Shell model description of dipole strength at low energy Kamila Sieja Institut Pluridisciplinaire Hubert Curien, Strasbourg 8-12.5.217 Kamila Sieja (IPHC) 8-12.5.217 1 / 18
Overview & Motivation Low energy enhancement of γ-sf E. Litvinova and N. Belov, Phys. Rev. C88 (213) 3132R Thermal continuum QRPA calculations Enhancement due to transitions between thermally unblocked s.p. states and the continuum Note the difference between T = (ground state) and T > (excited state) E1 strength distribution R. Schwengner et al., PRL111 (213) 23254 Shell model transitions between a large amount of states Enhancement due to the M1 transitions between states in the region near the quasicontinuum A general mechanism to be found throughout the nuclear chart Kamila Sieja (IPHC) 8-12.5.217 2 / 18
SM calculations in sd pf gds valence space s1/2 d3/2 d5/2 g7/2 g9/2 p1/2 p3/2 f5/2 f7/2 s1/2 d3/2 d5/2 Π 16 O fp 1p-1h 48 Cr 1.963.461 165.821.912 56 Fe 345.4.174 23.194.461.394 ν Full fp-calculations for positive parity states Full 1 hω calculations for negative parity states- all 1p-1h excitations from sd and to gds shells H SM = i ε i c i c i + i,j,k,l V ijkl c i c j c lc k + β c.m. H c.m. 6 states per spin and parity S M1/E1 = B(M1/E1) ρ(e i ) (6 5 iterations...) or Lanczos SF method with 5 iterations for upward S M1/E1 Kamila Sieja (IPHC) 8-12.5.217 3 / 18
Downward and upward strength calculations γ-decay (RSF) calculate desired number of low lying states using standard SM diagonalization techniques obtain the averages and radiative strength functions from relations: photoabsorption (PSF) use Lanczos Strength Function method with a large number of iterations: S = Ô ψ i = ψ i Ô2 ψ i f M1/E1 (E γ )=16π/9( hc) 3 S M1/E1 (E γ ) S M1/E1 = B(M1/E1) ρ i (E i ) The operator Ô does not commute with H and Ô ψ i is not necessarily the eigenstate of the Hamiltonian. But it can be developed in the basis of energy eigenstates: Ô ψ i = S(E f ) E f, f where S(E f )= E f Ô ψ i is microscopic strength function. If we carry Lanczos procedure using O =Ô ψ i as initial vector then H is diagonalized to obtain eigenvalues E f and after N iterations we have also the strength distribution. Kamila Sieja (IPHC) 8-12.5.217 4 / 18
Downward and upward strength calculations γ-decay (RSF) calculate desired number of low lying states using standard SM diagonalization techniques obtain the averages and radiative strength functions from relations: photoabsorption (PSF) use Lanczos Strength Function method with a large number of iterations: S = Ô ψ i = ψ i Ô2 ψ i f M1/E1 (E γ )=16π/9( hc) 3 S M1/E1 (E γ ) S M1/E1 = B(M1/E1) ρ i (E i ) The operator Ô does not commute with H and Ô ψ i is not necessarily the eigenstate of the Hamiltonian. But it can be developed in the basis of energy eigenstates: Ô ψ i = S(E f ) E f, f where S(E f )= E f Ô ψ i is microscopic strength function. If we carry Lanczos procedure using O =Ô ψ i as initial vector then H is diagonalized to obtain eigenvalues E f and after N iterations we have also the strength distribution. Kamila Sieja (IPHC) 8-12.5.217 5 / 18
Large scale shell model framework define valence space H eff Ψ eff = EΨ eff INTERACTIONS CORE build and diagonalize Hamiltonian matrix CODES Strasbourg shell model codes (E. Caurier, RMP77, 25, 427) NATHAN j-coupled basis large number of non-zero ME smaller dimensions tractable good quantum number: J can get many states of the same J used in the present calculations ideal for description of γ-decay downward strength (RSF) ANTOINE m-scheme basis sparse matrices larger dimensions tractable good quantum number: M numerical problems to get many states of the same J used in the present calculations ideal for description of photoabsorption upward strength (PSF) Kamila Sieja (IPHC) 8-12.5.217 6 / 18
