Supernova β decay of nuclide 53 Fe, 54 Fe, 55 Fe, and 56 Fe in strongly screening plasma

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1 Reearch in Atron. Atrophy. Vol. (2x) No., Reearch in Atronomy and Atrophyic Supernova β decay of nuclide,,, and in trongly creening plama Jing-Jing Liu and Dong-Mei Liu College of Electronic and Communication Engineering, Hainan Tropical Ocean Univerity, Sanya, 57222, China. Received 27 Augut 22; accepted 27 November 3 Abtract The electron creening trong effect on the electron energy and threhold energy of the beta decay reaction. in thi paper, we tudy theβ decay rate of ome iron iotope. The electron creening beta decay rate increae by about two order of magnitude. The trong creening beta decay rate due to Q-value correction are by more than one order of magnitude higher than thoe of without Q-value correction. Key word: phyical data and procee: nuclear reaction, nucleoynthei, abundance tar: upernova INTRODUCTION Beta decay play a key role in preupernova evolution. The cooling rate and antineutrino energy lo are trongly affected by the beta-decay rate. Some author (e.g., Fuller et al. 98; Aufderheide et al. 99, 994; Langanke et al. 998, Liu et al, 22, 23a, 23b, 23c, 23d, 23e, 23f, 24, 25, 26a, 26b) did a lot on the beta decay and electron capture. However, the effect of SES on weak interaction are not included. According to the linear repone theory model (LRTM) and hell model Fermi theory, we tudied the SES beta decay rate of nuclide,,, and in atrophyical environment, which are very important for numerical imulation of upernova exploion (e.g., Fuller et al. 982; Aufderheide et al. 99, 993, 994; Langanke et al. 23; Domingo-Pardo et al. 29). The article i organized a follow. In ection 2, we tudied the beta-decay rate by including and neglecting SES effect. In ection 3, the reult and dicuion are preented. The concluion are given in ection 4. 2 STUDY THEβ DECAY 2. The beta decay in no SES In no SES, theβ decay rate i given by (Fuller et al. 98; Aufderheide et al. 99, 994; Liu 26a). λ bd = ln2 (2J i +)e Ei k B T G(Z,A,T) f ψ(ρ,t,y e,q ij ) ft ij, () wherej i i the pin, ande i i excitation energie of the parent tate.k B i the Boltzmann contant.ft ij i the comparative half-life connecting tate of i and j, Q ij i the nuclear energy difference between the tate of i and j. Q = M p c 2 M d c 2, M p and M d are the mae of the parent nucleu and the

2 2 Jing-Jing. Liu daughter nucleu, repectively, E i and E j, are the excitation energie of the i th and j th nuclear tate. G(Z,A,T) i the nuclear partition function. The phae pace integral ψ(ρ,t,y e,q ij ) for the β decay i given by ψ(ρ,t,y e,q ij ) = c 3 Qij (m e c 2 ) 5 dε e ε e (ε 2 e ) /2 (Q ij ε e ) 2 F(Z +,ε e ) +exp[(u F ε e )/k B T], (2) where p,m e,u F and ε e are the electron momentum, ma, chemical potential and energy, repectively. F(Z +,ε e ) i the Coulomb wave correction. In no SES, a reaonable approximation for the electron chemical potential take the form (e.g., Bludman et al. 978) U F =.( Y e ) /3 [+( π. ) (k BT) 2 ( Y e ) 2/3] /3 MeV (3) According to dicuion from Zhou & Li et al. (27), the half-lifeft ij ha the label ln(ft ij ) = a +(α 2 Z 2 5+a 2 N Z A )ln(q if a 3 δ)+(a 4 α 2 Z 2 )+ 3 α2 Z 2 ln(a) αzπ+s(n,z), (4) where α i the fine tructrue contant with /37. The correction factor S(N,Z) will take the label (e.g., Zhou & Li et al. 27) S(N,Z) = a 5 exp( ((N 28) 2 +(N 2) 2 )/2)+a 6 exp( ((N 5) 2 +(N 38) 2 )/43) +a 7 exp( ((N 82) 2 +(N 5) 2 )/3)+a 8 exp( ((N 82) 2 +(N 58) 2 )/24) +a 9 exp( ((N ) 2 +(N 7) 2 )/244), (5) where a i (i =,2,3,...9) =.9,.7,.935, 5.398,3.6,3.879,.322,6.3,.669 in Eq.(4-5). In Eq.(4), the hell and pairing effect on the nuclear matrix element, which reflect the main information of the nuclear tructure. The factorδ i well decribed byδ = ( ) N +( ) Z (zhou & Li et al. 27). The Fermi and the Gamow-Teller matrix element for β decay i given by (e.g., Aufderheide et al. 994) M F (fi) 2 = ω D f n (τ ± ) ω P i 2 /(2J i +) = j p (τ ± ) j n 2 N n (2j n +)(2J i +) ( N p ), (6) 2j p + M GT (fi) 2 = ω D f n σ n (τ ± ) ω P i 2 /(2J i +) = j p σ n (τ ± ) j n 2 N n (2j n +)(2J i +) ( N p ), (7) 2j p + where ω P i i the initial parent tate and ωd f i the final daughter tate. N n, and N p are the number of neutron and proton within thej n, and j p hell, repectively. The Shell Model Monte Carlo (SMMC) method i ued to calculate the total amount of GT trength S GT and the repone function R A (τ) of an operator  at an imaginary-time τ. R A(τ) i given by (e.g., Langanke et al. 998; Langanke et al. 23) R A(τ) = if (2Ji +)e βe i e τ(e f E i ) f  i 2, (8) i (2Ji +)e βe i

