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General self-consistent method for the construction

J. Kvasil 1) , P. Vesely 1) W. Kleinig 2) V.O. Nesterenko 3) P.-G. Reinhard 4). 1). Inst itute of Part icle and Nucl ear Phys ics, Charles University, CZ-18000 Praha 8, Czech Republic Techn ical Univ ersiy of Dresden, Institut e f or Analysis, D-01062, Dresden, Germany

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General self-consistent method for the construction

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  1. J. Kvasil 1) , P. Vesely 1) W. Kleinig 2) V.O. Nesterenko 3) P.-G. Reinhard 4) 1) Institute of Particle and Nuclear Physics, Charles University, CZ-18000 Praha 8,Czech Republic Technical Universiyof Dresden, Institute for Analysis, D-01062, Dresden, Germany Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna,Moscow region, 141980, Russia Institute of Theoretical Physics II, University of Erlangen, D-91058, Erlangen, Germany 2) 3) 4) General self-consistent method for the construction of separable residual interaction - application for Skyrme functional

  2. Motivation • Brief formulation of the SRPA approach • Using the SRPA for the Skyrme functional • - analyses of E1 (T=1) and E2 (T=0) resonances • in 154Sm, 238U, 254No • - E1 (T=1) and E2 (T=0) resonance analyses in 134-158Nd • the influence of the residual interaction • the effect of T-odd current • the exploring the nuclear surface or interior of nucleus • depending on the radial dependence of the exciting • external fields

  3. motivation 1. Motivation • Effective n-n interactions (Skyrme or Gogny) are widely used for the description of the static characteristics of spherical and deformed nuclei • Dynamics of small amplitude vibrations is mainly described by the RPA. The solving of the RPA problem requires a diagonalization of matrices involving the matrix elements of the effective n-n interaction in the huge p-h (two-quasiparticle) space technical problemslimited number of papers with the RPA based on the Skyrme or Gogny effective interactions see e.g. M.Bender, P.-H.Heenen, P.-G.Reinhard, Rev.Mod.Phys. 75, 121 (2003) and citation therein Treating of deformed nuclei requires even higher dimensions of matrices to be diagonalized

  4. motivation • RPA problem becames simpler if the residual two-body interaction • is factorized: where and are ph- or two-quasiparticle parts of the s.p. operators • in the separable case it is possible to avoid the problem of • building and diagonalization of huge matrices • we will show that it is possible to formulate the separabilization • of residual interactions in a fully self-consistent manner and provide • a high accuracy with a small number (K < 3-4) of separable terms

  5. motivation • during last decades several “self-consistent” schemes were • proposed – see e.g. - D.J.Rowe, Nuclear Collective Motion ( Methueu, London, 1970) - A.Bohr, B.R.Mottelson, Nuclear Structure II (Benjamin, N.Y., 1975) - E.Lipparini, S.Stringari, Nucl.Phys. A371, 430 (1981) - T.Suzuki, H.Sagava, Prog.Theor.Phys. 65, 565 (1981) - T.Kubo, H.Sakamoto, T.Kammuri, T.Kishimoto, Phys.Rev. C54, 2331 (1996) however, they are connected with simple analytical or numerical estimations; or more promising: - N.Van Giai, Ch.Stoyanov, V.V.Voronov, Phys.Rev. C57, 1204 (1998) - A.P.Severyuchin, Ch.Stoyanov, V.V.Voronov, N.Van Giai, Phys.Rev. C66, 034304 (2002) but based only on the technical speciality of Skyrme interaction and leading to the diagonalization of matrices with dimension > 100 • we developed a general self-consistent separable RPA (SRPA) approach • applicable to any density- and current- dependent functional: • sperical nuclei: • deformed nuclei: V.O.Nesterenko, J.Kvasil, P.-G.Reinhard, Phys.Rev. C66, 044307 (2002) V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard Phys.Rev. C74, 064306 (2006)

