Gamow-Teller Strength Distribution in Proton-Rich Nucleus $^{57} $ Zn And

Home , Proton capture

arXiv:1203.4669v1 [nucl-th] 21 Mar 2012 fteelwligsae ndaughter of in ture states structure low-lying and location these the of to sensitive stel- are The rates calculation. weak lar model shell large-scale the in used for pn-QRPA strength The low-lying measured 57 the calculation. reproduced model model shell ear- the reported with lier differences shows agreement calcu- and good The measurements in with is nuclei. distribution proton-rich strength perfor- for B(GT) lated the model check the to of function) mance strength calcula- B(GT) pn-QRPA other of in tions made usually (as insertions made experimental were No theory. (pn-QRPA) Approxi- mation Phase Random Quasiparticle proton-neutron for The strength B(GT) for problem. nucleosynthesis and concomitant dynamics the supernova probing decay core-collapse weak the of for calculation the rates for and used ground microscopically are states from The excited distributions strength stages supernova. GT the Type-II in calculated core a stellar to of leading collapse the in role eminent Abstract ftepeitosfrtesnhsso rtnrc iso- rich proton of synthesis The the topes. for predictions modifications and the significant binding of to the lead can in energies changes be- excitation collapsing Small is the of and stars. dynamics the rp-process supermassive in the important be mechanism is to nuclei primary lieved such The producing nuclei. for proton-rich of thesis aelU Nabi Jameel-Un nucleu proton-rich in 57 distribution strength Gamow-Teller ooy oa 60,Kye ahukw,Pakistan Pakhtunkhwa, 1 Khyber 26000, Kohat nology, n cecsadTcnlg,Tp 34,Sai Khyber Swabi, 23640, Topi Technology, and Pakistan Pakhtunkhwa, Sciences ing aut fEgneigSine,GKIsiueo Engineer of Institute GIK Sciences, Engineering of Faculty Nabi Jameel-Un uebU Rahman Muneeb-Ur eateto hsc,KhtUiest fSineadTech and Science of University Kohat Physics, of Department orsodn uhreal:[email protected] : email author Corresponding nbte ncmaio oteK3 interaction KB3G the to comparison in better Zn nadisipiain nastrophysics in implications its and Zn 57 upasasmtosrl ntenucleosyn- the in role sumptuous a plays Cu ao-elr(T rniin lyapre- a play transitions (GT) Gamow-Teller β + dcyadeeto atr E)rates (EC) capture electron and -decay 1 57 • ni acltdi h oanof domain the in calculated is Zn uebU Rahman Muneeb-Ur 57 u h struc- The Cu. - - olradNwa FN.Tep-RAcalculated pn-QRPA The β (FFN). Newman and Fowler on h F aclto.Hwvra ihselrtempera- calculated stellar high the at tures However calculation. FFN the F ae.O h te ad o ppoescondi- rp-process ( for capture hand, electron other calculated the the tions, On rates. FFN iko nryadlpo ubrutltecr density core the until number lepton and energy of sink iiet h nrp n otenme flposper leptons of number sen- very the to is and baryon, star entropy the the or of to type-II collapse sitive as The they known then supernovae. commonly Ib/c massive spec- display too in pro- visual explode not the and tacular are bounce of consequently stars mass and the the collapse on If neutron depending a genitors. hole in either black is star or relic the The star of explosion. death core catastrophic the inner a the announces of un- consequently implosion and becomes the ap- it inaugurates iron the and limits of exceeds stable mass core composed inner Chandrasekhar mainly this is propriate when exploding core and the inner nuclei of group This core inner (1). the star of of energy ejecta fraction internal the core- a of and energy the of kinetic into conversion energy of gravitational the the incarnation is The mechanism con- subject collapse the computation. the and much been stars of has that mass which guru heavy nucleosynthesis, the of comitant are evolution gravity the and drive interactions weak The Introduction 1 distri- strength GT rp-process. theory; bution, pn-QRPA decay, beta and Keywords (1.5) consequences. 