Dipole strength in 44 Sc: theory vs exp <B(M1)> (µ Ν 2 ).12.1.8.6.4.2 SM, π + (hω states) SM, π - (1hω states) 44 Sc 1 2 3 4 5 6 7 8 f 1 (MeV -3 ) 1e-6 1e-7 1e-8 44 Sc EXP SM, E1 SM, M1 SM, E1+M1 <B(M1)> (µ N 2 ) No enhancement of the 1 hω M1 strength due to the complexity of wave functions (comparable to that of deformed nuclei).8.7.6.5.4.3.2.1 46 Ti 5 Ti 54 Ti 48 Cr 2 4 6 8 1 1e-9 1 2 3 4 5 6 7 8 -all states below S n 1MeV -86642 M1 matrix elements -6567 E1 matrix elements KS, submitted Kamila Sieja (IPHC) 8-12.5.217 7 / 18
Dipole strength in 44 Sc: E1 part B(E1) (e 2.fm 2 ).7.6.5.4.3.2.1 J i =-12 +/- J i =2 + 44 Sc 5 1 15 2 25 E exc (MeV) low energy part: 6567 E1 matrix elements higher energy part: LSF with 5 iterations (J i = 2 + 1,2,3 ) <B(E1)> (e 2.fm 2 ) f E1 (MeV -3 ).1.1 1e-5-5MeV 5-1MeV 44 Sc 8-1MeV -1MeV 1e-6 2 4 6 8 1 3e-8 2e-8 1e-8-5MeV 5-1MeV 44 Sc 8-1MeV -1MeV 1 2 3 4 5 6 7 8 Behaviour of the E1 strength at low energy independent on spin range and energy cut-off Kamila Sieja (IPHC) 8-12.5.217 8 / 18
Comparison to other models B(E1) (e 2.fm 2 ).7.6.5.4.3.2.1 J i =-12 +/- J i =2 + 44 Sc 5 1 15 2 25 E exc (MeV) f E1 (MeV -3 ) 1e-5 1e-6 1e-7 44 Sc 1e-8 QRPA(Sly4) SM, 2 + SM, 2 +,6 + 1e-9 5 1 15 2 25 1e-8 Reasonable agreement between QRPA and SM PSF up to 15MeV for higher energies, 3 hω excitations should be included in SM Microscopic SM strength has a non-zero limit for E γ =. Consistent with EGLO model. f E1 (MeV -3 ) 1e-9 44 Sc SM EGLO GFL SLO 1e-1 1 2 3 4 5 6 7 8 Kamila Sieja (IPHC) 8-12.5.217 9 / 18
Low energy E1 strength in SM & QRPA models QRPA describes formally photoabsorption from the ground state correction term is added to take into account collisions of quasiparticles: QRPA from S. Goriely, priv. comm. Γ (E)=Γ(E)(1+α 4πT 2 EE GDR ) the upbend of QRPA strength is not consistent with low energy strength from shell model shell model is the only microscopic tool that gives the downward strength Kamila Sieja (IPHC) 8-12.5.217 1 / 18
Downward vs upward strength: E1 RSF E1 (MeV -3 ) 1e-5 1e-6 1e-7 1e-8 1e-9 γ-decay (RSF) 1e-1 2 4 6 8 1 12 14 photoabsorption (PSF) 44 48 Ti 1e-5 Ti PSF E1 (MeV -3 ) 1e-6 1e-7 1e-8 1e-9 44 Ti 46 Ti 48 Ti 5 Ti 52 Ti 1e-1 2 4 6 8 1 12 14 E exc = Decay to all available states adds a broad component to the strength function with a non-zero E γ = limit. 1e-5 upward downward 1e-6 f E1 (MeV -3 ) 1e-7 1e-8 48 Ti RSF=PSF of the g.s. only for higher transition energies 1e-9 2 4 6 8 1 12 14 Kamila Sieja (IPHC) 8-12.5.217 11 / 18
Downward vs upward strength: M1 RSF M1 (MeV -3 ) 6e-8 5e-8 4e-8 3e-8 2e-8 1e-8 γ-decay (RSF) 44 Ti 46 Ti 2 4 6 8 1 12 14 photoabsorption (PSF) 6e-8 48 Ti PSF M1 (MeV -3 ) 5e-8 4e-8 3e-8 2e-8 1e-8 44 Ti 46 Ti 48 Ti 5 Ti 52 Ti 2 4 6 8 1 12 14 E exc = Decay to all available states can produce an enhancement effect at low energy. Kamila Sieja (IPHC) 8-12.5.217 12 / 18