  6. motivation • in the SRPA one – body operators and • and associated strengths and are derived from given • energy – functional without any adjustable parameters. The succes of • the SRPA becomes possible because of the following factors: • an efficient self-consistent procedure • proper inclusion of all parts of residual interaction, time-even as • well as time-od terms • incorporation of the symmetries (translation, particle number, etc.) • building the separable operators in such a way that they have • maxima at different slices of the nucleus and thus cover both the • surface and interior dynamics • the SRPA can be applied to a wide variety of finite Fermion • systems – e.g. for the Kohn-Sham functional in the description • of linear dynamics of valence electrons in spherical and deformed • atomic clusters – see: • - V.O.Nesterenko, W.Kleinig, V.V.Gudkov, N.Lo Iudice, J.Kvasil, • Phys.Rev. A56, 607 (1997) • - V.O.Nesterenko, P.-G.Reinhard, W.Kleinig, D.S.Dolci, • Phys.Rev. A70, 023205 (2004)

  7. SRPA equations 2. Brief formulation of the SRPA approach 2.1. SRPA equations Starting energy functional: where is the time-dependent Slater determinant or time-dependent quasiparticle vacuum. Time-dependent densities and currents are: Time-dependent Slater determinant can be related to the equilibrium Slater determinant by (see E.Lipparini, S.Stringari, Nucl.Phys. A371, 430 (1981)) with T-even generators and T-odd generators :

  8. SRPA equations and are T-even and T-odd periodically time- dependent deformations, respectively: The equilibrium Slater determinant (HF ground state) is given by the HF equation which gives also the HF mean field: In the small amplitude limit (up to the linear order in the deformations and the time- dependent Slater determinant is:

  9. SRPA equations Therefore for the time-dependence of densities: with and similarly for time-dependent (vibrating) s.p. mean field: where is the static HF mean field and time- dependent vibrating part is

  10. SRPA equations since: where where where enumerates T- even densities enumerates T- odd densities where enumerates T- even densities enumerates T- odd densities with and

  11. SRPA equations Similarly as for (see p.8) we can write for the time- dependent variations of and where we introduced inverse strength constants and

  12. For the determination of the vibrating shifts and : the TDHF or TDHFB approach can be used starting from the Thouless theorem for the the vibrating Slater determinant: Using we can express to obtain alternative expressions for and . By comparison through with previous ones we finallly have a system of equations for unknown amplitudes ,

  13. where we introduced following matrices: • the matrix of the eq. system for and is symmetric and • real • this eq. system has nontrivial solution only if the determinat of its • matrix is zero, - dispersion equation for

  14. SRPA equations It can be shown that the eq. system for and is the same as that one obtained from the standard RPA equations: with the RPA Hamiltonian: where is the HF average field and is the residual interaction (see p. 3): p-h ( two-qp ) part of corresponding operator

  15. SRPA equations • Advantages of the SRPA: • instead of the construction and diagonalization of huge matrices it • is sufficient to solve the system of eqs. for and with • matrix of the dimension 4K ( K is the number of modes) • in the opposite to the standard separable interaction RPA the SRPA • method gives the receipt for the determination of the strength • constant and . • only a few correctly chosen exciting modes are sufficient • for the description of each giant resonance of given type and • multipolarity

  16. Skyrme 2.2. Using the SRPA for the Skyrme interaction We use the Skyrme energy density for the energy functional - see e.g. J.Dobaczewski, J.Dudek, Phys.Rev. C52, 1827 (1995): with gauge invariance interaction parameters

  17. Skyrme Alternative expression for Skyrme energy density: (see e.g. V.O.Nesterenko, J.Kvasil, P.-G.Reinhard, PRC 66, 044307 (2002)) parameters b are unambiguously connected with parameters C for parameters b are unambiguously connected with parameters C for

  18. Skyrme The dependence of the energy density on goes through the following densities and currents: density kinetic energy density spin-orbit current current spin-current pairing density kinetic energy – spin current