2 astrophysical factor interesting a have than may more by and rates FFN than bigger are + dcyrtsaegnrlyi odareetwith agreement good in generally are rates -decay 57 naecmae otesmnlwr fFuller, of work seminal the to compared are Zn Y e 2 ) h etio r osdrda main as considered are neutrinos The 3). (2; ekitrcinrts lcrncapture electron rates; weak-interaction β + dcyrtsaeams afof half almost are rates -decay β + dcy rates -decay) s 2 reaches around 1010g − cm−3. At later stage of the col- that it is likely possible that the neutrino-induced rp- lapse this assumption is no longer legitimate as weak process takes place in all core-collapse supernovae and interaction rates increase with increase of stellar core other astrophysical sites such as collapsar jets or disk density. For densities > 1011g − cm−3, the neutrinos winds formed around a black hole. In reference (7) the mean free paths become shorter and consequently they authors discovered an extremely luminous X-ray out- proceed through all phases of free streaming, diffusion, burst that marked the birth of a supernova of Type Ibc. and trapping. The most tightly bound of all the nuclei They attributed the X-ray outburst to the break-out of in the inner core of star is 56Fe (4) and further fusing the supernova shock-wave from the progenitor. of the nuclei is highly endothermic. The second bot- The open shell nuclei with a few nucleons outside a tleneck to the synthesis of heavier elements is the high doubly magic shell closure are of colossal interest to test Z number which poses high coulombic barrier for the the nuclear model predictions. 57Cu is of paramount charged particles to initiate nuclear reactions at stellar importance in this regard to test the pn-QRPA predic- core temperatures. This impediment to further nucle- tions. The pf -shell nucleus 57Cu has a single proton osynthesis is rescinded with the help of neuron capture just above the closed core of even-even 56Ni with Z = processes in the stellar core to form heavier isotopes N = 28. This simple structure permits far more accu- beyond iron group nuclei. These processes are further rate model calculations than are possible in the mid- classified into slow (s-) and rapid (p-) neutron capture dle of a nuclear shell. In particular, a comparison of processes depending on the neutron capture time scale the low-lying levels of 57Cu with the well-determined 57 τn and beta decay time scale τβ for nucleus to endure excited states of its mirror nucleus Ni is important beta decay. As discussed earlier that weak interaction for studying the charge symmetry of the nucleus. The and gravity are the mentor that drives the stellar evo- doubly magic nature of 56Ni confers it a stable struc- lutionary process and its subsequent death in a cat- ture and elements beyond 56Ni are cooked in the stellar aclysmic explosion. The main weak interaction pro- pot only via 56Ni(p, γ)57Cu reaction. The authors in cesses that play an effective role in the stellar evolution Ref. (8) pointed out that the structure of 57Cu plays are beta decays, electron and positron capture processes a sumptuous role in the nucleosynthesis of proton-rich and neutrinos emission/capture processes subject to the nuclei. This points the fact that this reaction rate is physical conditions available for these processes in the susceptible to the structure of 57Cu and environs nuclei, stellar core. These weak decay processes are smitten by including their binding energies. The primary mecha- Fermi and Gamow-Teller (GT) transitions. Fermi tran- nism for producing such nuclei is the rp-process and is sitions are straightforward and are important only for believed to be important in the dynamics of the col- beta decays. For nuclei with N >Z, Fermi transitions lapsing supermassive stars and x-ray bursts (9). The are Pauli blocked and the GT transitions dominate and rp-process is characterized by proton capture reaction their calculation is model dependent. Spin-isospin-flip rates that are orders of magnitude faster than any other excitations in nuclei at vanishing momentum transfer competing process, specially β-decay rates. The reac- are commonly known as GT transitions. These transition path follows a series of fast (p,γ)-reactions until tions are ideal probe to test nuclear structure and play further proton capture is inhibited, either by negative preeminent role in the nucleosynthetic origin of the el- proton capture Q-values (proton decay) or small posi- ements in the late phases of the stellar life. When the tive proton capture Q-values (photodisintegration). As stellar matter is degenerate then the phase space for electrons captures in the stellar pot assist the cooking the electron in beta decay is Pauli blocked and electron process and gravity, therefore, it is crucial to know the captures become dominant in the stellar core producing nuclear structure properties of the doubly magic shell neutrinos which escape from the surface of the star and 56Ni and nuclei in its