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Achievements and Open Problems in Photon Strength Functions of Heavy Nuclei

This report discusses the achievements and open problems in the study of photon strength functions (PSFs) of heavy nuclei. Topics include fragmentation of photon strength, Brink hypothesis, models for E1 and M1 PSFs, comparison of high-energy and photonuclear data, and the use of the DICEBOX algorithm. Additional topics covered are two-step gamma cascades, scissors resonances of excited nuclei, and supporting results from gamma-calorimetric measurements.

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Achievements and Open Problems in Photon Strength Functions of Heavy Nuclei

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  1. Photon Strength Functions of Heavy Nuclei: Achievements and Open Problems František Bečvář Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  2. – FZK Karlsruhe – NPI Řež – FZK Karlsruhe – CU Prague –LANL Los Alamos – NPI Řež – FZK Karlsruhe – FZK Karlsruhe Michael Heil Jaroslav Honzátko Franz Kaeppeler MilanKrtička Rene Reifarth Ivo Tomandl Fritz Voss Klaus Wisshak The major part of the reported results were achieved thanks to hard work of my colleagues Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  3. The relation between the photon strength functions (PSFs) and the photoabsorption cross section. Fragmentation of photon strength. Brink hypothesis Models for E1 and M1 PSFs Comparison between the high-energy (n,) data and the photonuclear data DICEBOX algorithm as a tool for testing the validity of models for PSFs at intermediate -ray energies Two-step  cascades: the essentials of the method and the survey of the results obtained Evidence for scissors resonances of excited nuclei Supporting results from Karlsruhe -calorimetric measurements Comparison with the data from (,’), (3He,3He’,) and (3He,) reactions Conclusions Outline Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  4. Cornerstones of the extreme statistical model of g decay • A notion of strength function • (fragmentation of strength of simple nuclear states) • Porter-Thomas fluctuations of partial radiation widths • Brink hypothesis • The principle of the detailed balance … and rather problematic absence of width correlation Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  5. Fragmentation of the GDER and the paradigm of the photon strength function Photonuclear x-section Photoabsorption x-section (γ, x) (γ,γ’) γ abs=(γ, x) + (γ,γ’) ≈10 eV Fluctuations Eγ ≈0.1 eV Bn ≈15 MeV Eγ <γ abs> …an energy-smoothed photoabsorption x-section Bn Eγ Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  6. Fragmentation of the GDER and the paradigm of the photon strength function The principle of the detailed balance: i i n n n “equivalent to” γ γ g.s. g.s. Photonuclear x-section Photoabsorption x-section (γ, x) (γ,γ’) γ abs=(γ, x) + (γ,γ’) ≈10 eV Eγ ≈0.1 eV Bn ≈15 MeV Eγ <γ abs> Bn Eγ Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  7. Fragmentation of the GDER and the paradigm of the photon strength function The principle of the detailed balance: i i n n n “equivalent to” γ γ g.s. g.s. Photonuclear x-section GDR Photoabsorption x-section (γ, x) (γ,γ’) γ abs=(γ, x) + (γ,γ’) ≈10 eV Eγ ≈0.1 eV Bn ≈15 MeV Eγ <γ abs> Bn Eγ If E1 transitions dominate <iγg.s.> = f(E1)(Eγ) Eγ3/(Ei) f(E1)(Eγ) = <γ abs(Eγ)>/(3 π2 h2 c2 Eγ) — Photon strength function Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  8. Fragmentation of the GDER and the paradigm of the photon strength function The principle of the detailed balance: i i n n n “equivalent to” γ γ g.s. g.s. f(XL)(Eγ) = <γ abs(XL)(Eγ)>/[(2L+1)π2 h2 c2 Eγ] <γ abs (Eγ)> = XL <γabs(XL)(Eγ)> Photonuclear x-section Photoabsorption x-section (γ, x) (γ,γ’) γ abs=(γ, x) + (γ,γ’) ≈10 eV Eγ ≈0.1 eV Bn ≈15 MeV Eγ <γ abs> Bn Eγ The paradigm of PSF: f(XL)andρare primordial, not derived quantities In a general case <iγg.s(XL).> = f(XL)(Eγ) Eγ2L+1/(Ei) Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  9. Brink hypothesis Fictitious photoexcitation experiments NRF experiments Photoexcitation pattern does not depend on initial excitation energy of a target nucleus Quasicontinuum g.s. Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  10. Brink hypothesis i i n n = γ g.s. g.s. Quasicontinuum <iγg.s(XL).> = f(XL)(Eγ) Eγ2L+1/(Ei) f(XL)(Eγ) = <γ abs(XL)(Eγ)>/[(2L+1)π2 h2 c2 Eγ] <γ abs (Eγ)> = XL <γabs(XL)(Eγ)> Brink hypothesis: generalization “g.s.”  “f” i i n n n n Identical ! = γ γ γ f f Fictitious (γ,γ‘) and (γ,x) experiments g.s. <iγf(XL).