  19. Skyrme time - even time - odd These densities and currents are used for the construction of operators and .

  20. exc. modes 2.3. The choice of external exciting modes. we should start with operators taken in the form of multipole operators analysis of rezonance • if no long-wave approximation : • if long-wave approximation: In this paper – analyses of isovector E1 and isoscalar E2 resonances with , the main contribution to is expected from corresponding to with explore the nucleus surface corresponding to with explore the nucleus inetrior In this paper we analyse E1 and E2 resonance with alone or with simultaneously ( it is enough for the resolution of 2 MeV )

  21. 3. Results 3.1 analysis of isovector E1 and isoscalar E2 resonances in 154Sm, 238U, 254No ( this choice covers nuclei from rare-earth up to super-heavy region) • analyses were done with the representative set of Skyrme forces: • SkT6 (F.Tondeour, M.Brack, M.Farine, J.M.Pearson, • Nucl.Phys. A420 (1984) • SkM* (J.Bartel, P.Quentin, M.Brack, C.Guet, H.-B.Haakansson, • Nucl.Phys. A386, 79 (1982) • SLy6 (E.Chabanat, P.Bonche, P.Haensel, J.Meyer, R.Schaeffer, • Nucl.Phys. A627, 710 (1997) • SkI3 (P.-G.Reinhard, H.Flocard, Nucl.Phys. A584, 467 (1995) • with different features ( different isoscalar and isovector masses, • symmetry energies etc.) Force m0/m asym m1/m Q2[b] [MeV] 154Sm 238U 254No SkT6 1.00 30.0 1.00 6.8 11.1 13.7 SkM* 0.79 30.0 0.65 6.8 11.1 14.0 SLy6 0.69 32.0 0.80 6.8 11.0 13.7 SkI3 0.58 34.8 0.80 6.8 11.0 13.7 exp 6.8 11.0 13.7

  22. Calculations were done in the cylindrical coordinate-space grid with a • spacing of 0.7 fm. Pairing is treated at the BCS level. The collective • response for the GDR ( ) and GQR ( ) is computed with • two input operators (see p. 23) with k=1 and k=2. Both giant • resonances are calculated in terms of the energy-weighted (L=1) • strength function with the averaging parameter . • effective charges: • We use a large configuration space involving the s.p. spectrum from • the bottom up to 16 MeV 7000 – 10000 dipole and 11000 – 17000 • two-quasiparticle configurations in the energy interval 0 – 100 MeV. • The relevant energy-weighted sum rule is exhausted by 95-98 %.

  23. Isovector E1resonance ( data from photoabsorption) (dependence on Skyrme force type and on number of input operators) experiment -S.Dietrich, et al. At.Data.Nucl.Tables 38, 199 (1998) full E1 strength with two input operators (k=1+k=2) full E1 strength with one input operator (k=1) – only for 254No 2 – quasiparticle part of E1 strength function (residual interaction switched off ) proper right shift caused by collective part of the str. function for SkM* - artificial right shoulder (especially for 154Sm)

  24. Isoscalar E2resonance ( data from ( a,a ) reaction ) (dependence on Skyrme force type and on number of input operators) experiment –D.H.Youngblood, et al., Phys.Rev. C69, 034315 (2004) full E2 strength with two input operators (k=1+k=2) full E2 strength with one input operator (k=1) – only for 254No 2 – quasiparticle part of E2 strength function (residual interaction switched off ) • proper left shift caused by the • collective part of the str. function • two-qp. str. maximum is • shifted to higher E with decreasing • m*/m. Similar shift seen also for E1 • but in E1 case this was compensated • by right collective shift. two-qp. str. function maximum is shifted to higher E with decreasing . • ppp • for SkT6 the best agreement

  25. Isovector E1 and isoscalar E2 resonances in 238U (dependence on the presence of time-odd densities) full E2 strength with two input operators (k=1+k=2) and with time-even and time-odd densities Full E1 and E2 strength with Two input operators (k=1 + k=2) and with time-even densities only (time-odd densities are switched off) For SkT6 practically there is no time-odd shift (because the dominant terms in the functional giving a contribution to the time-odd shift are terms with b1, b’1 which are switched off in the case of SkT6 ).