vicinity. These nuclei drive the takes away the core energy and entropy as well. These electron capture processes in the dense core of heavy electron capture rates and β+ decay rates are very sen- mass stars. The beta decay rate calculations require sitive to the distribution of the GT+ strength which is the evaluation of the GT strengths for many levels per responsible for changing a proton into a neutron (the nucleus. The pn-QRPA theory gives us the emanci- plus sign is for the isospin raising operator (t+), present pation to use model space as big as 7 ~ω rather than in the GT matrix elements, which converts a proton truncated model spaces usually employed in some shell into a neutron). The authors in (5; 6) took the accurate model calculations. neutrino transport into account in their hydrodynamic Earlier the half-lives for β+/EC decays for neutron- studies of core-collapse supernovae and showed that the deficient nuclei with atomic numbers Z = 10 - 108 were bulk of the neutrino-heated ejecta is proton-rich during calculated up to the proton drip line for more than 2000 the early phase (≤ 1s). Their studies also provided nuclei using the same model (10). These microscopic 3 calculations gave a remarkably good agreement with the total electron capture rate and a microscopic cal- the then existing experimental data (within a factor culation of excited state GT strength distributions is of two for more than 73% of nuclei with experimental desirable. The electron capture rates play a pivotal role half-lives shorter than 1 s for β+/EC decays). Most in the dynamics of core collapse of stars. The pn-QRPA nuclei of interest of astrophysical importance are the model is used in the present work to calculate the GT ones far from stability and one has to rely on theo- strength functions and associated electron capture/β+- retical models to estimate their beta decay properties. decay rates for proton-rich nucleus 57Zn. The accuracy of the pn-QRPA model increases with in- The Hamiltonian of the pn-QRPA is given by creasing distance from the β-stability line (shorter half- QRPA sp pair ph pp lives) (10; 11). This is a promising feature with respect H = H + V + VGT + VGT, (1) to the prediction of experimentally unknown half-lives Hsp V pair (specially those present in the stellar interior), implying where is the single-particle Hamiltonian, is V ph that the predictions are made on the basis of a realistic the pairing force, GT is the particle-hole (ph) GT force, and V pp is the particle-particle (pp) GT force. Pairing physical model. GT BCS Nucleosynthesis in proton-rich ejecta occur at low is treated in the approximation, where an assumed densities. At high densities the composition is neutron- constant pairing force with force strength G (Gp and Gn rich. Under terrestrial conditions 57Zn β+ decays to for protons and neutrons, respectively) is applied, 57 Cu. We used the calculated ground and excited state − ′ ′ − ′ V pair = −G (−1)l+j mc+ c+ (−1)l +j m GT strength functions to microscopically calculate the jm j−m jmjX′m′ β+/EC rate of 57Zn in stellar matter. In proton-rich ′ ′ ′ ′ supernova environments 57Zn is produced at high tem- cj −m cj m , (2) peratures where nuclear statistical equilibrium is valid. ′ ′ where the sum over m and m is restricted to m, m > 0, According to studies by authors in Ref. (8) the peak and l donates the orbital angular momentum. conditions for rp-process are in the vicinity of T = In the present work, in addition to the well known (1–3) GK and ρ = (106 − 107) gcm−3. The current particle-hole GT force (13; 14; 15) analysis shows that the electron capture and positron 57 decay rates of Zn contribute roughly equally to the ph µ + VGT =2χ (−1) YµY−µ, (3) total weak rates at peak rp-process conditions (with Xµ the positron decay rates bigger roughly by a factor of 6–7). At still higher temperatures and densities the with 57 electron capture on Zn dominates . As a result both + + 57 Y = < j m |t−σ |j m >c c , (4) β and electron capture stellar rates on Zn are being µ p p µ n n jpmp jnmn X presented in this paper. jpmpjnmn Section 2 describes the theoretical formalism of pn- the particle-particle GT force (16; 17) QRPA calculation. Section 3 discusses the pn-QRPA calculated GT strength functions and its comparison pp µ + VGT = −2κ (−1) Pµ P−µ, (5) with measurements and previous calculation. The cal- Xµ culated weak rates are presented in Section 4. Here we also compare our results with the pioneering cal- with

