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PROCESSES OF NUCLEOSYNTESIS .

BETA-STRENGTH FUNCTION IN NUCLEOSYNTHESIS CALCULATIONS Yu.S. Lutostansky, I.V. Panov, and V . N . Tikhonov N a tional Research Center "Kurchatov Institute" Institute of Theoretical and Experimental Physics ITEP – 09.09.2013. PROCESSES OF NUCLEOSYNTESIS. Superheavy nuclei. β -decay.

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PROCESSES OF NUCLEOSYNTESIS .

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  1. BETA-STRENGTH FUNCTIONIN NUCLEOSYNTHESIS CALCULATIONSYu.S. Lutostansky, I.V. Panov, andV.N. TikhonovNational Research Center "Kurchatov Institute"Institute of Theoretical and Experimental PhysicsITEP – 09.09.2013

  2. PROCESSES OF NUCLEOSYNTESIS. Superheavy nuclei β-decay fission s-process track r-process track β-decay The tracks of elements synthesis ins (slow)- andr (rapid)- processes.

  3. NUCLEOSYNTHESISOF THE HEAVY NUCLEI NUCLEOSYNTHESIS OF THE HEAVY NUCLEI in s (slow)andr (rapid)- processes – nuclei withT1/2  1 y. ; О – T1/2 < 1 y.; + ‑ predictions.

  4. I - METHOD: r –Process equations for the concentration calculations Concentrations n(A,Z) are changing in time(may be more than4000 equations): dn(A, Z)/dt= – (A, Z).n(A, Z) –n(A, Z).n(A, Z) + n(A+1, Z).n(A+1, Z) + + n(A–1, Z).n(A–1, Z) – n(A, Z).n(A, Z) + + (A, Z–1).n(A, Z–1) × P(A, Z–1) + + (A+1,Z–1).n(A+1,Z–1) × P1n(A+1,Z–1)+ + (A+2,Z–1).n(A+2,Z–1)×P2n(A+2,Z–1) +(A+3,Z–1)n(A+3,Z–1) × P3n(A+3,Z–1)+ + (A, Z) + Ff (A, Z), n andn — rates of (n,γ) and (γ,n) -reactions,=ln(2/T1/2) —-decay rate, P - probability of (A, Z) nuclide creation after –-decay of (A,Z-1) nuclide. Branching coefficients of isobaric chains - P1n, P2n, Р3ncorresponds to probabilities of one-, two- and three- neutrons emission in–- decay of the neutron-rich nuclei; the total probability of the delayed neutrons emission is the sum: Ff (A, Z)describes fission processes — spontaneous and beta-delayed fission. (A, Z) - neutrino capturing processes. Inner time scale is strongly depends on the nuclear reactions rates.

  5. II. NUCLEOSYNTHESIS WAVE MOVEMENT Concentrations: nА= forthree time moments calculated for r-process conditions: nn=1024 сm-3, Т9=1.= 109K Lutostansky Yu.S., et al. Sov. J. Nucl. Phys. 1985, v. 42. s s s

  6. β-Delayed processes in very neutron-rich nuclei Delayed neutron emission -(β, n) ------------------------------------ Multi-neutron β – delayed emission -(β, kn) ------------------------------------ β – delayed fission - (β,f) GTR GTR AR “pigmy”-resonances

  7. Beta – Delayed Multi-Neutron Emission Probability for (β, 2n) - emission: Probability for (β, kn) - emission: U, I(U) – energies and intensities in the daughter nucleus, Wn(U, E) – probability of neutron emission: qi and qf – level densities of compound and final nucleus, Тn(Е) — transitivity factor Lyutostansky Yu.S., Panov I.V., and Sirotkin V.K. “The -Delayed Multi–Neutron Emission.” Phys. Lett. 1985. V. 161B. №1. 2, 3. P. 9-13.

  8. BETA-DELAYED NEUTRONS IN NUCLEOSYNTESIS exp Calculated abandancies:1–with out (β,n)-effect; 2–with (β,n)-effect; in the relative units (Т=109 К, nn =1024 см-3). Calc.: Lutostansky Yu., Panov I., et al. Sov. J. Nucl. Phys. 1986. v.44.

