QRPA calculations of stellar weak decay rates

Transcription

1 QRPA calculations of stellar weak decay rates P. Sarriguren Instituto de Estructura de la Materia CSIC, Madrid, Spain E. Moya de Guerra, R. Alvarez-Rodriguez, O. Moreno Universidad Complutense Madrid International School of Nuclear Physics. 36 th Course. Nuclei in the Laboratory and in the Cosmos. Erice-Sicily. September 16-24, 2014

2 Problem : Weak-decay rates determine late stages of stellar evolution. Experimental extrapolations or theoretical predictions: Reproduce exp information on T 1/2 and BGT under terrestrial conditions. Theoretical approach : Weak-decay rates Deformed HF+BCS+QRPA formalism with Skyrme forces and residual interactions in both ph and pp channels. Results : Weak-decay rates at various (,T) in stellar scenarios pf-shell nuclei. Main constituents of stellar core in presupernova formations: Sc, Ti, V, Mn, Fe, Co, Ni, Zn isotopes. BGT measured in laboratory (Charge exchange reactions) and compared with benchmark Shell Model calculations. Neutron-deficient waiting-point isotopes (Ni-Sn) at (,T) typical of rpprocess. Neutron rich Zr-Mo isotopes: r process.

3 Weak decay rates ln 2 ln 2 T ( ) if 1 1/2 Pi T Bif, D if T Initial states thermally populated 2Ji 1 PT i( ) e, G2J1 e G Nuclear structure Ei /( kt ) Ei /( kt ) i i 2 2 g A k k Bif ( GT ) f t i gv k eff P ( 0) 1 ig.. s T Phase space factors : EC if if if +, EC : - : if if are different from laboratory (P i, cec)

4 Q if Q if 1( Q ) F( Z 1, ) 1 S 1 S ( Q ) d 2 2 if 1 if p if cec 2 2 if if e if Phase space factors 1( Q ) F( Z 1, ) 1 S 1 S ( Q ) d 2 2 if 1 if e if 1( Q ) F( Z, ) S 1 S ( Q ) d oec qk gk BK ql g 1 L B 1 L 1 2 q Neutrino energy g Radial components of the e-wf at r=0 B Exchange and overlap corrections Distribution functions S p =S S e : Fermi-Dirac distribution 1 Se exp e / kt 1 1 Q M M E E if 2 p d i f mc e

5 Hartree-Fock method A H T k W k, l k1 kl1 How to extract a single-particle potential U(k) out of the sum of two-body interactions W(k,l) A H A A A Tk UkW k, luk H0 V k 1 k l 1 k1 res Hartree-Fock: Selfconsistent method to derive the single-particle potential Variational principle: Search for the best Slater det. minimizing the energy Assume that the resulting residual interaction is small

6 Skyrme effective interactions Two-body interaction: zero-range limit of a finite range force leading term: delta-function with strength t 0 and spin exchange x 0 leading finite range corrections t 1, x 1, t 2, x 2 (finite range = momentum dependence) short-range spin-orbit interaction with strength W 0 Short-range density-dependent two-body force t 3, x 3, V ij t0 1 x0p ri rj t1 1 x1p ri rj k k' 2 t 2 1 x2p k' ri rj k iw0 i j k ' ri rj k 1 r i r j t3 1x3P ri rj 6 2 Parameters (10) fitted using nuclear matter properties and ground state properties of a selected set of nuclei (binding energies, charge radii, ) Sk3, SG2, SLy4

7 2m r,, q m r U r W r 2m U q * q 2 2 r Skyrme Hartree-Fock Schroedinger equation 2 U q r W r i i eii Algebraic combinations of the densities 1 1 t1 x1 t2 x2 r t1 x1 t2 x2 r 2m q r t 0 2x012x0 q t3 2x32 2x3 12 q p n t x t x 8 t x t x q 16 t x t x t112x1t212x2 q t1t2jq t1x1t2x2j 16 qp, Vcoulr 8 8 Wq 2 1 r W0 q,,j 2 2 * r rst,,, r rst,,, J st r rs, ', ti rst,, st i st i i i i i i, s'

8 Axially symmetric deformed nuclei,,, i 1/2, i R q q r z e r ze 1/2 i i i i q i i Expansion into eigenfunctions of deformed harmonic oscillator 1 1 V(, r z) M r M z z i, ( ) ( ) e R 2 ( ) n r r n z z /2 /2 n () r N n 2 e L n () r r r 1/2 /2 n ( z) Nn z e Hn ( ) z z z 2 Optimal basis to minimize truncation effects N= 12 major shells 1/2 2 2 o M z z q z z 2 1/3 1/3 i R,, q C R,, n, n,, i qi r z

9 Pairing correlations in BCS approximation BCS ground state u va a BCS i i i i i0 ' H a a a a G a a a a k k k k k k k k' k k0 kk' 0 0 Variational equation constrained by the expectation value of particle number 2 2 vi i BCS eqs. Number eq. Gap eq. N 1 e v 1 ; E e 2 2 i 2 i i i 2 Ei G k 0 u v k k Fixed gaps taken from phenomenology 1 n B( N2, Z) 4 BN ( 1, Z) 6 BNZ (, ) 4 BN ( 1, Z) BN ( 2, Z) 8