culation of Fuller and collaborators (12). We finally + + l +j −m P = < j m |(t−σ ) |j m > (−1) n n n conclude our findings in Section 5. µ n n µ p p jpmXpjnmn c+ c+ , jpmp jn−mn (6) 2 Model Description is also taken into account. The nuclei involved in the stellar interior have finite The capture/decay rate of a transition from the ith probability of occupation of excited states and the state state of a parent nucleus (Z, N) to the jth state of the by state evaluation of the GT strength distribution daughter nucleus (Z − 1,N + 1) is given by from these excited states is a formidable task. The pn-

QRPA is considered an eﬃcient model to extract the fij(T,ρ,Ef ) GT strengths for the ground as well as excited states of λij = ln 2 , (7) (ft)ij the involved nuclei in stellar matter thanks to the huge available model space of seven major shells. The tran- fij is the phase space integral. The (ft)ij of an ordi- sitions from the excited states contribute eﬀectively to nary β± decay from the state |ii of the mother nucleus 4 to the state |fi of the daughter is related to the reduced

transition probability Bij of the nuclear transition by

pn-QRPA

-1 (ft)ij = D/Bij . (8) 10

-2

The D appearing in Eq. 8 is compound expression

of physical constants, -3

-4 2π3~7 ln 2 10 D = , (9) Experimental 2 5 4 -1 gV mec 10

-2

and the reduced transition probability of the nuclear 10

) Strength)

transition is +

-3

B(GT

-4

2 10 Bij = B(F )ij + (gA/gV ) B(GT )ij , (10)

Shell model

-1 The value of D = 6146 ± 6 s (18) is adopted and the 10

ratio of the axial-vector (gA) to the vector (gV ) cou- -2 10

pling constant is taken as -1.257. B(F )ij and B(GT )ij

-3 are reduced transition probabilities of the Fermi and 10

GT transitions, respectively. These reduced transition -4

probabilities of the nuclear transition are given by, 0 1 2 3 4 5 6 7 8

E (MeV)

j 1 k 2 B(F )ij = | < j|| t±|| i> | , (11) 2Ji +1 Xk Fig. 1 The comparison of pn-QRPA, experimental, and 57 shell model B(GT+) strength distribution of Zn as a func- 1 k k 2 57 B(GT )ij = | < j|| t±~σ || i> | . (12) tion of Cu excitation energy. 2Ji +1 Xk Calculation of phase space integrals can be seen from 3 Gamow-Teller Strength Distributions Ref. (19). The number density of electrons associated with pro- The charge exchange reactions with high resolution are tons and nuclei is ρYeNA, where ρ is the baryon density, important probe for the study of nuclear structure in Ye is the ratio of electron number to the baryon number, astrophysics. The GT strengths play an important role and NA is the Avogadro’s number. in electron capture and beta decay processes in the dynamics of stellar collapse (20; 21). Experimentally the ∞ (p, n), (n, p), (d, 2He), and (2He, t) reactions can be 1 mec 3 2 ρYe = ( ) (G− − G+)p dp, (13) used to probe the GT transitions (in both directions) π2N ~ Z A 0 at higher excitation energies. Turning to theory one where p = (w2 − 1)1/2 is the electron or positron sees that a large amount of calculations of weak inter- momentum. action rates for astrophysical applications have become The total capture/decay rate per unit time for a nu- available in recent times (e.g. (19; 22; 23; 24; 25)). cleus in thermal equilibrium at temperature T for any To allow the reader the judgment of quality achieved weak process is then given by in the pn-QRPA model underlying the stellar weak interaction rate calculation, the B(GT) strength calculated using the pn-QRPA theory is compared with ex- λ = Piλij . (14) perimental (18) and with B(GT) strengths predictions X ij obtained from a large-scale shell model calculation us- Here Pi is the probability of occupation of parent ing the effective interaction KB3G (26). The results excited states and follows the normal Boltzmann distri- are depicted in Fig. 1. Vieira et al. (27) for the first bution. The summation over all initial and final states time probed the structure of 57Cu in the beta decay of is carried out until satisfactory convergence in the rate 57Zn by employing the 40Ca(20Ne, 3n) fusion evapora- calculations is achieved. tion reaction and by Zhou et al. (28) in the 1H(58Ni, 5

