, located at energies MeV, multipolarity with π½ β€5. More specifically, for πΈπ = π = 3.686 π = 6.555 + MeV, and MeV, respectively (see 25 MeV the 1 multipolarity contributes to about 44%, the + β Figure 4). The other less important peaks are also shown in 0 contributes to about 26%, and the 1 contributes to about Figure 4. 11%. The contributions coming from other multipolarities Before closing, it should be mentioned that the π-capture are less important (smaller than 5%). cross sections presented in this work may be useful in In performing state-by-state calculations for the electron estimating neutrino-spectra arising from π-capture on nuclei capture cross sections, our code has the possibility to provide during supernova phase. The knowledge of ]-spectra at every separately the contribution of the polar-vector, the axial- point and time in the core is quite relevant for simulations of vector, and the overlap parts induced by the corresponding thefinalcollapseandexplosionphaseofamassivestar.As components of the electron capture operators. In Figure 4 it is known [21], in the collapse phase, neutrinos are mainly we illustrate the stellar differential cross sections of each produced by π-capture on nuclei and on free protons. The individual transition of the polar-vector and axial-vector energy spectra of the emerging neutrinos from both reactions components. are important ingredients in stellar modelling and stellar As mentioned before, our code gives separately the partial β simulations [21, 27]. π -capture cross sections of each multipolarity. In order to Furthermore, in core collapse simulations one defines the study the dependence of the differential cross sections on the reaction rate of electron capture on nuclei given by excitation energy π throughout the entire pn-QRPA spec- π = βπ π , trum of the daughter nucleus, a rearrangement of all possible β π π (12) excitations π in ascending order, with the corresponding π 8 Advances in High Energy Physics
β β42 β1 2 β Table 7: Total π -capture cross sections (in 10 MeV cm )forπΈπ =25MeV. The percentage of each multipolarity into the total π -capture cross section evaluated with our pn-QRPA code is also tabulated here.
Positive parity transitions Negative parity transitions π β42 2 β1 π β42 2 β1 π½ ππ(Γ10 cm MeV ) Portions (%)π½ ππ (Γ10 cm MeV ) Portions (%) + β 0 31.164 25.96 0 5.288 4.41 + β 1 52.779 43.98 1 13.409 11.14 + β 2 6.921 5.77 2 3.262 2.72 + β 3 5.499 4.58 3 0.905 0.75 + β 4 0.244 0.20 4 0.299 0.25 + β 5 0.208 0.17 5 0.042 0.04
66 wherethesumrunsoverallnuclearisotopespresentin studied in detail the electron capture process on Zn isotope β the astrophysical environment (ππ denotes the abundance and calculated original and stellar π -capture cross sections. of a given nuclear isotope and ππ isthecalculatedelectron We tested our nuclear model (the pn-QRPA) through the capture rate for this isotope). The rates of (12)mustbe reproducibility of orbital muon capture rates for this isotope. known for a wide range of the parameters: π (temperature) The agreement with experimental data and other reliable and π (nuclear density) of the studied star. Thus, for the theoretical results of partial and total π-capture rates and of calculation of the quantity πβ πof a specific nuclear isotope the percentage contributions of various low-lying excitations oneneedstoknowinadditiontothenuclearcompositionπ is quite good which provides us with high confidence level for the electron capture rates π calculated as we have shown in the obtained cross sections. our present work. The rates of electron capture on various Our