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Selfconsistent variational methods applied to double beta decay in deformed nuclei

Selfconsistent variational methods applied to double beta decay in deformed nuclei. E. Moya de Guerra. UCM and CSIC Madrid (Spain). Blaubeuren July 2007. Collaborators. A. Faessler F. Simkovic. P. Sarriguren R. Alvarez-Rodriguez O. Moreno. Contents. Theory: QATDHF

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Selfconsistent variational methods applied to double beta decay in deformed nuclei

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  1. Selfconsistent variational methods applied to double beta decay in deformed nuclei E. Moya de Guerra UCM and CSIC Madrid (Spain) Blaubeuren July 2007

  2. Collaborators • A. Faessler • F. Simkovic • P. Sarriguren • R. Alvarez-Rodriguez • O. Moreno

  3. Contents • Theory: QATDHF • Selfconsistency and charge-exchange excitations • Overview of numerical applications Test cases - Magnetic dipole excitations -Single beta decay: - Gamow-Teller strength distributions - Half-lives - Dependence on deformation • Application to double beta decay • Summary and outlook

  4. Moya de Guerra and Villars, NPA 285 (1977) 287; NPA 298 (1978) 109; PRL 48 (1982) 922 H is the many-body Hamiltonian UNDERLYING THEORY QATDHF The starting point is the variational principle V.P.:,, • = y(t) time dependent many-body nuclear wave function ,,y S.D.  ATDHF (or ATDHFB) V.P. set of self. cond. I- At equilibrium {qi=qi0} I- HF or HFB II- Normal modes II- RPA or QRPA of the system

  5. H ^ H H H QUANTIZATION OF CLASSICAL DYNAMICS OF COL. MODES V.P. Time dependent many body w.f. near equilibrium Microscopic many-body aspects contained in Q’s, P’s and f0 Collective aspects q’s, p’s classical to be quantized: , Define phonon operators: and phonon basis: Relevant Physical Quantities: Expectation values of operatorsqbetween quantal many body collective statesyn, ym: = qn,m

  6. Relevant Physical Quantities: Expectation values of operators q between quantal many body collective states yn, ym: = qn,m Quantal representation of the operator in collective space Matrix elements qn,m: z.p. correction

  7. From condition I: From condition II: Selfconsistency and charge-exchange excitations Axially deformed harmonic oscillator basis with Deformed Skyrme-HF with pairing with Equilibrium conditions: Migdal equations: reduce to separable form averaging over nuclear volume, giving separable forces consistent with Skyrme-HF-BCS mean field: H  E0 + Vph + Vpp   with

  8. Overview of numerical applications Test cases - Magnetic dipole excitations of deformed rare-earth nuclei - Single beta decay: - Gamow-Teller strength distributions Fe, Ni, etc. - Dependence on deformation Kr, Sr, etc. - Half-lives of waiting-point nuclei - GT predictions for Pb neutron-deficient isotopes

  9. ORBITAL M1 excitations in deformed rare-earth nuclei: Theory and experiment SPIN M1 excitations in deformed rare earth nuclei: Theory and experiment Scissors mode ~ 2 E* ≤ 3 MeV Moya de Guerra et al. PRC 49 (1994) 3354 47 (1993) 811 54 (1996) 690

  10. GT transitions schematic picture • Even-even nuclei 0+0g1+K • (0qpg2qp) Eex,2qp= w-Ep0-En0 • Odd-A nuclei IipKigIfp Kf • Phonon excitations : (1qpg3qp) • Odd nucleon acts as a spectator • Transitions involving the odd • nucleon state : (1qpg1qp) Eex,3qp=w+En, spect-Ep0 > 2D Eex,1qp=Ep-Ep0

  11. Stable nuclei in Fe-Ni mass region. GT strength: Theory and experiment Gamow-Teller properties. Comparison with experimental (n,p) Comparison with SM calculations. Test of our QRPA method Total B(GT+) strength Total B(GT-) strength SM: Caurier et al. NPA 653(99)439 QRPA: Sarriguren et al. NPA 716(2003)230 EXP: Frekers, Vetterly et al. (83-95)

  12. Medium-mass proton-rich nuclei : Ge, Se, Kr, Sr Minimization of the energy under the constraint of holding the nuclear deformation fixed Shape coexistence. Large Q values Isotopic chains approaching drip lines Beyond full Shell Model Waiting points in rp process Sarriguren et al. NPA 658 (1999) 13

  13. oblate oblate prolate prolate 76Sr 74Kr oblate prolate Exp: Poirier et al. 2003 Exp: Nácher et al. 2004 Test of b-decay shape dependence predictions 74Kr obl+prol shape coex. 76Sr prolate Gamow-Teller strength: Theory and Experiment