10 GT

1 SummedB

pn-QRPA

0.1

0 2 4 6 8 10 12 14

E (MeV)

Fig. 2 The pn-QRPA calculated integrated strength Fig. 3 Comparison of experimental and shell model inte- within the QEC window. Ej represents the energy in daugh- grated GT strength within the QEC window. The ﬁgure is ter nucleus 57Cu in units of MeV. reproduced from Ref. (18).

57Cu-γ)2n reaction by using the recoil mass spectrome- used the Q-value of 14.51 MeV for 57Zn from the recent ter MARS at the Texas A&M Cyclotron Institute. Con- compilations of Ref. (29; 30). siderable improvements were obtained in Ref. (18) con- It is evident that the GT strength distribution cal- cerning the quality of the experimental data, in partic- culated by the shell model KB3G, as compared to pn- ular with respect to source purity and proton energy QRPA and experimental results, tends to shift to higher resolution. excitation energies in daughter 57Cu. The pn-QRPA The authors in Ref. (18) investigated the beta- integrated GT strength over the Q window is shown delayed proton decay of 57Zn at GSI online isotope in Fig. 2. For comparison, the overall shift in B(GT) separator. 57Zn nuclei were produced in fusion evapo- strength of the shell model prediction along with the ex- ration reactions, 28Si(32S, 3n), by using a 150 MeV 32S perimentally extracted integrated GT strength is shown beam on a 28Si target and then the beta-delayed pro- in Fig. 3. In the present work we obtained a summed tons were measured with high resolution by employing B(GT) of 1.86 as compared to 1.25 in Ref. (18) within a charged-particle detector. The GT strength distribu- the excitation energy interval between 0 and 7 MeV in tions observed in this experiment is shown in the middle daughter 57Cu. The authors in Ref. (18) increased the panel of Fig. 1. The morphology of the pn-QRPA GT strength by the upper B(GT) limit for the 1.028 MeV strength (upper panel) for beta transitions between the state and the theoretical B(GT) value for the IAS to 57 +0.25 ground state of Zn to the ground state of daughter yield 1.44−0.17 and they estimated that less than 2% of 57Cu is in good agreement with the measured data of the total GT strength could be missed due to the un- Ref. (18). These transitions are of allowed nature in observed gamma-transitions. The shell model predicted daughter 57Cu. The experimental GT strength between strength for the same excitation of energies amounts to 2 and 3 MeV is well reproduced by pn-QRPA. These 1.68. peaks were missing in the large-scale shell model cal- In beta decay experiments the sensitivity of the ex- culation as shown in the bottom panel of Fig. 1. We periment strongly decreases as one moves to higher excitation energies in daughter. This puts a limit for 6 such experiments and thus one can only probe the low- 4 β+-Decay and Electron Capture Rates energy tails of the GT strength distribution. Conse- quently one has to rely on theoretical models for the At low stellar densities and temperatures, β+-decay is calculation of GT strength distributions in high excita- the dominant mode for 57Zn to transform to 57Cu. For tion energies region as well as for parent excited states. the calculation of the stellar weak interaction rate un- Within the Q window of the reaction, we extracted a der corresponding physical conditions, it is of colossal total GT strength of 15.23 and the KB3G calculation importance to reproduce the measured GT strength dis- amount it to be 8.63. The pn-QRPA results include a tribution as compared to the total GT strength due to quenching factor of 0.8 as usually employed for the pf - very strong energy dependence of the phase space fac- shell nuclei. The luxurious 7 ~ω model space of the pn- tors. As discussed earlier the pn-QRPA model repro- QRPA facilitates one to extract the state by state evalu- duced the low-lying strength for 57Zn better in com- ation of the these GT strength for higher excited states parison to the KB3G interaction used in the large shell as well without assuming the Brink’s hypothesis and model calculation. The stellar weak rates are sensitive model space truncation as usually employed in the shell to the location and structure of these