E1 STRENGTH FUNCTIONS FOR PHOTOABSORPTION AND EMISSION WITHIN CLOSED-FORM METHODS

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1 E1 STRENGTH FUNCTIONS FOR PHOTOABSORPTION AND EMISSION WITHIN CLOSED-FORM METHODS 1. Introduction V.A. Plujko *), O.M.Gorbachenko *), R. Capote &), V.Bondar *) *) NPD, Taras Shevchenko National University, Kiev, Ukraine &) NDS, International Atomic Energy Agency, Vienna, Austria. Average description of gamma-transitions by radiative (gamma-ray) strength functions. Closed-form models 3. Renewed data base of the GDR parameters 4. Calculations and comparisons with experimental data. Role of folding procedure in microscopic calculations of RSF 5. Brink s hypothesis violation 7. Conclusions

2 INTRODUCTION Gamma-emission is one of the most universal channels of the nuclear de-excitation processes which can accompany any nuclear reaction Schematic picture of the nuclear decay

3 Radiative (photon) strength functions (RSF) Gamma-decay strength functions for c-d gamma-transitions average partial gamma-decay width Г s і f і f E λ = λ + 1 E D і for c-c gamma-transitions (E1 example) average level spacing d E s Γ ρ U 3 E1 i f E E, 1/ D E1 = ρ = de 3ρ U = U E f i

4 Photoexcitation strength functions (E1) r f σ = E1 E1 ( ) 3E π hc r photoabsorption cross-section r s feλ = Φ E T i feλ = Φ E Tf (, ), (, ) s Ti, Tf =ϕ( Ti, E ) - the temperatures of initial and final states

5 CLOSED-FORM COMPONENT OF E1 RSF DUE TO GDR EXCITATION Standard Lorentzian (SLO) [D. Brink. PhD Thesis(1955); P. Axel. PR 16(196)] Гr su f E Г r r + r ur = f ~ 0 ( E E ) E Г E 0 E r E Г = const ϕ ( E ) ~ 5 M ev ( T = 0) r Enhanced Generalized Lorentzian (EGLO) [J. Kopecky, M. Uhl, PRC47(1993)] [S. Kadmensky, V. Markushev, W.Furman, Sov.J.N.Phys 37(1983)] su EГ ( E, Tf ) 0.7 Г ( E = 0, Ti) f = + 3 ( E Er) + E Г ( E, Tf ) Er su f const 0[ E 0] + 4π Tf E E Г ( E, T ) = Г KE ( ) f r Infinite fermi- liquid (twobody dissipation) K ( E ) T f = U E a empirical factor from fitting exp. data ;

6 Weak points of the approximations Shapes of RSF within the models of realistic shape are, in fact, interpolations s f models { KMF } ( 0 ), SLO E = s s F fe E fe ( Er ) λ λ λ inconsistent with general relation between gamma-decay RSF of heated nuclei and nuclear response function on electromagnetic field ( with detailed balance principle for gamma-transitions between C-C states) (J.L.Egido, P.Ring, J.Phys.G:Nucl.Part.Phys.19(1993)1) Dependences of shape parameter ( width ) on gamma-ray energy and the final state temperature Γ ( E ) ( ) r T => Γ E Tf,,??? in expressions of EGLO and GFL models were introduced in phenomenological way by substitutions of the gamma-ray energy instead of GDR energy and the temperature of final states instead of initial states

7 RSF within modified Lorentzian (MLO) based on expression for gamma-width averaged on microcanonical ensemble of initial states dгif Г λ(j i,e ) = / Ni, N i = ρ(e,n,z,j i)( J i+ 1) E Z N de ν f,j f Z, N,M, ν i i dг de if λ + 1 λ( ) if δ ( i f ), λ ~ = d E B E E E d E * if = < f f fν f λµ i i iνi >, λµ = νν ' ν ν ' νν ' B J M E Q J M E Q q a a

