Instantaneous Shape Sampling for Calculating the Electromagnetic Dipole Strength in Transitional Nuclei S. Frauendorf Department of Physics University of Notre Dame, USA Institut fuer Strahlenphysik, Forschungszentrum Rossendorf Dresden, Germany
Collaborators F. Doenau (FZD) B. Kaempfer (FZD) R. Schwengner (FZD) A. Wagner (FZD) S. G. Zhang (FZD, Peking U.) I. Bentley (ND)
Photo disintegration Experimental situation In hot stellar medium for (γ,p), (γ,n), (γ,α) is only a narrow window available. T 9 =2.5 92 Mo(γ,p) 91 Nb Dipole strength close to the particle separation energy is important 98 Mo 1 2 (γ,n) σ γ / mb 1 1 corrected Gamov window 1 uncorrected from analysis of the peaks 4 5 6 7 8 9 1 11 12 13 14 15 16 E x / MeV
Static deformation + Hydrodynamic/oscillator approximation σ γ / mb 3 25 2 15 1 154 Sm "Extra strength" 144 Sm σ abs / mb 14 12 1 8 6 4 Oscillator+RPA K= 1/1 K=+1,-1 1/1 5 N 2 8 1 12 14 16 18 2 22 E / MeV 6 8 1 12 14 16 E / MeV Lorentzian widths Γ K=±1 = 2Γ K= = 6 MeV. Shape parameter ε from Moeller&Nix Deformation increase ε= (A=144) to ε=.25 (A=154) leads to increasing strength in the tail region. Standard in reaction networks (width related to quadrupole softness)
Why a microscopic theory? There may be still fragments of the nucleonic orbitals Low level density-discreteness Ratio M1/E1 Predictions for neutron-rich nuclei, based on developments of mean field theory
Data from ELBE
RPA with static deformation Nilsson or Woods-Saxon potential+ isovector dipole-dipole interaction H = h 2 o 2 Nilsson / WS ( β, γ,...) + η( Rπ Rυ ) η = 1.5-3 (ratio np/nn of effective force) R π,ν = center of mass for Z protons or N neutrons, resp. Solve RPA equation of motion for the E1 and M1 modes [ ] + + H, Ω = Ω i E i i mω A to find the vibrational modes E i (i 3-5) and their E1 and M1 strengths. Method: Contour Integration in complex plane. Generates strength function Lorentzian-smoothed with narrow width
RPA solution for 154 Sm Oscillator: split of strength Nilsson: Landau fragmentation and split of E1 strength 14 12 Oscillator+RPA K=+1,-1 1/1 12 1 Nilsson + RPA σ abs / mb 1 8 6 4 K= 1/1 σ abs / mb 8 6 4 2 2 6 8 1 12 14 16 6 8 1 12 14 16 E / MeV E / MeV
E1 strength the individual particle-hole excitations B(E1,ph) = <p M(E1) h> 2 no interaction between the 1p1h excitations 1 2 144 Sm Oscillator One line for spherical Two lines for axial deformed (K= and K=±1) 1 1 1 1-1 ε = Nilsson Many lines according to the deformation splitting, intermediate Structure B(E1) (e 2 fm 2 ) 1-2 154 Sm 1 1 ε =.25 1 K = 1-1 1-2 1 1 1 154 Sm ε =.25 K = 1 1-1 1-2 5 1 15 2 E (MeV)
Nuclear shape is not rigid RPA for fixed shape QPM for spherical shape Many nuclei have large amplitude shape fluctuations Several show shape coexistence
Instantaneous Shape Sampling - ISS Many nuclei have a soft, fluctuating shape. The quadrupole dynamics is slow compared to the dipole vibrations: t quadrupole t dipole E(2 E(1 + ~ ) ) 1 1 The absorbed quant sees a snapshot of the fluctuating shape. ~ γ 1 2 + + Probability for instantaneous shape n. Calculated by RPA for instantaneous shape n.