  14. QEC and T1/2 : Theory and Experiment When z.p. energy corrections are taken into account QthQexp and TthTexp

  15. Beta-decay half-lives of waiting point nuclei Good agreement with experiment : Reliable extrapolations MNK: Moeller et al. At. Data and Nucl. Data. Tables 66 (1997) 131; Sarriguren et al. Eur. Phys. J. A 24 (2005) 193

  16. ... (MeV) 6+ 4+ 186Pb 1 2+ 0+ 0+ e- e- g 0+ 0 186Pb Pb neutron-deficient isotopes • Unique laboratory to study the phenomenon of nuclear shape coexistence • Search for signatures of deformation on their b-decay patterns Andreyev et al., Nature 405 (2000) 430 triple shape coexistence at low excitation energy

  17. Influence of Skyrme force and pairing • Influence: Relative energy of minima • Little influence : location of minima Energy-deformation curves : Pb isotopes Sarriguren et al. PRC 72 (2005) 054317

  18. QEC QEC Could GT strength distributions be used as a bar code to identify the deformation of the b-decaying nucleus? b+/EC decay  GT strength distributions show specific signatures, different for each equilibrium deformation, which hardly change when reasonable interactions are used

  19. 2n 0n Double beta-decay: Test of the Standard Model One of the most rare events in nature T ~ 1020 years 2 decay modes (A,Z)  (A,Z+2)+2e-+2n (A,Z)  (A,Z+2)+2e- 2n : 2 successive bdecays through intermediate virtual states. Second order process in the weak interaction. Observed in 48Ca 76Ge 82Se 96Zr 100Mo 116Cd 128Te 130Te 136Xe150Nd (T ~ 1019-1021 years) 0n : Lepton number not conserved. Forbidden in the SM. n emitted is absorbed: Massive Majorana particle. (T (76Ge) >1025 years)

  20. Double beta-decay: Two neutrino mode (A,Z)  (A,Z+2)+2e-+2n 2n : 2 successive bdecays through intermediate virtual states. Second order process in the weak interaction. Observed in 48Ca 76Ge 82Se 96Zr 100Mo 116Cd 128Te 130Te 150Nd(T1/2~ 1019-1021 years)

  21. Int. Parent Daughter Double beta-decay and deformation Calculation of matrix elements involved in the process Int. Parent Daughter

  22. Parent Daughter bexp Qexp Qexp Binding energy profiles of DBD partners Theoretical and experimental values of Qbb and deformation Alvarez-Rodriguez et al. PRC 70 (2004) 064309

  23. Double beta-decay and Deformation GT strengths of DBD partners

  24. GT strength in the single-beta branches of double beta partners

  25. Comparison of experimental and theoretical values for Gamow-Teller strengths. Also shown are the r.m.s charge radii

  26. Double beta matrix elements as a function of deformation Suppression mechanism due to deformation

  27. 2n 2b M vs. kpp and deformation Dependence on particle-particle interaction Dependence on deformation

  28. HF-QRPA WS-QRPA Double beta matrix elements

  29. kpp = 6/A = 0.05 MeV SGT+ = 0.80 SGT - = 0.32 SGT + = 0.58 SGT- = 0.60 116In 116In 116Cd 116Cd (0.1164 – 0.1818) MeV-1 0.12 MeV-1 Rule of thumb ~ 0.08 MeV-1 0.058 MeV-1 116Sn 116Sn GT+ : Rakers et al., Phys. Rev. C 71, 054313 (2005) GT - : Akimune et al., Phys. Lett. B 394, 23 (1997) Single and double beta decay: experiment and theory EXPERIMENTAL THEORETICAL

  30. SGT+ SGT- E* (MeV) Intermed. Parent Qbb (MeV) Daughter Simple rule of thumb for M2n

  31. Rotational spectra and intermediate 1+ excitations of prolate Nd and Sm ground states 1+ QRPA states from Nd from Sm Rotational states I th=21.21 MeV-1 I exp=23.06 MeV-1 Rotational states I th=10.87 MeV-1 I (2+exp)=9.00 MeV-1 I (4+exp)=12.90 MeV-1 12+ 10+ 8+ 8+ 6+ 4+ 150Pm 2+ 6+ 0+ 150Nd 4+ 2+ 0+ 150Sm HF(Sk3)+BCS(D) gs E= -1229.5 MeV HF(Sk3)+BCS(D) gs E= -1231.5 MeV