low-lying states in model calculations. The truncation level, i.e, the num- daughter 57Cu. At the lowest temperature considered ber of nucleons which are to be excited from the f7/2 in this work (T9[K]=0.01), excited parent states are orbital to the rest of the pf -shell, in the KB3G calcu- not appreciably populated (where T9 is the stellar tem- 57 9 lation was restricted to 4 nucleons for parent Zn and perature in units of 10 K) , while at low density (ρYe 5 nucleons for daughter 57Cu. The pn-QRPA model = 10 gcm−3), the continuum electron density is quite extracted a bigger total strength as compared to the low and the stellar rates should be close to the terres- KB3G interactions employed in the shell model calcula- trial values. At this value of temperature and density tion. This certainly affects the calculation of β+-decay the calculated half-life for β+-decay on 57Zn is 39.6 ms and electron capture rates in the stellar core at high which is in excellent agreement with the measured value densities. Except at higher temperatures, the double of 40 ms (27). shell closure of 56Ni confers it stability compared to Table 1 shows the calculated β+-decay rates on 57Zn any of its neighbors and the main path of the rp-process as a function of stellar temperature and density. In Ta- passes through 56Ni. These properties pointed to the ble 1 the first column gives the stellar density in units fact that heavier nuclei may be produced by the rate of gcm−3. The second column gives the value of tem- of the radiative proton capture reaction 56Ni(p, γ)57Cu perature in units of 109K. The third column gives the and confirm the candidature of 56Ni as a waiting point value of the calculated β+-decay rates in units of sec−1 nucleus in the reaction network. The rate of this re- whereas the final column shows the ratio of the calcu- action in the stellar kilns is due almost entirely to the lated β+-decay rates to the electron capture rates for first four resonances (28) and particularly very sensitive physical conditions given in first two columns. One to the location and structure of the low-lying states in should note that even though terrestrially 57Zn β+- daughter 57Cu. The pn-QRPA accounted well for these decays to 57Cu with a 100% ratio (no electron cap- low-lying states in 57Cu. The proton separation energy ture), we do calculate a finite ratio of β+-decay to elec- of 57Cu is merely Sp = 694.87 keV (29; 30), and so tron capture even at low temperature and density (see all of its excited states are resonances in this reaction. Table 1). This is because we calculate only continuum The authors in Ref. (28) stressed that small changes electron capture and no bound states capture. It can be in the binding and excitation energies lead to signifi- seen from the table that there is no appreciable change cant modifications of the predictions for the synthesis in the β+-decay rates as the core stiffens from low den- of proton rich isotopes with A > 56 and possibly for sity to high density region. However the rates increase the time evolution of cosmic x-ray bursts. considerably with increasing stellar temperature due to We calculated the ground state and excited states a considerable increase in the available phase space. GT strength functions of 57Zn using the pn-QRPA At the later stages of the collapse, β+-decay becomes model. The calculation was performed for a total of unimportant as an increased electron chemical poten- 259 excited states in 57Zn covering an excitation en- tial, which grows like ρ1/3 during in fall, drastically re- ergy of around 30 MeV. The density of states were cho- duces the phase space. These results in increased elec- sen accordingly and contributions to GT strength from tron capture rates during the collapse phase. Electron all excited states were incorporated in the rate calcula- capture rates on 57Zn hence become important during tion. The ASCII files of these GT strength functions are the very late phases of stellar evolution of massive stars available and can be requested from the corresponding (prior to collapse) and at high stellar temperatures. At −3 11 author. ρYe[gcm ]=10 the calculated electron capture rates 7