8 General expression for gamma-decay RSF Transformations by Green-function method with the use of saddle point approximation lead to s ( ) 1 E f E = s =, T, MeV 1 exp h 8 3 ω f ( E ) Tf 1 s Tf T π ( ω, ) = χ'' ( ω, ) Peculiarity presence of low-energy enhancement factor 1 1 Ti N11 ph dε1dε f0( ε1) ( 1 f0( ε) ) δ( ε1 ε ω) ( E 0) 1 ω + h = = = >> h 1 exp E E1 Zero-energy limit su f ( E = 0) ~ T Φ ( ω >+ 0), Φ ( ω) χ ( ω)/ ω i E1 Xλ Xλ f ( E T ) f

9 Response function within semiclassical Landau-Vlasov approach is used in approximation of one strong collective state (spherical nuclei) ( T ) f Im χω, E Γ ( E, ) Tf ( ) (, ) r + Γ f E E E T E Parameter of line spreading ( energy-dependent width ) Γ E ω, T = h =Φ( h ) τ ( ω, T) Σ

10 METHOD OF INDEPENDENT SOURCES OF LINE SPREADING (Kolomietz, Plujko, Shlomo, PRC 54 (1996) 3014) h h h = + τ ( ω, T) τ ( ω, T) τw Σ coll Two-body collisional damping is relaxation due to coupling of 1p-1h to more complicated states (ph and so on) lying at the same excitation energy ( in the kinetic theory simulated by the collision integral. One-body dissipation corresponds to strength-function fragmentation caused by interaction of particles with the time-dependent selfconsistent mean field. In the RPA calculations this contribution to the damping indicates a redistribution of 1p-1h excitations in a vicinity of collective state ( in kinetic equation approach imitated by the one-body ("wall") relaxation). Deformation splitting is considered in approximation of axiallydeformed nuclei (-component RSF) with β, eff = f( Q[{ β j}])

11 PECULARITIES OF TWO-PARTICLE COLLISIONS IN PRESENSE OF AN EXTERNAL FIELD Energy conservation law is changed and becomes dependent on field frequency due to possibility of exchange of the energy between the particles and field hω It means changing arguments of function which is ensured the conservation of the energy of the colliding particles δ δ + => ) ( ε ε 1 ε ε 3 ε 4 ) δ ( ε ±hω As a result, a probability of two-body collisions is also dependent on field frequency

12 RSF for gamma-decay Modified Lorentzian approximation (MLO) (spherical nuclei) s f ( E ) E ( E, T ) 8 σ rγr Γ f exp ( ) f, = ( ) ( ) r + Γ f E T E E E T E Shape parameters ( widths ) Γ E T = a+ be α δ (, f ) + ct f MLO1( SMLO1) => Γ ( E T ) = b( E, + U ) = bu f f i MLO(3) => Γ ( E, Tf ) = a + be + ctf

13 RSF for photoabsorption r f n ( ) 8 j 3 E1 E = σ r, jγr, j, MeV j= 1 E E + Γ j( E) E E ( ) r, j Γ ( E ) MLO1( SMLO1) => Γ ( E ) = E MLO(3) => Γ ( E ) = a + be Γ Γ = Γ = =Γ j( E) j( E, T 0), j( E Er) r E, Γ, σ fitted parameters rj, rj, rj, n= in axially deformed nuclei

14 GDR parameters with uncertainties from renewed database V.A.Plujko, R.Capote, O.M. Gorbachenko, At.Data Nucl.Data Tables, 010, in press Shape parameters of the models were obtained by fitting the theoretical calculations for photoabsorption cross sections to reanalyzed experimental and estimated data from the EXFOR library and library of Moscow Photonuclear Data Center [with allowance for (, p) reaction ] Adjustment was performed by the least square method with minimizing δ δ ( ) min χ = ( E, ) σ exp ( E, ) σ ( i ) N 1 σ N N E theor i i par i= 1 exp, Energy dependent errors were used for estimated data Spherical nuclei Deformed nuclei E = δ + b E E min ( E,min ) = 0.1; δ = 0.5 r δ ( ) ( ) δ min + b E1 E, E < E1, E = δmin, E1 E E, δ min + b ( E E ), E > E.

