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Outline : towards effective superfluid local density approximation (SLDA)

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##### Outline : towards effective superfluid local density approximation (SLDA)

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**Probing effective NN interaction**at band termination M. Zalewski/saturday H. Zduńczuk/poster W. Satuła IFT Univ. of Warsaw in collaboration with R.A. Wyss, H. Zduńczuk, M. Zalewski, M. Kosmulski, G. Stoicheva, D. Dean, W. Nazarewicz, H. Sagawa (+exp), A. Bhagwat, J.Meng ... • Outline: • towards effective superfluid local density approximation (SLDA) • - general remarks • pairing: volume-, mixed- or surface-type • - selectivity/resolution of nuclear data • odd-even staggering (OES) in high-spin isomers – new opportunities to • study: • - blocking of pair correlations • - single-particle, time-odd, and residual p-n effects • termination in N~Z and N=Z nuclei in A~50 mass-region • - fine tuning of particle-hole field • - fine tuning of shell-model interaction • 73Kr – dynamical manifestation of T=0 pn-pairing at high spins?**Skyrme-force as a particular realization of effective ph**interaction Fourier Long-range part of NN interaction (must be treated exactly!!!) local correcting potential infinite number of equivalent effective theories**Gogny:**a 0 density-dependent Y | H | Y Slater determinat (number conserving) Skyrme interaction: lim da 10(11) spin-orbit parameters LEDF: 20 parameters**Saturation point of symmetric infinite nuclear matter:**- saturation density ( ~0.16fm-3) - energy per nucleon (-16 0.2MeV) - incompresibility modulus (210 20MeV) - isoscalar effective mass (???) + + + + Finite nuclei [masses,radii,sp levels]: Asymmetric infinite nuclear matter and the isovector properties: entire ZOO of parameterizations !!! Fitting the Skyrme force parameters or the nuclear LEDF - symmetry energy ( 30 2MeV) • isovector effective mass (GDR sum-rule enhancement) - neutron-matter EOS (Wiringa, Friedmann-Pandharipande) - surface properties (semi-infinite nuclear matter) - realistic mean level-density (masses)**TOWARDS EFFECTIVE (local) PAIRING THEORY**(I) Low-momentum transfer expansion in particle-particle channel: v(k,k’) = g + g2(k-k’)2 + g4(k-k’)4+..... 2.0 S. Hilaire et al. PLB531 (2002) 61 ...in r-space: contact term D1S Gogny exp. th. 1.5 Dn (MeV) three-point filter 1.0 Gaussian Dirac-delta model leads to Gogny pairing • offers excellent agreement with data 0.5 (200MeVfm)2 1.4fm-1 • no cut-off is needed 0 20 40 60 80 100 120 140 neutron number (hc)2kF 1 x = dx = >> interparticle distance However, the so called coherence length i.e. spatial extension of the nucleonic Cooper pair x: (mc2)D kF 0 1000MeV 1MeV In this context the use of finite range pairing force can be viewed as rather unnecessary complication.**TOWARDS EFFECTIVE (local) PAIRING THEORY**(II) resolution Gogny versus local(DDDI) pp interaction: E. Garrido et al. PRC60, 064312 (1999) PRC63, 037304 (2001) DDDI: dots represent the Gogny gap Cut-off!!! (otherwise divergent!)**TOWARDS EFFECTIVE (local) PAIRING THEORY**(IIIa) in-medium effects towards the SLDA approach: [Superfluid Local Density Approximation] Major obstacle in constructing SLDA is: we use cut-off!!! (usual, but not at all satisfactory solution) anomalous density (pairing tensor) ultraviolet divergence In particular, in infinite homogenous system (example): regular „regularize” means in practice simply „remove divergent part” (relate to scattering amplitude; use dimensional regularization; introduce counter-terms [regulators] with explicit cut-off) isolate and regularize divergent term ;**LOCAL