Table 1 β+-decay rates (in units of sec−1) and ratio of β+- decay to electron capture rates on 57Zn for diﬀerent selected

densities and temperatures in stellar matter. ρYe denotes 3 the stellar density in units of g/cm and T9 represents the temperature in 109 K. 1.7

FFN

+ + 3 ρYe T9 λ + R(β /EC) ρYe T9 λ + R(β /EC) 1.6 β β 10

7 10 0.01 17.50 5.6E+05 107 0.01 17.50 6.5E+00 10

7 1.5 10 1 19.54 5.8E+04 10 1 19.54 7.0E+00 11 7 10

10 3 21.09 1.8E+02 10 3 21.09 7.0E+00 ]) 7 10 5 22.18 2.7E+01 10 5 22.23 6.4E+00 -1 10 10 26.67 2.4E+00 107 10 26.73 1.9E+00 [s 1.4 10 30 137.09 1.3E-01 107 30 137.09 1.3E-01 3 9

10 0.01 17.50 7.9E+03 10 0.01 17.50 3.5E-02 (

3 9

10 1 19.54 3.4E+04 10 1 19.54 3.8E-02 10 103 3 21.09 1.8E+02 109 3 21.09 3.9E-02 3 9 1.3 10 5 22.18 2.7E+01 10 5 22.28 4.0E-02 log 3 9 10 10 26.67 2.4E+00 10 10 27.10 4.1E-02 (pn-QRPA) 3 9 10 30 137.09 1.3E-01 10 30 143.88 6.7E-02 3 5 11 10 0.01 17.50 3.2E+02 10 0.01 17.50 5.8E-05 10 5 11 1.2 10 1 19.54 5.4E+02 10 1 19.54 6.4E-05 7 105 3 21.09 1.6E+02 1011 3 21.09 6.8E-05 10 5 11 10 5 22.23 2.7E+01 10 5 22.28 7.1E-05 11 105 10 26.67 2.4E+00 1011 10 27.10 8.3E-05 10 5 11

10 30 137.09 1.3E-01 10 30 151.36 4.0E-04 1.1

9 9.2 9.4 9.6 9.8 10 10.2

log (T[K])

Fig. 4 ( Color on line) Comparison of the pn-QRPA and FFN (12) β+ decay rates of 57Zn nucleus as a function of temperature for selected densities (in units of g.cm−3 shown in legends) in stellar matter.

(25) which lead to the suppression of pn-QRPA β+ de- are bigger than the competing β+-decay rates by more cay rates at high stellar temperatures (when the prob- than four orders of magnitude. For the relevant peak ability of occupation of high-lying parent excited states 7 −3 rp-process conditions (T9[K] ∼ 3 and ρ ∼ 10 gcm ), increases substantially). Regarding the calculation of the pn-QRPA calculated β+-decay rates are around a electron capture rates (Fig. 5) one notes that at high factor seven bigger than the corresponding electron cap- stellar densities (where electron capture rates surpass + −3 11 ture rates. Therefore we present our results for both the β decay rates), ρYe[gcm ] ∼ 10 , the FFN rates β+-decay rates of 57Zn and electron capture rates on are in good comparison with the pn-QRPA electron 57Zn and compare them with previous calculations. capture rates. At lower densities the pn-QRPA elec- To the best of author’s knowledge, there are no shell tron capture rates are bigger by a factor 2 (T9[K] ∼ 1) model weak rates available in the literature for 57Zn nu- to a factor of 3 when T9[K] = 30. This enhancement is cleus. Therefore, pn-QRPA rates were compared with attributed to the low placement of the GT centroid in the pioneer work of Fuller et al. (12). This comparison the daughter nucleus, 57Cu, by the pn-QRPA model. for the β+ decay and electron capture rates are shown The β+-decay and electron capture rates on 57Zn in Fig. 4, and Fig. 5, respectively. Fig. 4 shows that were calculated on a fine grid temperature-density scale the pn-QRPA β+ decay rates are in good comparison suitable for simulation codes and for necessary interpo- with the FFN rates. However at high stellar tempera- lation purposes. The ASCII files of these rates may be tures (T9[K] = 30) the FFN rates are almost twice the requested from the corresponding author. calculated rates. The authors in Ref. (12) put no constraints on the threshold values of parent excitation energies while in the present calculation such constraints were incorporated through particle emission processes 8 pn-QRPA theory as a preferred model for the calculation of weak rates for proton-rich nuclei. The choice of nuclear model can affect the prediction and synthe- 6 sis of proton-rich isotopes as well as the time evolution of the cosmic x-ray burst and dynamics of the collaps- 5