future plans are to extend the application of this nuclear isotopes and the corresponding emitted neutrino method and make similar calculations for other interesting spectraintherangeoftheparameters(π, π, ππ) describing nuclei [52]. Also this method could be applied to other the star until reaching equilibrium during the core collapse semileptonic nuclear processes like beta-decay and charged- arecomprehensivelystudiedin[21, 26, 27]foragreatnumber current neutrino-nucleus processes important in nuclear of nuclear isotopes by using the large scale shell model. Weare astrophysics and neutrino nucleosynthesis. currently performing similar calculations for a set of isotopes by employing the present pn-QRPA method [52]. Furthermore, the average neutrino energy, β¨πΈ]β©,ofthe Appendices neutrinos emitted by π-capture on nuclei can be obtained by dividing the neutrino-energy loss rate (defined by an A. Nuclear Matrix Elements expression similar to (12)byreplacingtherateππ with the πΈ π The eight different tensor multipole operators entering the energy loss rate π) with the reaction rate for -capture Μ above equations (see Section 3), refered to as Coulomb Mπ½π, on nuclei π β. Assuming, for example, power-law energy LΜ TΜel distribution for the neutrino spectrum produced by the π- longitudinal π½π, transverse electric π½π,andtransverse Μmagn capture in supernova phase, the average neutrino-energy magnetic Tπ½π , are defined as follows: β¨πΈ]β© determines a specific supernova-neutrino scenario. In addition, the neutrino emissivity is obtained by multiplying Μ Μcoul Μcoul5 Μ Μ Μ5 the electron capture rate at nuclear statistical equilibrium Mπ½π (ππ) = ππ½π + ππ½π , Lπ½π (ππ) = πΏπ½π + πΏπ½π, with the neutrino-spectra [21, 26, 27]. Finally we note that Μel Μel Μeπ5 Μmagn Μmagn Μmagn5 the rates for the inverse neutrino absorption process are Tπ½π (ππ) = ππ½π + ππ½π, Tπ½π (ππ) = ππ½π + ππ½π . also determined from the electron capture rates obtained as (A.1) discussed in this section [21].
These multipole operators contain polar-vector and axial- 4. Summary and Conclusions vector parts and are written in terms of seven independent The electron capture on nuclei plays crucial role during the basic multipole operators as follows: β presupernova and collapse phase (in the late stage π -capture on free protons is also significant). It becomes increasingly coul π 2 π½ πΜ (πr) =πΉ (π ) π (πr) , (A.2) possible as the density in the starβs center is enhanced and it is π½π 1 π π π accompanied by an increase of the chemical potential (Fermi πΏΜ (πr)= 0 πΜcoul (πr), energy) of the degenerate electron gas. This process reduces π½π π π½π (A.3) the electron-to-baryon ratio ππ of the matter composition. π π½ 1 In this work, by using our numerical approach based Μel π 2 σΈ π 2 π½ ππ½π (πr) = [πΉ1 (ππ) Ξ π (πr) + π (ππ) Ξ£π (πr)] , on a refinement of the pn-QRPA that describes reliably all ππ 2 the semileptonic weak interaction processes in nuclei, we (A.4) Advances in High Energy Physics 9
π 1 π½ Μmag π 2 π½ π 2 σΈ πππ½π (πr)= [πΉ1 (ππ)Ξπ (πr)β π (ππ)Ξ£ π (πr)] , B. Nuclear Form Factors ππ 2 In (A.2)β(A.9) the standard set of free nucleon form factors (A.5) 2 π 2 πΉπ(ππ), π=1,π΄,πand π (ππ) reads Μ5 π 2 π½ πππ½π (πr)= [πΉπ΄ (ππ)Ξ©π (πr) π 2 β2 π π 2 π πΉ1 (ππ) = 1.000[1 + ( ) ] , 1 2 2 σΈ σΈ π½ 840 MeV + (πΉπ΄ (ππ)+π0πΉπ (ππ)) Ξ£ π (πr)] , 2 2 β2 π 2 π (A.6) π (ππ) = 4.706[1 + ( ) ] , 840 MeV 2 5 2 π 2 σΈ σΈ π½ (B.1) βππΏΜ (πr)=[πΉ (π )β πΉ (π )] Ξ£ (πr), 2 β2 π½π π΄ π 2π π π π (A.7) 2 π π πΉπ΄ (ππ)=ππ΄[1 + ( ) ] , 1032 MeV π½ Μel5 2 σΈ βπππ½π (πr)=πΉπ΄ (ππ)Ξ£ π (πr), (A.8) 2 2πππΉπ΄ (π ) πΉ (π2 )= π , Μmag5 2 π½ π π π2 +π2 ππ½π (πr)=πΉπ΄ (ππ)Ξ£π (πr), (A.9) π