  9. BETA-STRENGTH FUNCTION CALCULATIONS-1: COLLECTIVE ISOBARIC STATES protons neutrons G-T - SELECTION RULES: Δ j =0;±1 Δ j =+1: j =l+1/2 → j =l–1/2 Δ j =0: j =l+1/2 → j=l+1/2 Δ j = –1: j =l–1/2 → j =l+1/2 j =l–1/2→ j =l–1/2

  10. BETA-STRENGTH FUNCTION CALCULATIONS-2: MICROSCOPIC DESCRIPTION - 1 The Gamow–Teller resonance and other charge-exchange excitations of nuclei are described in Migdal TFFS-theory by the system of equations for the effective field: where Vpnand Vpnh are the effective fields of quasi-particles and holes, respectively; Vpnω is an external charge-exchange field; dpn1 and dpn2 are effective vertex functions that describe change of the pairing gap Δ in an external field; Γωand Γξare the amplitudes of the effective nucleon–nucleon interaction in, the particle–hole and the particle–particle channel; ρ, ρh, φ1 and φ2 are the corresponding transition densities. --------------------------------------------------------------------------- Effects associated with change of the pairing gap in external field are negligible small, so we set dpn1 = dpn2 = 0, what is valid in our case for external fields having zero diagonal elements [Migdal].Pairing effects are included in the shell structure calculations: ελ → Eλ = -------------------------------------------------------------------------- The selfcosistent microscopic theory used for the beta-strength function calculations.

  11. BETA-STRENGTH FUNCTION CALCULATIONS-3: MICROSCOPIC DESCRIPTION - 2 For the GT effective nuclear field, system of equations in the energetic λ-representation has the form[Migdal, Gaponov]: G-Tselectionrules: Δ j =0;±1 Δ j =+1: j=l+1/2 → j =l–1/2 Δ j =0: j=l±1/2 → j=l±1/2 Δ j = –1: j=l–1/2 → j=l+1/2 j =l–1/2→ j =l–1/2 where nλand ελare, respectively, the occupation numbers and energies of states λ. --------------------------------------------------------------------------------------------- Local nucleon–nucleon δ-interaction Γωin the Landau-Migdal formused: Г = С0 (f0′ + g0′σ1σ2) τ1τ2 δ(r1- r2) where coupling constants of: f0′ –isospin-isospinandg0′ –spin-isospin quasi-particle interaction with L = 0. ------------------------------------------------------------------------------------ Constants f0′and g0′ are the phenomenological parameters. Matrix elementsMGT :where χλν – mathematical deductions G-T values are normalized in FFST: Standard sum rule for στ-excitations: Effective quasiparticle charge is the “quenching” parameter of the theory.

  12. BETA-STRENGTH FUNCTION CALCULATIONS-1: MICROSCOPIC DESCRIPTION - 3 RESONANCE STRUCTURE OF BETA-STRENGTH FUNCTION 1. Discrete structure of beta-strength function. Partial function:Ci (old variant) = 2. Resonance structure of beta-strength function. Partial function: = The Bright-Wigner form for E > Sn Sn Гivalue up to Migdal is: Г = – 2 Im [∑ (ε + iI)] andГ =.ε | ε | + βε3 + γ ε2 | ε | + O(ε4)…, where Гi(i) = 0,018Ei2 МэВ • Exp. Krofcheck D., et al. Phys. Rev. Lett. 55 (1985) 1051. • - -Borzov I. Fayans S.,Trykov E. Nucl. Phys. А. 584 (1995) 335. • Borovoi A., Lutostan- sky Yu., Panov I., et al. JETP Lett. 45 (1987) 521 71Ge Yu. V. Gaponov and Yu. S. Lyutostansky, Sov. J. Phys. Elem. Part. At. Nucl. 12, 528 (1981).