Probability distribution for the shapes Te quadrupole shape dynamics is described by the IBA1. Number of bosons: 1 Diagonalize H excitation energies, E2 transition rates Fitted to experiment parameters & Ground state >
Diagonalize Localized states n> P( β, γ ) = < n> n n 2
Shape coexistence Large fluctuations
RPA for the set instantaneous deformations Woods-Saxon potential+ isovector dipole-dipole interaction H = h 2 o 2 2 WS ( β n, γ n ) + η( R π Rυ ) + Σ η adjusted to reproduce the GDR position in the region R π,ν = center of mass for Z protons or N neutrons, resp. total spin of all nucleons, & adjusted to reproduce the spin flip resonance Solve RPA equation of motion [ ] + + H, Ω = Ω i E i i mω A r r κ v 2 to find the vibrational modes E i (i 3-5) and their E1 strengths. Method: Contour Integration in complex plane. Generates (smoothed with narrow width D).
(mb) 12 8 94 42Mo 52 (a) EXP ISS MM MM 4 n 5 1 15 2 25 E x (MeV) Elbe data + (n, ) Equilibrium deformation (Micro-Macro - MM) Instantaneous Shape Sampling - ISS
(mb) 12 8 4 94 42Mo 52 (a) n EXP ISS MM Landau fragmentation + Shape fluctuations give too weak damping in GDR region 5 1 15 2 25 E x (MeV) There are other degrees of freedom that cause Collisional damping Taken into account phenomenologically de' σ ISS Γ( E) = αe ( E') 2π 2, σ ISS CD ( E) = Γ( E') 2 2 ( E E') + ( Γ( E) / 2) ) α =.14MeV 1 =.15MeV ( N 1 > 5), ( N = 5)
Collision damping does not shift dipole strength into the threshold region It only smoothes the cross section
Composition of the width near the GDR peak If there are several mechanisms, each generating a Lorentz spread, the result is again a Lorentzian with the total width being the sum of the individual widths. N Γ ΓTOTAL[MeV ] Γ LD + ISS CD 5 52 1.7 4. 5.7 54 2.7 3.6 6.3 56 2.5 3.5 6. 58 4. 3.9 7.9
(mb) 2 1 94 42Mo 52 (b) n 1/( Γ ΓISS 1/ Γ EXP ISS-CD MM-CD 5 1 15 2 25 ) E x (MeV) Γ Γ Γ Γ LD ISS CD = 1.1MeV =.6MeV = 4.MeV = 5.7MeV
Dipole Strength near the threshold Landau Damping + Dynamic Deformation can explain the strength These doorways are mixed with more complex states-> smoothing (collsional damping) Some properties of the individual phexcitations survive (isospin dependence) About 7% M1 (need to be compared with experiment) Indication for pygmy collectivity so far???
Ongoing Other nuclei to test the ISS More sophsticated RPA (Skyrme, Kvasil) Properties of the ph-states in the low-energy range Inclusion of two-phonon states More predictive description of quadrupole dynamics (IBA+PES, GCM)
More stringent condition for validity of ISS dω1 β dβ 1MeV >ω 2.2 >.6MeV OK Scatter around 1 MeV incoherently better Displacement of the dipole states by the deformation fluctuations must be larger than the energy of the collective quadruple excitation.
Problem of photo excitation by Bremsstrahlung The scatter spectrum measured by Bremsstrahlung contains all deexcitations, i.e. ground state transitions together with feeding and branching transitions. Cascade simulation for feeding and branching need to be performed.
Monte Carlo code for simulations of γ cascades E x B f B B f = I N i= f I B 1 i = g.s. Γ f Γ Level scheme for J=,1,2 using level densities from the systematics given in Phys. Rev. C 72, 44311 (25) and Wigner distribution for the nearest-neighbour spacings. 1. Creation of a nuclear realisation: Partial decay widths using strength functions from RIPL2 systematics and Porter-Thomas distribution of the partial widths. 2. Monte Carlo method: Excitation of a level with J=1 according to σ γ. Deexcitation according to B f. same strategy as DICEBOX code: F. Bečvář, Nucl. Instr. and Meth. A 417, 434 (1998)
Low-lying dipole strength-theory Pygmy resonance: Possible existence a low lying extra strength - few percents of the TRK sum - due to a soft oscillation of a core against a neutron skin. Pygmy Effect ~ N-Z (neutron excess). Pygmy GDR Our point to be investigated: Nuclear deformation generates extra E1 strength in the tail region This is a shell effect i.e. NOT directly correlated to N-Z!