10 1 E Vph r r 16 ( ) ( ) ph Residual interactions Particle-hole residual interaction consistent with the HF mean field 2 ss' tt' 1 ( 1) ( 1) sts'' t st r1 s ' t ' r2 2 1 ss' tt' E ( 1) ( 1) sts'' t st ( r1 ) s ' t '( r2) V r r t 0 2t1kF 2t2kF t r1r Average over nuclear volume Separable forces V 2 ( 1), t a a ph ph K GT GT K K K K K K 0, 1 pp 2 pp ( 1) K GT GT K K, K K K 0, 1 V P P P a a ph GT t0 kf t1t2 t3 8 R 2 6

11 pnqrpa with separable forces Phonon operator pnqrpa equations K K X A Y A, A K K K K K K n p K 0 0, 0 K K K t M K K Transition amplitudes 0 K K K K M q X qy M q X q Y K K K q u v, q v u, K K K K Single-particle states from deformed Skyrme Hartree-Fock u, v: Occupation amplitudes from BCS pairing B(GT) in the lab. system IiKi00I fk f 1 K B ( GT ) t 0 2 t 0

12 Stable nuclei in Fe-Ni mass region: Theory vs experiment Main constituents of stellar core in presupernovae. Comparison with : exp. (n,p), (p,n) SM calculations GT properties: Test of QRPA SM: NPA 653, 439 (1999) QRPA: NPA 716, 230 (2003)

13 Weak decay rates in pf-shell nuclei Calculations for several isotopes: Sc, Ti, V, Fe, Mn, Ni, Co, Zn exp: charge exchange reactions QRPA (n,p) B ( GT ) f t i if 2 Shell Model Accumulated GT strength A.L. Cole PRC 86, (2012) P.S. PRC87, (2013)

14 Weak decay rates in pf-shell nuclei, T P i ( T) B if if, T if Electron Capture from e - plasma cec 1( Q ) F( Z, ) S e if 2 2 if Se 1 S ( Qif ) d 1 exp e / kt 1 1 Q M M E E if 2 p d i f mc e Si burning Ia Supernova

15 Weak decay rates in pf-shell nuclei (Q<0) large: sensitive low-lying exc. P.S. PRC87, (2013)

16 Weak decay rates in pf-shell nuclei Q small

17 Exotic Nuclei : Nuclear Astrophysics X-ray bursts: Source of intense X-ray emissions generated by thermonuclear runaway in the H-rich environment of an accreting n-star, fed from a binary red giant companion. Nucleosynthesis mechanism: rp capture process rp-process: Proton capture reaction rates are orders of magitude faster than the competing -decays. Waiting point nuclei: When the dominant p-capture is inhibited, the reaction flow waits for a slow beta-decay to proceed further. Time scale Isotope abundances Light curve profile Tc (43) Mo (42) Nb (41) Zr (40) Y (39) Sr (38) Rb (37) Kr (36) Br (35) Se (34) As (33) Ge (32) Ga (31) Ag (47) Pd (46) Rh (45) Ru (44)

18 Medium mass neutron deficient isotopes Constrained HF+BCS Waiting point nuclei in rp-processes Shape coexistence Beyond full Shell Model

19 Gamow-Teller strength distribution Calculations for Ni, Zn, Ge, Se, Kr, Sr, Zr, Mo, Ru, Pd, Cd, Sn (N=Z and neighbors)

20 Gamow-Teller strength distribution

21 Gamow-Teller strength: Theory and Experiment ISOLDE: Total absorption spectroscopy Exp: Poirier et al. PRC69, (2004) Exp: Nacher et al. PRL92, (2004)

22 + /EC half-lives: Theory and Experiment g / g A V eff / EC 1/ f T f i g / g 0.74 g / g A V eff A V bare 1( Q ) F( Z, ) d if 1 Q if 2 2 if EC qkgkbk ql g 1 L B 1 L 1 2 Good agreement with experiment: Reliable extrapolations to high and T

23 Weak decay rates in rp-process 74 Sr Deformed pn-qrpa B(GT) and T 1/2 : Good agreement with experiment Competition +/EC P.S. PRC83, (2011) 78 Sr

24 Weak decay rates in rp-process 74 Sr 76 Sr (WP) 78 Sr

25 Weak decay rates in rp-process 70 Kr (Large Q) P.S. Phys. Rev. C 83, (2011) 74 Kr (Small Q)

26 Weak decay rates in rp-process 70 Kr (Large Q) 72 Kr (WP) 74 Kr (Small Q)

27 Zr-Mo neutron-rich isotopes Zr Mo

28 Zr-Mo B(GT)

29 Zr-Mo B(GT)

30 Zr-Mo Half-lives

31 Shape dependence of GTdistributions in neutron-deficient Hg, Pb, Po isotopes... Triple shape coexistence at low excitation energy Search for signatures of deformation on their beta-decay patterns (MeV) e - e Pb

32 Shape dependence of GTdistributions in neutron-deficient: Pb isotopes B(GT) strength distributions Not very sensitive to : Skyrme force and pairing treatment Sensitive to : Nuclear shape Signatures of deformation PRC 72, (2005), PRC 73, (2006) prolate oblate spherical QEC

33 Shape dependence of GTdistributions in neutron-deficient Hg, Po isotopes

34 Conclusions Nuclear structure model (deformed Skyrme HF+BCS+QRPA) o o Nuclear structure in different mass regions, astrophysical applications. Reproduce half-lives and main features of GT strength distributions extracted from beta-decay and/or charge exchange reactions. o Weak-decay rates at (,T) typical of astrophysical scenarios: pf-shell nuclei (presupernova): EC rates from QRPA comparable quality to benchmark SM calculations. Neutron-deficient WP nuclei (rp-process): EC/ + compete Neutron-rich Zr-Mo isotopes (r process).