TEST OF CLOSED-FORM GAMMA-STRENGTH FUNCTIONS. Vladimir Plujko. Taras Shevchenko National University; Institute for Nuclear Research, Kyiv, Ukraine

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1 TEST OF CLOSED-FORM GAMMA-STRENGTH FUNCTIONS Vladimir Plujko Taras Shevchenko National University; Institute for Nuclear Research, Kyiv, Ukraine 1. Radiative (photon, gamma) strength functions (RSF)- definitions. Modified Lorentzian approximation 3. Calculations, comparisons and conclusions

2 Radiative strength functions (RSF=PSF=GSF) d Γ ( E ) s ( U U E ) ρ = E1 3 3 ( ) f f i = E fe1 E de ρi i s T ( E )~ E f ( E ) 3 E1 π E1 ( U ) σ ( E ) = 3 E ( πhc) f ( E ) E1 E1 r s f ( E ) = F E, T f ( E ) = F ( E, T ), T =ϕ( T, E ) ( ), αλ αλ i αλ αλ f f i r

3 Statistical method of RSF calculations dг if Г λ(j i,e ) = / Ni, N i = ρ(e,n,z,j i)( J i+ 1) E Z N de dг de based on expression for gamma-width averaged on microcanonical ensemble of initial states if ν f,j f Z, N,M, ν i i 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 V.A.Plujko (Plyuiko), Sov.J.Nucl.Phys. 5 (1990) 639; Proc. 9th Inter. Conf. Nucl. Reaction Mechanisms, Varenna, June 5-9, 000, Universita degli Studi di Milano, Suppl. N.115 ( 000)113

4 s 8 ( ) E f E = χ'' ω =, Tf π 1 exp h ( E T) ( E ) Tf Low-energy enhancement factor 1 1 T = ( )( ( )) 11 i N ph dε1dεn ε1 1 n ε δ( ε1 ε+ hω) E 0 => >> 1 1 exp hω E i RSF for gamma-decay Transformations within Green-function method and saddle point approximation lead to Non zero low-energy limit MeV 3 su 4 e χ Xλ( ω, Ti) f E1( E hω = 0) = 9 ( ) T 0 3 i h hc ω ω => 0

5 MODIFIED LORENTZIAN MODELS (MLO) ---- CLOSED-FORM EXPRESSIONS FOR AVERAGE E1 RSF Based on approximation that only one strong collective state determines response nucleus on electromagnetic field and due to this can be suited to describe RSF component resulted from GDR excitation ( ω E h T) χ" = /, Γ E, T ( E, T) ( ) + Γ(, ) r - parameter of line spreading ( energy-dependent width ) There are many variants of derivations; see for references: R. Capote et al, Nucl.Data Sheets, 110(009)3107; V.A.Plujko, O.M. Gorbachenko, E.V.Kulich, Int.J.Mod.Phys. E18 (009) 996 E Γ E E E T E

6 METHOD OF INDEPENDENT SOURCES OF LINE SPREADING Γ E = hω, T =Γ E = hω, T +Γ coll frag Energy-dependent component results from two-nucleon scattering in external E1 field (spreading width) GDR (1p1h coherent state) For kinetic equation description of nuclear dynamics, energy-dependent component of width results from memory-dependent collision integral ( r, p, ω) rr t r δ n J ( prt,, ) = dt R( t t ) δn( r, p, t ) => => Γ E =, T τ ( ω, T ) c r r > ph It caused by frequency dependence of energy conservation law for scattering in external field due to possibility of energy exchange between the particles and field δ ( ε ε 1 + ε ε 3 ε 4 ) => δ ( ε± hω) ( hω )

7 Energy-dependence of spreading width j Γ E ω, T ae = h = + bg( T) => GDR ->ph coll j 1 j Fragmentation ( almost energy-independent ) component Γ => frag GDR -> 1p1h [wall formala] + β-vibrations β-vibrations: J. Le Tourneux. Mat. Fys.Medd. Dan. Vid. Selsk. 34 (1965) 1 S.F. Mughabghab,C.L. Dunford, PL B487(000) 155 GENERAL SHAPE OF ENERGY-DEPENDENT WIDTH E T n a j j ct k 1 Γ, = hω = + b E + j

8 r f RSF for photoabsorption APPROXIMATION OF AXIALLY-DEFORMED NUCLEI RSF for gamma-decay ( MeV 3 ) RSF for photoabsorption on cold nuclei ( ) 8 j E1 E = σ r, jγr, j j= 1 E E + Γ j( E,0) E ( ) r, j Normalization condition for widths Γ ( E = E T= 0) =Γ E j r, j, r, j Γ ( E,0) (, ) s 8 ( ) E Γ, = j E Tf f E T σ ( ) r, jγr, j 1 exp E Tf j= 1 E E E, T E ( ) ( ) r, j + Γ j f

9 j Energy-dependent width within MLO4 Γ = a E + a β E r, j 1 r, j r, j j systematics V.A.Plujko, R.Capote, O.M. Gorbachenko, At.Data Nucl.Data Tables 97(011) 567; Nucl. Phys. At.Energy 13(01)341 1, sph. nucl. = ( ) 1.6 R0 / R j, def.nucl.[b.bush,y.alhassid, NPA 531 (1991) 7] 7/3 = A E β 14 / + [L. Esser, U. Neuneyer, R. F. Casten, P. von Brentano, PRC 55(1997) 06] 1 E + 1 from experimental data-base (RIPL3) or systematics by 5/6 S.Hilaire, S.Goriely, NPA 779(006)63 E + = 65 A /( E shell ) Γ j( E, T) = bj { a1 [ E + U( T)] + a β Er, j j} SLO, EGLO, GFL, MLO1-3, SMLO: β = ϕ Q β j 1 ( [{ }]