3 Beta decay rate of nuclide,,, and 3 where E i and E f are energie correponding to the final tate i and f. The total trength for the operator i given by R(τ = ). The trength ditribution i given by if S GT +(E) = δ(e E f +E i)(2j i +)e βe i f  i 2 i (2Ji +)e βe i = S A(E), (9) which i related to R A (τ) by a Laplace Tranform, R A (τ) = S A(E)e τe de. Note that here E i the energy tranfer within the parent nucleu, and that the trength ditributions GT +(E) ha unit of MeV and β = /T N,T N i the nuclear temperature. 2.2 The beta decay in SES Electron creening for nuclear reaction in atrophyical environment play an unexpected and important role in enhancing reaction cro ection. In our previou work (e.g., Liu. 23d, 26c, 27a, 27b), we dicued thi intereting problem. Baed on the linear repone theory model (hereafter LRTM), Itoh et al.(22) alo tudied the influence of the creening potential on the weak interaction. The electron i trongly degenerate in our coniderable regime of the denity-temperature, which i given by T T F = {[+.8( Z A )2/3 () 2/3 ] /2 }, () here the electron Fermi temperature and the denity aret F and (in unit of 7 g/cm 3 ), repectively. Jancovici et al. (962) tudied the tatic longitudinal dielectric function for relativitic degenerate electron liquid. The electron potential energy in SES i given by V(r) = Ze2 (2k F ) 2k F r 2 π in[(2k F r)]q dq, () qǫ(q, ) where ǫ(q,) i Jancovici tatic longitudinal dielectric function and k F i the electron Fermi wavenumber. For relativitic degenerate electron and baed on LRTM, the creening potential i calculated a D = Z( z A ) 3 J(r,R)(MeV) (2) where the parameter J(r,R), r and R are dicued by Itoh et al.(22)in detail. The Eq. (2, 4) are atified for 5 r, R 5, which fulfill in the pre-upernova environment. When we take account into the influence of SES, the beta decay Q-value change by (Fuller et al(982)) Q Z 2/3 (ρy e ) /3 MeV. (3) Thu, the Q-value of beta decay change from Q if toq if = Q if Q. We can not neglect it influence at high denity when electron i trongly creened due to the creening energy i o high. The electron creening make electron energy increae from ε e to ε e = ε e + D beta decay. The creening alo decreae the threhold energy from Q if to Q if (I) = Q if + D, and Q if (II) = Q if + D = Q if Q + D correponding to the SES model(i) and model(ii). The SES model(i) and model(ii) are correponding to the cae without and with the correction of Q-value. So the phae pace integral ψ (ρ,t,y e,q ij ) replace ψ(ρ,t,y e,q ij ) in Eq.(2) for the SES beta decay rate, and i calculated a ψ (ρ,t,y e,q c 3 Q ij ij) = (m e c 2 ) 5 dε eε e((ε e) 2 ) /2 (Q ij ε e) 2 F(Z +,ε e) +D +exp[(u F ε e)/k B T], (4) Therefore, according to Eq.(), the beta decay rate in SES i given by λ bd = ln2 (2J i +)e Ei k B T G(Z,A,T) ψ (ρ,t,y e,q ij ), (5) f ft ij