Impact of the realistic M1 fragmentation on the neutron capture cross sections M1 microscopic strength functions in iron chain ( 53 Fe- 7 Fe) and impact on (n,γ) cross sections f M1 [1-8 MeV -3 ] f M1 [1-8 MeV -3 ] f M1 [1-8 MeV -3 ] 4. 3.5 3. 2.5 2. 1.5 1..5. 4. 3.5 3. 2.5 2. 1.5 1..5. 4. 3.5 3. 2.5 2. 1.5 1..5. J= + 2 4 6 8 1 12 14 J=2 + H.-P. Loens K. Langanke, G. Martinez-Pinedo and KS, EPJ A48 (212) 48 E γ [MeV] 68 Fe 2 4 6 8 1 12 14 J=4 + E γ [MeV] 68 Fe 2 4 6 8 1 12 14 E γ [MeV] 68 Fe σ [b] 1-3 1-4 1-5 67 Fe(n,γ) 68 Fe J= + J=2 + J=4 + 1-6.1.1.1 1 E n [MeV] Using SF of excited states instead of that of + leads to larger cross sections Kamila Sieja (IPHC) 8-12.5.217 13 / 18
Outlook: A word on shell model calculations SM is a powerful tool that can be used in many mass regions. Nevertheless, to get the proper physics, large valence spaces and time-consuming calculations are necessary. A valence space can be adequate to describe some properties and completely wrong for others: 48 Cr (f 7/2 ) 8 (f 7/2 p 3/2 ) 8 (fp) 8 Q(2 + )(e.fm 2 ). -23.3-23.8 E(4 + )/E(2 + ) 1.94 2.52 2.26 B(E2;2 + )(e 2.fm 4 ) 77 15 216 B(GT).9.95 3.88 The GT (magnetic) properties require model spaces that include all spin-orbit partners. Some effects can be incorporated via renormalization of the magnetic operator (quenching), but some other cannot... Kamila Sieja (IPHC) 8-12.5.217 14 / 18
B(M1) in 87 Kr: model space dependence <B(M1)> (µ N 2 ).12.1.8.6.4.2 π +, fpgd (LNPS) π +, pf-gds 87 Kr 1 2 3 4 5 6 7 from V. W. Ingeberg talk fpgd model space: full pf-shell for protons, 2p-2h excitations from p 3/2,f 5/2 to p 1/2,g 9/2,d 5/2 neutron orbits pf gds model space: full pf-shell for protons, 2p-2h excitations from g 9/2 to d 5/2,s 1/2,d 3/2,g 7/2 neutron orbits Kamila Sieja (IPHC) 8-12.5.217 15 / 18
Outlook: M1 strengths and correlations within the model space <B(M1)> (µ N 2 ).8.7.6.5.4.3.2.1 full ν=4 46 Ti 2 4 6 8 1 12 14 B(M1; + ->1 + ) (µ N 2 ).6.5.4.3.2.1 68 Fe, fpgd model space 2p-2h, q=1. 6p-6h, q=.75 8p-8h, q=.75 2 4 6 8 1 Not enough correlations too much enhancement, not enough strength at low energy... Kamila Sieja (IPHC) 8-12.5.217 16 / 18
Summary The large-scale shell model can describe both, upward and downward M1/E1/E2... strengths. Employed in model spaces encompassing the necessary degrees of freedom can give a valuable insight into physics mechanisms behind the low-energy behavior of calculated strengths. M1 dipole strength functions show an upbend for the low γ-energies. However, the mixing destroys the enhancement. No enhancement of the 1 hω M1 is present for the same reason. More complexity of the wave function less probability for the upbend. E1 dipole strength functions do not show the low energy upbend. Given the nature of the E1 operator, one can expect no much upbend throughout the nuclear chart to be verified in very neutron-rich nuclei. Shell model E1 strength at low energy exhibits different trend from QRPA approaches with a flat, non-zero E γ = limit. Consistent with EGLO model. Systematic, reliable SM calculations needed to characterize the regions where the enhancement should be present and the evolution of strengths in neutron-rich nuclei. Kamila Sieja (IPHC) 8-12.5.217 17 / 18
Effective Hamiltonian Interaction from V lowk based on the CD-Bonn potential Monopole corrections to fix the s.p. and s.h. energies pf-shell TBME from the LNPS fit (PRC82, 5431, 21) Good reproduction of low lying levels in considered nuclei and accurate position of the first 1p-1h states Quenching of.75 on magnetic spin operator Accurate reproduction of known magnetic moments of f 7/2 -shell nuclei E exc (MeV) E exc (MeV) 2.5 2 1.5 1.5 4.5 4 3.5 3 2.5 2 1.5 1.5 EXP SM 44 Sc 2 4 6 8 1 12 2*J 44 Ti EXP SM 2 4 6 8 1 12 2*J Kamila Sieja (IPHC) 8-12.5.217 18 / 18