> = f(XL)(Eγ) Eγ2L+1/(Ei) NRF experiments f(XL)(Eγ) = <γ abs(XL)(Eγ)>/[(2L+1)π2 h2 c2 Eγ] <γ abs (Eγ)> = XL <γabs(XL)(Eγ)> Photoexcitation pattern of an excited target nucleus A target in photonuclear/photoabsorption experiment is in the ground state Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  11. Photon strength functions: problems The Brink hypothesis and the paradigm of the photon strength function: do they hold? (γ,x) – g.s. only (n,γ) at isolated neutron resonances, ARC (γ,γ') – NRF, g.s. only (n,γ) – two-step & n-step cascades (thermal and keV neutrons) (3He,3He’γ), (3He,αγ) (p,p’γ) at small angles of scattering Singlesγ-ray spectra from (n,γ), total radiation widths γ tot Bn (γ,γ') Is the E1 part of <γ abs(Eγ)> for energies near and below neutron threshold an extrapolation of a Lorentzian GDR? Does the energy dependence of <γ abs(Eγ)> display additional, statistically significant resonance-like structures in the region near or below the neutron threshold? γ abs To which extent the other multipolarities (M1, E2, …) contribute to photoabsorption cross section <γ abs(Eγ)> ? The main obstacles in studying PSFs: - strong Porter-Thomas fluctuations of iγf - limited knowledge of other quantities (γ,x) Eγ Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  12. First evidence for validity of Brink hypothesis k = 4.90.5 Data of L. M. Bollinger and G. E. Thomas, PRL 25 (1967) 1143 195Pt(n,g)196Pt at filtered neutron beam Transitions to 0+ and 2+ levels Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  13. First evidence for validity of Brink hypothesis L. M. Bollinger and G. E. Thomas, Phys. Rev. Lett. 18 (1967) 1143: “There is no adequate, theoretically based explanation of the Eg5 dependence of the experimental widths, although an idea introduced by Brink and developed by Axel may be relevant.” at <Glgf>l Eg5 For Lorentzian E1 GDR with a constant damping width for a typical heavy nucleus: … the birth of the extreme statistical model for g decay of nuclear levels implications for complete spectroscopy of low-lying levels (ARC method) Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  14. Further evidence … ARC data from 147Sm(n,g)148Sm reaction D. Buss and R. K. Smither, PRC 2 (1970) 1513 ... convincing? Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  15. Models used for E1 PSF 1. Axel-Brink model: The damping width due to strongly interacting quasiparticles Kadmenskij, Matkushev, Furman (1982) 2. Model of Kadmenskij, Markushev and Furman: Landau-Migdal parameters … energy and temperature dependence An approximation for low -ray energies Not justified for deformed nuclei At T>0 it gives a non-zero limit for E→0 Diverges for E→ EG Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  16. Models used for E1 PSF 3. Model GLO: R. E. Chrien, Dubna Report No. D3, 4, 17-86-747 (1987) Divergence for E→ EG removed 4. Semi-empirical model EGLO: , adjustable parameters J. Kopecky, M. Uhl and R. E. Chrien, PRC 47 (1993) 312 A good description for spherical, transitional and deformed nuclei achieved at energies E > 6 MeV Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  17. The damping width due to strongly interacting quasiparticles Kadmenskij, Matkushev, Furman (1982) The spreading width due to the interaction between dipole and quadrupole vibrations J. Le Tourneux, Mat. Fys. Medd. Dan. Selsk. 34 (1965) 1 The condition imposed on the modified width Summing the damping and spreading widths not fully legitimate For a typical deformed nucleus with β2>0.27, EG (low)≈12 MeV and ΓG(low)<3.2 MeV the damping width becomes negative Model of Mughabghab and Dunford a generalization of KMF model for description of PSFs of deformed nuclei S. F. Mughabghab and C. L. Dunford, PLB 487 (2000) 155 Problems: … the model is not applicable to well deformed nuclei Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  18. Models for M1 PSF 3. The scissors-resonance temperature-independent model Photon strength assumed not to depend on nuclear temperature F. Becvar et al., PRC52 (1995) 1278 1. The single-particle model A constant PSF 2. The spin-flip model Analogous to AB model for E1 PSF Parameters deduced from the (p,p’) data Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  19. Comparison between photonuclear and (n,) data Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  20. Comparison between photonuclear and (n,) data Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  21. Comparison between photonuclear and (n,) data Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  22. Comparison between photonuclear and (n,) data Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  23. Comparison between photonuclear and (n,) data Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  24. Comparison between photonuclear and (n,) data Double-Lorentzian fit Single-Lorentzian fit Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  25. Comparison between photonuclear and (n,) data Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  26. Behavior of photonuclear data for nuclei with N ≈ 50 No (n,) data Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  27. Comparison between photonuclear and (n,) data main conclusions PSFs of spherical and transitional nuclei in -ray energy region of 6-8 MeV are appreciably lower compared to predictions of the Axel-Brink model For spherical and transitional nuclei near A = 100 the photonuclear data also indicate a deficit of PSF with respect to predictions of the Axel-Brink model The KMF model for E1 PSF agrees with experimental data only in case of nuclei near A=100 Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  28. For nuclear levels below certain “critical energy” spin, parity and decay properties are known from experiments Energies, spins and parities of the remaining levels are assumed to be a discretization of an a priori known level-density formula – a random discretization with the Wigner-type long-range level-spacing correlations A partial radiation width if(XL), characterizing a decay of a level i to a level f, is a random realization of a chi-square-distributed quantity the expectation value of which is equal to f(XL)(Eγ) Eγ2L+1/(Ei), where the PSFs f(XL)and the density ρare also a priori known Selection rules governing the  decay are fully observed Any pair of partial radiation widths if(XL) is statistically uncorrelated Depopulating intensities are given by Iif = [ΣXLfΓif (XL)]/[ΣX΄L΄f ΄if ΄(X΄L΄)] Simulation of  cascades by means of DICEBOX algorithm Assumptions: Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  29. Specificity: in line with the basic assumptions the branching intensities Iif = P(f| i) are realizations pertinent to the corresponding random variables ~ Probability of reaching a level after n-th step of a cascade: , where level energies . 