  26. Demonstration of the connection between the exploring the nucleus surface and interior and the radial dependence of the exciting operator in cylindrical coordinates:

  27. 3.2 analysis of isovector E1 and isoscalar E2 resonances in 134-158Nd following Skyrme forces were used: SkT6 , SkO , SkM* , SGII , SIII , SLy4 , SLy6 , SkI3 effect of time-odd density with the correlation with respect to isoscalar effective mass was analysed SkT6 SkM* SIII SLy6 SkO SGII SLy4 SkI3

  28. E1 (T=1) giant resonance in 150Nd experiment P.Carlos et al., NPA 172, 437 (1971) B.L.Bergman et al, RMP 47, 713 (1971) A.V.Varlamov et al., Atlas of Giant R., INDC(NDS)-394, 1999 JANIS database with time-odd current without time-odd current

  29. E2 (T=0) giant resonance in 150Nd with time-odd current without time-odd current

  30. E1 (T=1) giant resonance experiment P.Carlos et al., NPA 172, 437 (1971) B.L.Bergman et al, RMP 47, 713 (1971) A.V.Varlamov et al., Atlas of Giant R., INDC(NDS)-394, 1999 JANIS database right panels: with time-odd current without time-odd current left panels: - left small peak - right big peak

  31. E1 (T=1) giant resonance full E1 strength including T-odd current dotted lines: - left small peak - right big peak

  32. Centroid energy (upper panel), width (middle panel) of E1 (T=1) resonance, and quadrupole moment in dependence on A for Nd isotopes

  33. -- Self-consistent SRPA is an effective and economical method suitable for systematic RPA calculations and analysis for deformed nuclei. -- Goodagreement with experimentaldata for isovector E1 giant resonance (especially for SkI3) and for isoscalar E2 resonance (especially for SkT6). -- Structure of E1resonance (especially its right flank) isvery sensitive to Skyrme forcesand related effective masses. Hense this resonance can serve as an additional test for selection of Skyrme parameters. -- Correlation between , and time-odd impacts. These factors originate from one term of Skyrme functional and so their correlation seems to be quite natural. Good point for further studies. Conclusions: m0*/m m1*/m

  34. References: 1) V.O. Nesterenko, J. Kvasil, and P.-G. Reinhard, "Separable random-phase-approximation for self-consistent nuclear models", Phys. Rev. C66, 044307 (2002). 2) W. Kleinig, V.O. Nesterenko, and P.-G. Reinhard, "Electric multipole plasmons in deformed sodium clusters ", Ann.Phys. (N.Y.) 297, 1 (2002). 3) V.O. Nesterenko, J. Kvasil, and P.-G. Reinhard, "Practicable factorized TDLDA for arbitrary density- and current-dependent functionals", Progress in Theor. Chem. and Phys., 15,127 (2006);ArXiv: physics/0512060. 3) V.O. Nesterenko, J. Kvasil, W. Kleinig, P.-G. Reinhard, and D.S. Dolci, "Self-consistent separable RPA approach for Skyrme forces: axial nuclei", ArXiv: nucl-th/0512045. 4) V.O. Nesterenko, W. Kleinig, J. Kvasil, P. Vesely, P.-G. Reinhard, and D.S. Dolci, "Self-consistent separable RPA for Skyrme forces: giant resonancesin axial nuclei” Phys.Rev C74, 064306 (2006). 5) V.O. Nesterenko, W. Kleinig, J. Kvasil, P. Vesely, P.-G.Reinhard, Int.Jour.Mod.Phys. E16, 624-633 (2007)

  35. Appendix: Transition probabilities and strength functions Reduced transition probability from the RPA ground state to the one- phonon excited state with the eigen energy is where and are solutions of RPA system of Eqs., (el) or (mag), and where , for electric type ( ) transition and , for magnetic type ( ) transition

  36. transitions Reduced transition probability can be rewritten: where energy weighted strength function: with the averaging function: The substitution for into the expr. for and using the Cauchy theorem gives:

  37. transitions where the first term is a contribution going from the residual interaction and the second term is the contribution of the mean field.

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