  13. BETA-STRENGTH FUNCTION FOR127Xe Dependence from eg 1 - Breaking line – experimental data (1999):M. Palarczyk, et. al. Phys. Rev. 1999. V. 59. P. 500; 2 –Solid red line TFFS calculations with еq= 0.9 ; 3 - Solidblackline– calculations with еq= 0.8:Yu.S. Lutostansky, N.B. Shulgina. Phys. Rev. Lett. 1991. V.67. P.430;

  14. QUENCHING EFFECTfor127Xe 1 -Breaking line – experimental data:M. Palarczyk, et. al. Phys. Rev.59(1999) 500; 2 -line –TFFS calculations with еq= 0.9; Yu.S. Lutostansky, and V.N. Tikhonov. Bull. Russ. Acad. Sci. Phys. 76, 476 (2012). 3 -- - - –TFFS calculations with еq= 0.8: Yu.S. Lutostansky, and N.B. Shulgina. Phys. Rev. Lett. 67 (1991)430;

  15. QUENCHING EFFECT – EXPERIMENT Standard sum rule for στ-excitations: For G-T beta-strength function: In FFST [Migdal] theory: For experimental data sum rule:ΣB(GT) = must be =3.(N – Z). Ideal Emax= ∞ 127Xe 71Ge

  16. INTERACTION CONSTANTS For the (ττ) coupling constant f0/ the value f0/ = 1.35 was used, taken from comparison of calculating energy splitting between the analog and anti-analog isobaric states (IS) with the experimental data for the large number of nuclei [Gaponov, Lutostansky 1970 - 1972]. Three main parameters of FFST theory: eq, f0/, g0/ are taken from exp. and calc. data comparison. -------------------------------------------- eq – from “quenching” effect f0/ and g0/ – from energy splitting data 112-124Sn (3He, t) reaction For (στ) coupling constant g0/ value g0/ = 1.22 ± 0.04received from comparison of calculated energy differences between GTR and the low-lying “pigmy”-resonance with the experimental data for nine Sb isotopes[K. Pham, J. Jänecke, D. A. Roberts, etal., Phys. Rev. C 51 (1995) 526].

  17. BETA - DELAYED FISSION

  18. Beta – Delayed Fission Calculations Probabilities - Pβf : Beta Strength function: # Г(Е) widths approximation: Г(Е) = α·E2 + β·E3 + … where α ≈ 1/εF and β « α, so we used only the first term. # As Гf «Гn so neutron emission dominates when this energetically possible. # Sub-barrier fission probabilities in the daughter nucleus are small to gamma decay of exited states (barrier was taken in standard parabolic form). # Main dependence of Pβf is from barrier energy Bf .

  19. Neptunium Beta – Delayed Fission Calculations Yu.S. Lutostansky, V.I. Liashuk, I.V. Panov. “Influence of the delayed fission on production of transuranium elements in the explosive nucleosynthesis.” Preprint ITEP 90-25. 1990 Moscow.

  20. Dubnium Beta – Delayed Fission Calculations I. Panov, Yu. Lutostansky, F.-K. Thielemann 2013 Upper panel: the neutron beta-delayed emission probabilities Pβdn(dashed line), beta-delayed fission probabilities Pβdf(line) and number of delayed neutrons per one decay (in percents) In(dotted line) for isotopes of Dubnium (Z=105); down panel: total energy of beta-decay Qβ(line), neutron separation energy Sn(dashed line) and fission barriers (bold line) for the same isotopes (in MeV).

  21. Factor of the concentration losing in Prompt-process Npβ-delayed fission probabilities

  22. MODEL DESCRIPTION OF Sβ(E) - 1 Mat. model developed for the approximate solutions of equations of the FFST theory by the quasi-classical method. ----------------------------- 2 new parameters: E = EF(n) – EF(p) = Els – average energy of the spin–orbit splitting Wigner’s SU(4) super-symmetry restoration in the heavy nuclei Calculated (circles – ○) and experimental (■) dependencies of the relative energy y(x)=Δ(EGTR-EAR)/Els from the dimensionless value x=E/Els. Black circles (●) connected by line – calculated values for Sn isotopes.

  23. MODEL DESCRIPTION OF Sβ(E) – 2. T1/2 calculations 1988. Time of new nuclei synthesis β-decay time: Fermi-function: The dependence of r-process duration time on mass A-value under different external conditions: curve 1) – constant nn=1026 cm-3, T=1.5 109K; 2) – the same nn, T=1.109K; 3) – dynamical calc. with ρ0=2.105 g/cm5, T=1.109K [(t) = 0.ехp (-t/H), Т(t)= Т0.ехр(-t/3H)]. Yu.S. Lutostansky, and I. V. Panov. Astron. Letters. 14, no 2 (1988) 168.

  24. Neutrino capturing En GT and IAS Resonances in Sβ(E )-function n M2GTR≈ 3 (N-Z) eq2 M2IAS≈(N-Z)

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