15 E r GDR energies ( E E ) ( E E ) E1σ1+ Eσ 1+ 3; β > 0 ( σ = σ1) = = σ1+ σ 1+ 3; β < 0 ( σ = σ1/) σ 1 Systematic for renewed data : X 1/3 1/6 E = 4.755( I ) / A ( I ) / A ( MeV); I = ( N Z) / A r ----S.S. Dietrich, B.L. Berman(1988), O new values 1/3 1/6 E = 7.47 / A +.06 / A ( MeV) r 1/3 1/6 E = 31. / A / A ( MeV) r

16 GDR widths Curves: X - Γ r = 0.37Er 0.14Erβ 0.6 E + ( MeV) Γ 1 r = 0.06 Er ( MeV ), Γ r = 0.07 Er ( MeV ), S.S. Dietrich, B.L. Berman(1988) O new values

17 Energy weighted sum rule for isovector E1 transitions S = const S, S = σ( E ) de, S ( SLO) = π σ Γ EWSR r r r r, j r, j 0 j= 1 50 MeV Sr (SMLO) = σ ( E) de 0 Sr (TRK) = 60 NZ/ A (mb MeV) Mean value of enhancement factor of TRK sum rule ~ 1. n

18 Volume(J) and surface(q) coefficients of the symmetry energy 1/3 1/3 1 + E = a A / 1 a A, r a = c J; a = d J/ Q 1 bv 1/3 bs 9 sym = v /(1 + ), =, v =, = s v 4 I N Z J E b A I b J b A b Q J, J/ Q Myers et al. ( ) Lipparini et al. Abrosimov et al. c = 3 ( c =15/ 4) c = 15/ ( ) Used previously 36.8,.18 33, , , , 0.9 Sph. + axial def. nuclei 37.09(7),.4(03) 4.39(33), 1.91(0) 4.39(33), 0.95(01)

19 The photoabsorption cross sections and RSF Comparison of calculated photoabsorption cross-section on Nd with experimental data

20 Comparison of the photoabsorption cross section calculated with different database for GDR parameters: (a) - old systematics (Berman&Fultz); (b) - renewed GDR parameters. (a) Averaged HFB-QRPA microscopic approach by S. Goriely et al NP A706 (00) 17; A739 (004) 331 MSA - semiclassical moving surface method by V.I. Abrosimov, O.I.Davidovskaya Izvestiya RAN. 68 (004)00; Ukrainian Phys. Jour. 51 (006)34 Exp.data - V.A. Erokhova et al Izvestiya RAN. Seriya Fiz. 67 (003) 1479 (b)

21 Folding procedure within microscopic calculations without ph states ( ) + f E = f ( E, E ) f ( E ) de fold, E1 L E1 f ( E, E ) = L 1 Γ( E ) π E E ( E ) ( E )/4 ( ) +Γ R.D. Smith et al. P RC38 (1988)100; S.Drozdz et al. P Rep. 197 (1990)1; F.T.Baker et al. P Rep. 89 (1997)35

22 Width and energy shift for averaging HFB+QRPA results Γ ( E ) = a + a E + a E + a E 0 1 ( E ) = b + b E + b E S. Goriely et al. NPA 706, 17 (00); 739, 331 (004); S. Goriely, private communication

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24 Comparisons of gamma-decay RSF The E1 gamma-decay strength function on 144 Nd. The experimental data: Yu.P. Popov, in Neutron induced reactions, Proc. Europhys. Topical Conf., Smolenice, 198, Physics and Applications, Vol. 10, Ed.P.Oblozinsky (198) 11; M1 s f = const. Model EGLO SLO GFL MLO1 MLO MLO3 SMLO χ

25 Dipole strength functions of E 1 and M 1 gamma-decay for 90 Zr (а) and 100 Mo (b): U = S n. Experimental data are taken from R. Schwengner, et al. // Phys. Rev. C78, (008); Phys. Rev. C81, (010) Values of χ deviation of calculated gammadecay strength functions from estimated experimental data for nuclei 90 Zr, 9 Mo, 94 Mo, 96 Mo, 98 Mo, 100 Mo, 139 La. (a) Nucleus EGLO SLO GFL MLO1 SMLO 90 Zr Mo Mo Mo Mo Mo (b) average