where ππ is the nucleon mass and ππ΄ is the axial vector free π where the form factors πΉπ, π=1,π΄,π,andπ are functions nucleon coupling constant (see the text). 2 of the 4-momentum transfer ππ and ππ is the nucleon mass. These multipole operators, due to the Conserved Vector Conflict of Interests Current (CVC) theory, are reduced to seven new basic operators expressed in terms of spherical Bessel functions, The authors declare that there is no conflict of interests spherical harmonics, and vector spherical harmonics (see regarding the publication of this paper. [24, 31, 44]).Thesingleparticlereducedmatrixelementsof π½ π½ the form β¨π1βππ βπ2β©,whereππ represents any of the seven Acknowledgments π½ π½ π½ σΈ π½ σΈ σΈ π½ π½ basicmultipoleoperators(π , Ξ© , Ξ£ , Ξ£ , Ξ£ , Ξ , π π π π π π This research has been cofinanced by the European Union ΞσΈ π½ π)of(A.2)β(A.9), have been written in closed compact (European Social Fund-ESF) and Greek national funds expressions as follows [31]: through the Operational Program βEducation and Lifelong Learningβ of the National Strategic Reference Framework π (NSRF)βResearch Funding Program: Heracleitus II, invest- σ΅© σ΅© max β¨(π π ) π σ΅©ππ½σ΅© (π π ) π β© =πβπ¦π¦π½/2 β ππ½π¦π, ing in knowledge society through the European Social Fund. 1 1 1 σ΅© σ΅© 2 2 2 π (A.10) π=0 References ππ½ where the coefficients π are given in [31]. In the latter [1] K. Langanke and G. Martinez-Pinedo, βNuclear weak- summation the upper index πmax represents the maximum interaction processes in stars,β Reviews of Modern Physics,vol. harmonic oscillator quanta included in the active model 75,p.819,2003. space chosen as πmax =(π1 +π2 β π½)/2,whereππ =2ππ +ππ, [2]G.M.Fuller,W.A.Fowler,andM.J.Newman,βStellarweak- π=1,2,andπ½ is related to the rank of the above operators interaction rates for sd-shell nuclei. I. Nuclear matrix element [31]. systematics with application to Al-26 and selected nuclei of In the context of the pn-QRPA, the required reduced importance to the supernova problem,β Astrophysical Journal + Supplement Series, vol. 42, pp. 447β473, 1980. nuclear matrix elements between the initial |0ππ β© and the final |πβ© state entering the rates of (6)aregivenby [3]G.M.Fuller,W.A.Fowler,andM.J.Newman,βStellarweak interaction rates for intermediate-mass nuclei. IIβA = 21 to A=60,βThe Astrophysical Journal,vol.252,pp.715β740,1982. σ΅© π½σ΅© σ΅© σ΅© β¨π σ΅©πΜ σ΅© π β© [4]G.M.Fuller,W.A.Fowler,andM.J.Newman,βStellar σ΅©Μπ½σ΅© + 2 σ΅© σ΅© 1 π π π π β¨π σ΅©π σ΅© 0ππ β©= β [ππ π π’π ππ +ππ π ππ π’π ], weak interaction rates for intermediate mass nuclei. IIIβrate σ΅© σ΅© [π½] 2 1 2 1 2 1 2 1 π2β₯π1 tables for the free nucleons and nuclei with A = 21 to A = 60,β (A.11) Astrophysical Journal Supplement,vol.48,pp.279β319,1982. [5] M. B. Aufderheide, I. Fushiki, S. E. Woosley, and D. H. Hartmann, βSearch for important weak interaction nuclei where π’π and ππ are the probability amplitudes for the π-level in presupernova evolution,β AstrophysicalJournal,Supplement to be unoccupied or occupied, respectively (see the text) [31, Series,vol.91,no.1,pp.389β417,1994. 32]. [6]H.A.Bethe,βSupernovamechanisms,βReviews of Modern These matrix elements enter the description of various Physics, vol. 62, p. 801, 1990. semileptonic weak interaction processes in the presence of [7] K. Langanke and G. Martinez-Pinedo, βSupernova electron 55 56 nuclei [31β38, 44, 53, 54]. capture rates for Co and Ni,β Physics Letters B: Nuclear, 10 Advances in High Energy Physics
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