4 4 Jing-Jing. Liu where the half-lifeft ij i given by ln(ft ij) = a +(α 2 Z 2 N Z 5+a 2 A )ln(q if a 3 δ)+(a 4 α 2 Z 2 )+ 3 α2 Z 2 ln(a) αzπ+s(n,z), (6) We compare the reult (λ bd ) in SES with thoe of the rate ( ) without SES by defining an enhancement factor C, which i given by 3 NUMERICAL RESULTS AND DISCUSSIONS C = λ. (7) bd Baed on proton-neutron quai particle RPA model, Nabi(2) invetigated the beta decay rate in upernova. Under the ame condition, FFN (Fuller et al.982) and Aufderheide et al. (99, 994) alo dicued the beta rate. Their tudie how that the beta decay rate play an important role in the core collape calculation and evolution. However, they neglected the effect of SES on beta decay. Here baed on LRTM, we dicu the beta decay for SES model (I) and (II). The model (I) and (II) are correponding to the cae without and with correction and correction of Q-value. Figure and 2 preent the influence of denity on beta decay rate of ome iron group iotope for the two SES model. The no SES and SES rate correponded to olid and dotted line are compared. We detailed the GT tranition contribution for beta decay according to SMMC method. For a given temperature, we find that the beta decay rate decreae by more than ix order of magnitude a the denity increae. The trong creening rate alway higher than thoe of no SES. For example, at = 5, = 7.79,Y e =.45 the rate for are.76 7 and correponding to thoe of no SES and SES for model (I) in Fig.(b), but are.29 6 and for model (II) in Fig.2(b). The SES beta decay rate of model (II) are by more than one order of magnitude higher than thoe of model (I). Figure 3 how the creening enhancement factorc a a function of. Due to SES, the rate may increae by about two order of magnitude. For intance, the creening enhancement factorc increae from.55 to 7.8 when the denity increae from 3 to 4 for at =.79,Y e =.48 for model (II)in Fig.3(a). The lower the temperature, the larger the effect of SES on beta decay rate i. One poible caue that the SES mainly increaed the number of higher energy electron. Thee electron can actively join in the beta decay reaction. Moreover, the SES can alo make the beta decay threhold energy greatly decreae. Thu, SES trong encourage the beta decay reaction. One alo find that the SES enhancement factor C of model (II) are higher than thoe of model (I). For example, at = 7, =.79,Y e =.48 the creening factor C for 53,54,55, i 88.9, 86.8, 7., 7.53 for model (I), and are 98.36, 95.89, 82.56, 84.2 for model (II) in Fig.3(a), repectively. Table and 2 how the creening enhancement factorc at =, for model (I), and (II). From Table, the creening rate for 53,54,55, increae by a factor of.59,.43, 9.288, at =, =.79,Y e =.48 for model (I), and by a factor of.56,.46,.9,.32 for model (II), repectively. From Table 2, the creening rate for 53,54,55, increae by a factor of 55.7, 5.6, 8.9, 2.6 at =, =.79,Y e =.48 for model (I), and by a factor of 7.8, 66.4, 32.2, 35.2 for model (II), repectively. But the difference of the creening enhancement factor C between model (I), and (II) i mall at the higher temperature. Thi i becaue that the higher the temperature, the larger the electron energy i for a given denity. So the higher temperature weaken the effect of SES on beta decay. The beta decay rate are trong depended on the decay Q-value. The higher the energy of outgoing electron, the larger the rate become when the electron energy i more than the threhold energy. When we take account the Q-value correction in model (II), according to Eq.(4), the half-life will increae a the the Q-value increae. The nuclear binding energy increae becaue of interaction with the dene electron ga in the plama. The beta decay Q-value (Q if ), change at high denity due to the affect of the charge dependence of thi binding. Baed on Eq.(), Q-value of beta decay decreae from Q if to