26 Dipole strength functions of E 1 and M 1 gamma-decay for 9 Mo, 94 Mo, 96 Mo, 98 Mo: U = S. n Experimental data are taken from R. Schwengner, et al. // Phys. Rev. C81, (010)

27 Dipole strength functions of E 1 and M 1 gamma-decay for 166 Er (а) and 171 Yb (b): U = S n. Experimental data are taken from E. Melby, M. Guttormsen, J. Rekstad, A. Schiller, and S. Siem // Phys. Rev. C63, (001) and U. Agvaanluvsan, A. Schiller, J. A. Becker, L. A. Bernstein, et al. // Phys. Rev. C70, (004) (a) Values of χ deviation of calculated gammadecay strength functions from experimental data for nuclei 160 Dy, 16 Dy, 166 Er, 171 Yb, 17 Yb. Model EGLO SLO GFL MLO1 SMLO 160 Dy Dy Er Yb Yb (b) average

28 Averaging procedure in the Oslo method of gamma-decay strength function measurements U* i + / s exp = E1 U* i / f E 1 f E, U du E, U* i = * i * i * i 4, < * i 4, U E, U > E U 4, E U MeV i i E < U *, * i U i Sn A.V.Voinov & M. Guttormsen, private communications, 009

29 Effect of energy averaging within MLO1 (а) and EGLO (b) models at different excitation energies (a) (b) Open colored squares - averaged values of the RSF

30 Dipole strength functions of E 1 and M1 gamma-decay for 138 Ba (а) and 150 Sm (b): U = S n. Experimental data are taken from Vasilieva E.V., Sukhovoj A.M., Khitrov V.A. // Yad.Fiz., 001. V. 64. P. 3. (a) Values of χ deviation of calculated gammadecay strength functions from experimental data for nuclei 118 Sn, 138 Ba, 150 Sm, 146 Nd, 14 Te. Nucleus EGLO SLO GFL MLO1 SMLO 118 Sn Ba Sm Nd Te average (b)

31 A violation of the Brink hypothesis The gamma-decay RSF depends on excitation energy. The dependence is the most important for transitions with low gammaray energies where values of RSF are growing with excitation energy

32 Dipole strength functions of E 1 gamma-decay within MLO1 and EGLO models at different excitation energies Low-energy part of gamma-decay RSF increase with excitation energy (EGLO, GFL, MLO) s f ( E 0~ ) T = i const

33 Main conclusions Most reliable simple description of E1 RSF for gamma-decay can be obtained by the use of models of asymmetric shapes with dependence of line spreading on excitation energy The energy dependence of parameter of line spreading ( width ) is governed by complex mechanisms of nuclear dissipation&splitting (two-body collisions, strength fragmentation, deformation splitting ) and contributions of different relaxation channels are still open problem Renewed values of GDR parameters with uncertainties should be used for more reliable description of gamma-transitions and studies of GDR properties

34 Main conclusions Among other simple models, MLO approach potentially can lead to more reliable predictions of RSF for E1 gammatransitions between c-c states because it is based on general relations between RSF and nuclear response function in hot nuclei To better understand role of the temperature and energy dependence of the RSF, experimental data, especially in low gamma-ray energy range, are needed as the functions of both gamma-ray and excitation energies R.Capote, M.Herman, P.Oblozinsky, P.G.Young, S.Goriely, T.Belgya, A.V.Ignatyuk, A..J..Koning, S. Hilaire, V.A.Plujko et al., Nucl. Data Sheets 110 (009) 310; V.A.Plujko, R.Capote, O.M. Gorbachenko, At.Data Nucl.Data Tables, 010, in press

35 I want to thank the Organizers, especially Dr. Ronald Schwengner, for invitation, support and warm hospitality THANK YOU!!!

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