  32. Single and double beta decay: experiment and theory 1+ QRPA states from Ndfrom Sm kpp = 0.04 – 0.05 MeV S<3MeVGT - = 0.86 12+ S<3MeVGT + = 0.30 10+ 8+ 8+ 6+ 4+ GT- 2+ 6+ 150Pm 0+ 150Nd 4+ Theory and experiment (0.0654-0.1021) MeV-1 2+ Rule of thumb 0.062 MeV-1 0+ 150Sm

  33. Summary of M2n results Accumulated sum of the 2n2b matrix elements as a function of the excitation energy of the intermediate nucleus HF(Sk3) + BCS + + QRPA (kpp=6/A) Horizontal lines: experimental values with gA=1.00 and gA=1.25

  34. Summary of M2n results

  35. 2n 0n Double beta-decay: Zero neutrino mode ▼ (A,Z)  (A,Z+2)+2e- 0n : Lepton number not conserved. Forbidden in the SM. n emitted is absorbed: Massive Majorana particle. (T (76Ge) >1025 years) Observation not yet confirmed Interesting candidates: 100Mo, 116Cd, 150Nd (SuperNEMO)

  36. Matrix elements of zero neutrino mode In the 0n case closure approximation is good because average int. nuclear E* ( 10 MeV) small in comparison with virtual neutrino energy  ( 100 MeV). In closure approximation

  37. Check of closure approximation with 2n case &ED estimate Compare with closure approximation ED can be deduced using wif

  38. 136Cs 136Xe 136Ba Closure approximation Closure approximation

  39. Closure approximation = Important short range correlations

  40. Closure approximation [ ]2 (simplified) short-range correlation (a = 0.8 fm-2) (a = 1.1 fm-2 , b=0.68 fm-2) = new ground state norms (including correlation)

  41. = = Assuming equal deformation of parent and daughter Detailed structure of matrix elements Closure approximation 

  42. Detailed structure of matrix elements.Different parent and daughter deformations  Approximate BCS pseudo-overlap Proton single particle state c is missing Neutron single particle state d is missing

  43. Exact BCS pseudo-overlap Single particle state c is missing (the double-beta operator has acted on it) Different deformation is included in the overlap matrices (M): Different parent and daughter deformations

  44. = S c,d CA1,A2,B1,B2 = SA1,A2,B1,B2 A1,A2,B1,B2 Structure of 2-body matrix elements inSpherical H.O. basis Harmonic oscillator basis Factor coming from the spin operators Spherical harmonic oscillator eigenfucntions Factor coming from the coefficients of the expansion of the 4 single particle states in spherical harmonic oscillator basis

  45. f an,l,m Structure of 2-body matrix elements inSpherical H.O. basis Analytical expressions for short-range correlations and basis functions Short-range correlation: Spherical harmonics (Clm) suitable for the angular integration Spherical harmonic oscillator eigenfunctions:

  46. GT strength in the single-beta branches of double beta partners Nd and Sm

  47. Rotational spectra and intermediate 1+ excitations of prolate Nd and Sm ground states 1+ QRPA states from Nd from Sm Rotational states I th=21.21 MeV-1 I exp=23.06 MeV-1 Rotational states I th=10.87 MeV-1 I (2+exp)=9.00 MeV-1 I (4+exp)=12.90 MeV-1 12+ 10+ 8+ 8+ 6+ 4+ 150Pm 2+ 6+ 0+ 150Nd 4+ 2+ 0+ 150Sm HF(Sk3)+BCS(D) gs E= -1229.5 MeV HF(Sk3)+BCS(D) gs E= -1231.5 MeV

  48. E*= 8 MeV Preliminary!! Single and double beta decay: experiment and theory 1+ QRPA states from Ndfrom Sm kpp = 0.04 – 0.05 MeV S<3MeVGT - = 0.86 12+ S<3MeVGT + = 0.30 10+ 8+ 8+ 6+ 4+ GT- 2+ 6+ 150Pm 0+ 150Nd 4+ Theory and experiment (0.0654-0.1021) MeV-1 2+ 0+ 150Sm

  49. Summary We havetestedour selfconsistent approach on GT strengths and related properties along the nuclear chart. Wefind: • Good agreement with experiment: • * GT strength distributions : Fe-Ni and 2nbb emitters. • * Half- lives : Proton-rich nuclei. Waiting point nuclei. • * Spin M1 strengths : Rare earth and actinide. • Important dependence on nuclear shapes. Suppression mechanism for DBD matrix elements. • Signatures of deformation in GT strengths of neutron-deficient Pb. Fair agreement on 2n DBD M.E. for reasonable values of kpp and nuclear deformation selfconsistently determined.

  50. APPENDIX

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