Nd series A=142-152 4 3 Nilsson + RPA 4 σ γ / mb 2 1 152 Nd N 142 Nd 6 8 1 12 14 16 18 2 E / MeV / mb MeV Σ γ 3 2 1 Lorentzian widths Γ K=±1 = Γ K= = 2.5 MeV. Reduced Γ due to splitting of Nilsson levels Shape parameters ε from Moeller&Nix Deformation increase ε= (A=142) to ε=.255 (A=152) increasing E1 strength in the tail region. 152 Nd N 8 9 1 11 12 E / MeV 142 Nd
Triaxial shapes: Mo series A=92-14 Shapes: calculated by Nilsson-Strutinsky (Rossendorf TAC program) 92 Mo 94 Mo 96 Mo 98 Mo 1 Mo A 92 94 96 98 1 12 14 ε..2.1.18.21.24.25 TAC γ /deg - 6 6 37 32 25 16 ε.3.5.18.2.22.24.25 FRLD γ /deg - - 32 28 25 25 16 Experimental information for A=92-1 from a) RIPL2 data for (γ,x) cross section for E above S x and b) (γ,γ ) data for E below S x from experiments with Bremsstrahlung at e-linac ELBE at Dresden-Rossendorf
92-14 Mo with Nilsson + RPA Deformation effect: Increase of E1 strength at lower-energies σ / mb 3 25 2 15 1 Nilsson + RPA 14 Mo 5 N 92 Mo 6 8 1 12 14 16 18 2 E / MeV Σ abs / mb MeV 3 25 2 15 1 5 Nilsson + RPA 14 Mo 92 Mo N 6 7 8 9 1 11 12 13 E / MeV A 92 94 96 98 1 12 14 ε..2.1.18.21.24.25 γ /deg - 6 6 37 32 25 16
92-14 Mo with Woods-Saxon + RPA WS+RPA shows qualitatively the same increase of E1 strength at lower-energies with deformation σ / mb 3 25 2 15 1 5 WS + RPA 14 Mo 92 Mo 6 8 1 12 14 16 18 2 N E / MeV Σ abs / mb MeV 3 25 2 15 1 5 WS + RPA 14 Mo 92 Mo N 6 7 8 9 1 11 12 13 E / MeV
12 Mo spherical versus deformed shape σ / mb 3 25 2 15 1 Nilsson + RPA 12 Mo ε =.25 ε = σ / mb 3 25 2 15 1 WS + RPA 12 Mo ε =.25 ε = 5 5 6 8 1 12 14 16 18 2 E / MeV 6 8 1 12 14 16 18 2 E / MeV Compared to Nilsson the Woods-Saxon yields a smoother and broader distribution
Comparison: Experiment versus Nilsson+RPA Σ / mb MeV 3 25 2 15 1 Experiment 92 Mo 98 Mo 1 Mo Σ / mb MeV 3 25 2 15 1 RPA Γ=.1 MeV 92 Mo 98 Mo 1 Mo 5 5 6 8 1 12 14 E / MeV 6 8 1 12 14 E / MeV Σ E ( σ E) exp ( E ) = γ i> 4 MeV The dipole strength increases with the deformation.