11 -3

Ye = 10 g-cm ing supermassive stars. For comparison, in the present

4 work we obtained a summed B(GT) of 1.86 as compared to 1.25 in Ref. (18) within the excitation energy 3 interval between 0 and 7 MeV in daughter 57Cu. The

]) shell model predicted strength for the same excitation 2 -1 energies amounts to 1.68. [s ec 1 7 -3 The calculated GT strength functions were further

Ye = 10 g-cm used to calculate electron capture and β+-decay rates of (

0 57 10 Zn in stellar matter, particularly for rp-process conditions. At high stellar temperatures the calculated log -1 + 57 -3 β -decay rates on Zn are half of the corresponding

-2 FFN calculated rates. For typical peak rp-process con-

QRPA = 10 g-cm 7 −3 Y e ditions, T9[K] ∼ 3 and ρ ∼ 10 gcm , the pn-QRPA -3 FFN calculated β+-decay (electron capture) rates on 57Zn are bigger by a factor of 1.5 (2). This may have inter-

-4 esting astrophysical consequences for collapse simula- 9.0 9.2 9.4 9.6 9.8 10.0 tors.

log (T[K]) Realistically speaking weak interaction mediated

10 rates of hundreds of nuclei are involved in the complex dynamics of supernova explosion . Incidently, the most abundant nuclei tend to have small weak rates as Fig. 5 Comparison of the pn-QRPA and FFN (12) electron they are more stable and the most reactive nuclei tend capture rates on 57Zn nucleus as a function of temperature for selected densities in stellar matter. to be present in minor quantities. Thus, the most important in the stellar core is the rate times abundance of a particular specie. We are in the process to cal- 5 Summary culate microscopically the weak decay rates for nuclei which are considered to be important in astrophysical The GT strength function is an important ingredient in environment as part of this on-going project. Few of the complex dynamics of presupernova and supernova such important weak rates were recently presented (e.g. explosion since these GT transitions partly determine (31; 32; 33; 34; 35; 36)). Work is still in progress for the + the β -decay and electron capture rates in the stel- microscopic calculation of weak rates of remaining key 57 lar core. The GT strength distributions in Zn were iron-regime nuclei. Core-collapse simulators are urged calculated within the domain of the pn-QRPA theory. to test run the pn-QRPA reported weak rates in typi- The pn-QRPA calculated GT strength showed differ- cal stellar conditions to check for probable interesting ences with the earlier reported shell model calculation. outcomes. The pn-QRPA results for 57Zn were in good agreement with the experimental results (18) and reproduced well the measured GT strength between 2-3 MeV. The stellar weak decay rates are sensitive to the location and structure of these low-lying states in daughter 57Cu. Small changes in the binding and excitation energies can lead to significant modifications of the predictions for the synthesis of proton-rich isotopes. The primary mechanism for the production of the proton-rich nuclei is the rp-process and is believed to be important in the dynamics of the collapsing supermassive stars. The good agreement of the reported low-lying GT strength with the experimental results validates the choice of the 9

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