1-3 × 106 levels A complete decay scheme is known for at most 100 levels, i.e. for a 10– 2 % fraction Necessity to generate the rest part of the decay scheme Problem: the need to generate an enormous number of depopulating intensities Iff’ = P(f ’| f), up to 1013 ! ~ A straightforward approach to simulating  cascades “Distant past of  cascading is irrelevant given knowleldge of the recent past” … Markovian process Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  30. An infinite number of NRs exist for a given level-density ρand a set of photon strength functions f(XL) Of these NRs only one of characterizes the behavior of a given nucleus Simulations of  cascades based on the use of various NRs lead to mutually different predictions of the cascade-related quantities Simulations of  cascades based on the use of various NRs lead to mutually different predictions of the cascade-related quantities To assess uncertainties of these predictions simulations of  cascades are to performed for a large number of NRs A notion of a nuclear realization A set of realizations{Ifi} for all possible i,f and a set of realizations {Ef , Jf , πf} for all levels f ≡ A nuclear realization Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  31. Getting a “nuclear realization” Bn 0 • Generate a level system (excitation energies, spins and parities); • take experimentally known levels and add the levels from • discretization according a chosen level-density formula Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  32. Getting a “nuclear realization” Bn 0 • Ascribe at random a precursors (a random generator seed) • to each level of the system Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  33. Getting a “nuclear realization” • Ascribe at random a precursors (a random generator seed) • to each level of the system Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  34. γ-cascading - Start with the neutron capturing level - Pick up its precursor Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  35. γ-cascading ... - Using this precursor generateGigf - Deduce Ig’s Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  36. γ-cascading ... - Generate a random number from the uniform distribution in (0,1) Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  37. γ-cascading ... - Get the energy of the depopulating transition Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  38. γ-cascading ... • Determine the first intermediate level - Pick up its precursor Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  39. γ-cascading ... - Using this precursor generateGigffor the intermediate level - Deduce Ig’s Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  40. γ-cascading ... - Generate another random number from (0,1) Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  41. γ-cascading ... - Get the energy of the next depopulating transition Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  42. γ-cascading ... • Determine the next intermediate level - Pick up its precursor Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  43. γ-cascading ... - Using this precursor generateGigffor the intermediate level - Deduce Ig’s Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  44. γ-cascading ... - Generate another random number from (0,1) Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  45. γ-cascading ... - Get the energy of the next depopulating transition Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  46. γ-cascading ... - Determine the next intermediate level - Pick up its precursor Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  47. γ-cascading ... - Using this precursor generateGigffor the intermediate level - Deduce Ig’s Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  48. γ-cascading ... - Generate a random number from (0,1) Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  49. γ-cascading ... - Get the energy of the next depopulating transition - This transition reaches the ground state – the cascade ends Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

  50. The outcome of γ-cascading - The number of steps of a given cascade is known • So also the energies of the individual transitions and the • intermediate levels involved • The procedure described is reiterated many times. Typically a set • of 100 000 cascades are obtained for a given nuclear realization • The whole this process in repeated for an enough • large number of other nuclear realizations • Various cascade-related quantities can be deduced and • their residual Porter-Thomas r.m.s. uncertainties estimated Workshop on Photon Strength Functions and Related Topics, Prague, June 17-20, 2007

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