The spherical 88Sr: Nilsson vs. Woods-Saxon 88 Sr Nilsson 1 2 WS Rauscher (γ,n) σ γ / mb 1 1 (γ,γ) Talys continuum 1 5 6 7 8 9 1 11 12 13 14 15 16 17 18 E x / MeV
Kr series A=78-84 Deformation decrease from A=78 to 86 due to N=5 shell closure ε= -.225 (oblate) in 78 Kr ε=. (spherical) in 86 Kr Prediction: low energy E1 strength is supposed to decrease with N-Z According to Pygmy interpretation the E1 strength should increase Experiment? Shell effects can counteract the effect of the neutron excess σ γ / MeV 2 15 1 5 78 Kr N 84 Kr 78Kr Ripl2 8Kr Ripl2 82Kr Ripl2 84Kr Ripl2 8 1 12 14 16 18 2 22 24 E / MeV
/ mb Nd: RPA-exp σ γ / mb 4 3 2 4 3 2 142 Nd 144 Nd 146 Nd 148 Nd 15 Nd 1 152 Nd N 142 Nd 6 8 1 12 14 16 18 2 E / MeV 1 7 8 9 1 11 12 13 14 15 16 17 18 19 2 21 E x / MeV
Nilsson vs. Woods-Saxon spherical deformed σ γ (mb) 4 3 2 WS Nilsson 144 Nd σ γ (mb) 4 3 2 WS Nilsson 152 Nd 1 1 6 8 1 12 14 16 18 2 E (MeV) 6 8 1 12 14 16 18 2 E (MeV) (mb MeV) Σ γ 4 3 2 1 Nd WS Nilsson 144 Nd (mb MeV) Σ γ 4 3 2 1 WS Nilsson 152 Nd 8 9 1 11 12 E (MeV) 8 9 1 11 12 E (MeV)
db/de (e 2 fm 2 MeV -1 ) 5 4 3 2 1 166Er Γ =.6 MeV ws(vesko) + rpa N=8 ws Gareev N=8 ws universal N=1 ws universal 4 6 8 1 12 14 16 18 2 E (MeV) Akulinichev, Malov, J.Phys. G. 3, 625 (1977)
Future developments Find a good deformed potential Dynamic deformation More realistic residual interaction
Dynamic deformation The collective quadrupole modes are slow compared with the dipole mode in the region of interest. h ω.5mev << h ~ 8MeV 2 ~ ω1 Adiabatic approximation: 1) Find dipole response for given 2) Average over probability P( in ground state P( ): 1) IBA systematics 2) Micro approach
Malov, Meliev, Soloviev, Z. Phys. A32, 521 (1985) RPA RPA+other phonons
Generating P( ) from IBA: 1) Find the IBA parameters 2) Diagonalize H -> gs> 2 3 3) Diagonalize [ Q] β, [ Q Q Q] β cos3γ -> set of 4) P( β, γ ) < gs β, γ > i i = i i Q β, > 2 i γ i
Improvements of the RPA planned Simple dipole-dipole interaction replaced by Migdal s interaction Treatment: Expansion of the delta-interaction in separable form = =,1,2,... ' 2 2 ' ') ( n x x n x x n e x x H e x x H x x δ Generate strength function keeping few seperable terms by same method as for the dipole-dipole force.
Summary Astrophysical network calculations need more precise photo cross sections. This is a challenge for both high resolution experiments and reliable theoretical predictions. A new reconstruction analysis of the (γ,γ ) spectra provides for the first time photo cross sections which connect these data smoothly to the measured (γ,x) data. The deformation is an important structure effect for producing low energy E1 strength. RPA with seperable interaction is a fast method for deformed nuclei, suitable for network calculations. First application to mass 1 nuclides gives promising results for the for low-lying E1 strength. Problems with the E1 strength function of well deformed rare earth nuclei Extension to transitional nuclei by means of adiabatic approximaton
σ γ / mb 1 2 1 1 1 1 2 92 Mo 94 Mo (γ,γ) (γ,p) (γ,n) σ γ / mb σ γ / mb 1 1 1 1 2 1 1 1 1 2 96 Mo 98 Mo σ γ / mb σ γ / mb 1 1 1 1 2 1 1 1 1 Mo corrected uncorrected 4 6 8 1 12 14 16 18 2 E x / MeV
Details of nucleosynthesis Sr Zr Mo α n Ru p s-process (n,γ) p-process γ-process (γ,n) (γ,p) (γ,α) Kr r-process (n,γ) Se Ge Zn