5 Beta decay rate of nuclide,,, and 5 Table The trong creening enhancement factor C for model (I) and (II) at = in ome typical atronomical condition. =.79,Y e =.48 = 7.79,Y e =.45 =.33,Y e =.43 = 9.33,Y e =.48 nuclei C(I) C(II) C(I) C(II) C(I) C(II) C(I) C(II) Table 2 The trong creening enhancement factor C for model (I) and (II) at = in ome typical atronomical condition. =.79,Y e =.48 = 7.79,Y e =.45 =.33,Y e =.43 = 9.33,Y e =.48 nuclei C(I) C(II) C(I) C(II) C(I) C(II) C(I) C(II) Q if Q. Thu, the beta decay will increae due to correction of Q-value in model (II) according to Eq.() and Eq.(4). In upernova evolution the ditribution of Gamow-Teller trength play a key role. A example for the excited tate GT ditribution of 55,, Fig.4 preent ome information about the comparion of our reult by SMMC with thoe of Nabi (Nabi et al. 2) for beta decay. We the firt two excited tate ditribution are only hown. From fig. 4, one find that our reult of GT trength ditribution calculated are lower than thoe of Nabi. For example, the GT ditribution for are.65mev,.362mev correponding to Nabi and our at E i = 3.86MeV,E j = 7.46MeV, and are.7265mev,.5865mev for ate i = 5.8MeV,E j = 6.55MeV. Baed the pn-qrpa theory, Nabi et al. (2) analyzed nuclear excitation energy ditribution by conidering the particle emiion procee. They calculated Gamow-Teller trength ditribution and only dicued the low angular momentum tate. By uing the method of SMMC, actually we dicu GT intenity ditribution and adopt an average ditribution. Syntheize the above analyi, the charge creening trong effect the beta decay. The influence may be mainly come from following everal factor. Firt, the electron Coulomb wave function i trongly changed by the creening potential in nuclear reaction. Second, the energy of outgoing electron increae greatly due to the electron creening potential. Third, the energy of atomic nuclei alo increae becaue of the electron creening (i.e., increae the ingle particle energy). Finally, the electron creening effectively make the number of the higher-energy electron increae. So the electron energy i more than the threhold of beta decay. SES relatively decreae the threhold needed for beta decay. 4 CONCLUDING REMARK Baed on LRTM and Fermi theory, we dicu the beta decay proce for two typical SES model (i.e., model (I) and (II)). We detailed the GT tranition contribution to the beta decay according to SMMC method. For a given temperature, the beta decay rate decreae by more than ix order of magnitude with the increaing of the denity. The trong creening rate alway higher than thoe of no SES. The SES beta decay rate of model (II) are by more than one order of magnitude higher than thoe of model (I). Our reult how that the beta decay rate increae by about one order of magnitude due to SES. For intance, the creening enhancement factor C increae from.55 to 7.8 when the denity

6 6 Jing-Jing. Liu 5 =.79,Y e =.48 (a) 53 Fe 54 Fe 5 =7.79,Y e =.45 (b) 53 Fe 54 Fe ( ) 56 Fe ( ) 56 Fe ( ) 5 =.33,Y e =.43 (c) 53 Fe ( ) =9.33,Y e =.4 (d) 53 Fe Fig. The beta decay rate of,,, and a a function of electron denity in and not in SES for model (I). increae from 3 to 4 for at =.79,Y e =.48 for model (II). The beta decay rate and the antineutrino energy lo are quite relevant for numerical imulation of tellar thermal evolution. Our reult may be helpful to the future tudie of burt mechanim of upernova, and cooling numerical imulation. Acknowledgement Thi work wa upported in part by the National Natural Science Foundation of China under grant 5652, and the Counterpart Foundation of Sanya under grant 26PT43, the Special Foundation of Science and Technology Cooperation for Advanced Academy and Regional of Sanya under grant 26YD28, the Scientific Reearch Starting Foundation for 55 Talented Project of Hainan Tropical Ocean Univerity under grant RHDRC27, and the Natural Science Foundation of Hainan Province under grant 42.