10 Comparisons of gamma-decay RSF Comparisons of gamma-decay strength functions for 98 Mo and 167 Er : experimental data -- E. Melby, M. Guttormsen, et al., Phys. Rev.C. 63, (001); U. Agvaanluvsan, A. Schiller, et al., Phys. Rev.C. 70, (004); Sn 1 s (, = ), 1< 4, 4 f E U f Ui E dui E s Sn 4 faver ( E ) = Sn 1 s (, = ), 4 <, f E U f Ui E dui E Sn S n E E

11 168 Er 187 W Comparisons of gamma-decay strength functions for and : U = S n ; experimental data -- A.M. Sukhovoj et al. Izvestiya RAN. Seriya Fiz. 69, 641 (005); A.M. Sukhovoj et al. in Proc. of the XV Int. Seminar on Interaction of Neutrons with Nuclei. (Dubna, May 007), 9 (007).

12 9 Mo 96 Mo Comparisons of gamma-decay strength functions for and : U = S n ;experimental data -- R. Schwengner, G. Rusev, et al., Phys. Rev. C. 78 (008) ; Phys. Rev. C. 81, (010)

13 Average ratio of chi-square deviations of the theoretical calculations from experimental data: n ( χ ) =1 i ( model)/ χi (SLO) / n i n - number of nuclei Model Exp.Data n MLO4 SMLO MLO1 GFL EGLO Oslo [1] Dubna [] Dresden [3] E. Melby, M. Guttormsen, et al., Phys. Rev.C. 63, (001); U. Agvaanluvsan, A. Schiller, et al., Phys. Rev.C. 70, (004); A.M. Sukhovoj et al. Izvestiya RAN. Seriya Fiz. 69, 641 (005); A.M. Sukhovoj et al. in Proc. of the XV Int. Seminar on Interaction of Neutrons with Nuclei. (Dubna, May 007), 9 (007). 3. R. Schwengner, G. Rusev, et al., Phys. Rev. C. 78 (008) ; Phys. Rev. C. 81, (010)

14 Gamma-ray spectra from (n, x) reactions at En=14 MeV and excitation function on iron isotopes. Calculations performed by EMPIRE code

15 CONCLUSION I Rather reliable simple description of E1 gammadecay strength can be obtained by the use of models with dependence of line spreading on gammaray&excitation energies. It seems that the MLO4 is best candidate for good overall description of the RSF. R.Capote et al, Nucl. Data Sheets 110 (009) 310; V.A.Plujko, R.Capote, O.M. Gorbachenko, At.Data Nucl.Data Tables 97(011) 567; V.A.Plujko, R.Capote, V.M.Bondar, O.M. Gorbachenko, J. Kor. Phys.Soc. 59(011) 1514 V.A.Plujko et al, Nucl. Phys. At.Energy 13(01)341; Proc. of Int. Conf. Nucl. Data for Science and Technology ND013 (013) (in press).

16 MLO approximation is based on general relation between gamma-decay RSF of heated nuclei and nuclear response function on electromagnetic field. Opposite to other approaches, it is not interpolation of different methods with empirical expressions for width s s s f = F f E f E Γ E T { KMF ( ) } ( 0 ), SLO= HOA ( r, r, ) models Eλ Eλ Eλ Γ => Γ ( E, T) ( E, T ) r f

17 CONCLUSION II For better understanding the temperature and energy dependences of the RSF, experimental data are necessary as functions of both gamma-ray energy and excitation energy, especially at low gamma-ray energy. It can help, for example, to understand the sources of low-energy enhancement of RSF.

18 POSSIBLE SOURCES OF LOW-ENERGY ENHANCEMENT OF RSF Increasing excitation energy of the decaying states corresponding to low-energy transitions Low-energy part of gamma-decay RSF is proportional to the temperature s f ( E 0~ ) T = i const ( V.A. Plujko, NP A649(1999)09c)

19 Low-energy part of gamma-decay RSF increases with excitation energy of decaying states (EGLO, GFL, MLO) RSF of E 1 gamma-decay within MLO, EGLO and GFL models at different excitation energies

20 POSSIBLE SOURCES OF LOW-ENERGY ENHANCEMENT OF RSF Special shape of energy-dependent width (parameter of line spreading of gamma-strength ) S. Goriely: (HM) Phys. Lett. B436, 10 (1998); Level densities and -strength functions for astrophysics applications. nd Workshop on Level Density and Gamma Strength, May 11-15, 009, Oslo Γ 0 E T + EE E π T HM 0, ~ Γ 4 ; Γ 0 E E π T E0 E Γ E, T =

21 POSSIBLE SOURCES OF LOW-ENERGY ENHANCEMENT OF RSF Effects of low-energy doorway states (possible candidate - ph states) Calculations beyond QRPA(1p1h) are necessary for careful investigation of contributions from excitation of the states on the slops of GDR peak (N.Tsoneva - QPM)

22 Crutial role of folding procedure in microscopic QRPA calculations (phenomenological allowance for ph) + ( E1 ) ( ) = Q RPA L(, ) E1 ( ) f E f E E f E de 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

23 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)

24 Calculations within QRPA with folding procedure too pure to get correct behavior of RSF on the slops of GDR peak

25 COLLABORATORS O.M.Gorbachenko Taras Shevchenko National University of Kyiv E. P. Rovenskykh Kyiv Institute for Nuclear Research V.A. Zheltonozhskii Kyiv Institute for Nuclear Research

26 I want to thank the Organizers, especially Prof. Sunniva Siem, for invitation, warm hospitality and support THANK YOU VERY MUCH FOR ATTENTION!