7 Beta decay rate of nuclide,,, and 7 ( ) 5 =.79,Y e =.48 (a) 53 Fe 54 Fe 56 Fe ( ) 5 =7.79,Y e =.45 (b) 53 Fe 54 Fe 56 Fe ( ) 5 =.33,Y e =.43 (c) 53 Fe ( ) =9.33,Y e =.4 (d) 53 Fe Fig. 2 The beta decay rate of,,, and a a function of electron denity in and not in SES for model (II). Reference Aufderheide, M. B., Brown, G. E., kuo, T. T. S., Stout, D. B., & Vogel, P., 99, ApJ, 62, 24 Aufderheide, M. B., Fuhikii, I., Wooely, S. E., & Hartmanm, D. H.,, 994, ApJS,, 9, 389 Aufderheide, M. B., Bloom, S. D., Reler, D. A., Mathew, G. J., 993, Phy.Rev. C, 9, 389 Bludman S.A. and van Riper K.A., 978, Atrophy. J., 224, 63 Domingo-Pardo, C., Dillmann, I., Faetermann, T., et al., 29, Conference Proceeding, 9, 23 Fuller, G. M., Fowler, W. A., & Newman, M. J., 982, ApJS, 48, 279 Itoh, N., Tomizawa, N., Tamamura, M. et al., 22, ApJ., 579, 38 Jancovici, B., 962, Nuovo Cimento, 25, 428 Langanke, K., & Martinez-Pinedo, G., 998, Phy. Lett. B, 436, 9

8 8 Jing-Jing. Liu 2 5 (I) (I) (I) (I) =.79,Y e =.48 (a) (I) (I) (I) (I) =7.79,Y e =.45 (b) C (II) (II) C 2 (II) (II) (II) (II) 5 (II).5 (II) C (I) (I) (I) (I) (II) (II) (II) (II) =.33,Y e =.43 (c) C (I) (I) (I) (I) (II) (II) (II) (II) =9.33,Y e =.4 (d) Fig. 3 The creening enhancement factor C for beta decay rate of,,, and a a function of electron denity for model (I) and (II). Langanke, K.; Teraaki, J.; Nowacki, F.,et al.,23, Phy.Rev. C, 67, 4434 Liu, J. J. & Luo, Z. Q., 27a, ChPhL, 24, 86 Liu, J. J. & Luo, Z. Q., 27b, ChPhy,. 6, 3624 Liu, J. J., Luo, Z. Q., Liu, H. L. & Lai, X. J., 27c, IJMPA,. 22, 335 Liu, J. J. & Luo, Z. Q., 27d, ChPhy, Liu, J. J. & Luo, Z. Q., 28a, ChPhC, 32, 67 Liu, J. J. & Luo, Z. Q., 28b, ChPhC, 32, 8 Liu, J. J. & Luo, Z. Q., 28c, CoTPh, 49, 239 Liu, J. J., 2a, ChPhC, 34, 7 Liu, J. J., 2b, ChPhC, 34, 7 Liu, J. J. & Kang, X. P. et al., 2, ChPhC, 35, 243

9 Beta decay rate of nuclide,,, and E i =3.86MeV (a) Nabi our E i =7.28MeV (b) Nabi our GT.5 GT GT E (MeV) j E i =5.8MeV (c) Nabi our GT E j (MeV).5.5 E i =5.57MeV (d) Nabi our E j (MeV) E j (MeV) Fig. 4 Comparion of the excited tate Gamow-Teller trength ditribution for, and between our and Nabi, E i(e j) repreent parent (daughter) energy tate. Liu, J. J., 22, ChPhL,29,223 Liu, J. J., 23a, Ap&SS,343, 7 Liu, J. J., 23b, Ap&SS,343, 579 Liu, J. J., 23c, RAA,3, 99 Liu, J. J., 23d, MNRAS., 433, 8 Liu J. J., 23e, RAA., 3, 945 Liu J. J., 23f, Chin.Phy. C., 37, 85 Liu, J. J., 24, MNRAS., 438, 39 Liu, J. J., 25, Ap&SS, 357, 93 Liu, J. J., et al., 26a, RAA, 6, 74 Liu, J. J., et al., 26b, ApJS, 224, 29

10 Jing-Jing. Liu Liu, J. J., et al., 26c, RAA, 6, 83 Liu, J. J., et al., 27a, RAA, 7, 7, eprint arxiv: Liu, J. J., et al., 27b, ChPhC, 4, 5 Nabi, J-U., 2, AdSpR, 46, 9 Zhou, Y., Li, Z. H., Wang, Y. B., et al., 27, SCIENCE CHINA Phyic, Mechanic & Atronomy.,.7/ Thi paper wa prepared with the RAA LATEX macro v.2.