EFFECTIVE PAIRING THEORY**(IIIb) Bulgac-Yu SLDA approach: A.Bulgac, Y.Yu, PRL88, 042504 (2002) A.Bulgac, PRC65, 051305(R) (2002) In fact, for sufficiently large Ec gap is cutoff independent local HFB 40MeV 45MeV 110Sn 50MeV 35MeV 30MeV Ec=20MeV Formally, gap depends on both the effective (running) coupling constant and on QP cut-off energy interaction distance & ropc~h Ec~h2/mro2 ~ 40MeV Ec~pc2/2mr**Pairing/resolution (1)**0.6 2.0 0.4 0.25 1.5 0.2 0.20 0 0.15 1.0 80 20 40 60 0.10 0.05 0.5 3.5 1.0 1.5 2.0 0 3.0 Dn [MeV] 2.5 2.0 Dn,p(MeV) 1.5 1.0 0.5 spherical HFB b2 surface mixed volume J.Dobaczewski & W.Nazarewicz Prog. of Theor. Phys. Suppl. No. 146 (2002) 50Cr SLy4 No selectivity!!! Dn Dn Eexp-Eth [MeV] dD (MeV) deformed N,Z spherical 40 50 28 14 82 126 N,Z 20 40 60 80 100 120**Pairing/resolution (2)**Hilaire et al. PLB531 (2002) 61 5.0 4.5 surface mixed 4.0 volume 3.5 3.0 46Ti 2.5 SLy4 1.0 1.5 2.0 Vo(1-r(r)/ri) 0.20 EXP 0.15 b2 0.10 Beautiful example of selectivity!!! 0.05 deformed Eexp-Eth [MeV] spherical Dn [MeV] M. Kosmulski – licentiate thesis**High-spin isomers - new opportunities to study pairing**correlations ~ ~ ~ ~ We expect: Odd-Even Binding Energy Effect in the High-Spin Isomers: Are Pairing Correlations Reduced in Excited States? A. Odahara, Y. Gono, T. Fukuchi, Y. Wakabayashi, H. Sagawa, WS, W. Nazarewicz, PRC72, 061303(R), (2005) Stretched configurations: N=83 DE 8.5MeV const. Hence, the OES: is similar in GS and HSI!!!! no blocking??? Dracoulis et al. PLB419, 7 (1998)**HF-SLy4**GS 0.5 full 3.0 0.8 0.0 D(Z) [MeV] TE p-Fermi energy -0.5 2.5 0.4 -1.0 esp D(Z) [MeV] 0 2.0 -1.5 Z d3/2 61 63 -2.0 s1/2 1.5 h11/2 1.0 64 fixed occup. 0.5 0 { d5/2 Z SkO WS SLy4 DIPM 61 62 63 64 65 66 hole in [402]5/2 Are Pairing Correlations Reduced in Excited States? (II) Isomerism of the same type! Oblate shapes at HSI (-0.2) Nearly spherical GS Spherical sp spectrum data HF SLy4/HSI HF SkO/HSI { GS EXP HSI • sp contribution to OES • time-odd terms (within • self-consistent models)**OES/DPES**2.0 dpn Excitation energy 1.5 1.0 |dpn| [MeV] D(Z) [MeV] HSI Dexp(Z) GS Dproj 2.0 0.25 10 Z 1.5 9 0.20 1.0 DIPM DPES EXP DPES+dpn DPES DIPM 8 0.15 Z 61 62 63 64 65 66 Sm Eu Gd Tb Dy Ho Nd Pm 149 143 145 147 61 63 65 67 144 146 148 150 Strutinsky calculations with pairing: GAP/OES DEHSI [MeV] +15% +10% Enhanced pairing is needed (see also Xu et al. PRC60,051301 (1999)) Blocking is too strong!!!**This study reveals that many effects can contribute**to OES in particular: pairing single-particle effects time-odd effects (nuclear magnetism) residual pn interaction (odd-odd) RMF ...and the question is... Is blocking under control? Time-odd fields Rutz et al. PLB468 (1999) 1**Consider the energy difference between stretched**(terminating) configurations in A~50 mass region • the best examples of almost unperturbed sp motion • uniquely defined (in N=Z) • config. mixing beyond mean-field is expected to be mariginal • (in particular all pairs are broken) • shape-polarization effects included already at the level of the SHF • time-odd mean-fields (badly known) can be tested n f7/2 energy scale (bulk properties) E( ) f7/2 Imax p-h -1 n+1 20 d3/2 f7/2 E( ) - Imax d3/2 spin-orbit dominates!!! ~ 0 light ~ ½ heavy nuclei DE = these are ideal for fine tune particle-hole interaction!!!!**Mean-field versus Shell-Model**Isospin symmtery „restoration” in N=Z nuclei: DZ 1.8 1.5 dET [MeV] SkO 1.4 T=1 1.0 1.0 nph dET DEth-DEexp [MeV] centroid 0.5 pph dET T=0 0 Can be evaluated from mirror-symmetric nuclei e.g. SM -0.5 42Ca 43Sc 45Sc 46Ti 40Ca from 40K and 42Sc etc. A 40 42 44 44Ca 44Sc 45Ti 47V 42Sc 46V 40Ca 44Ti N=Z T=0 pn pairing??? „isospin symmetry restoration” in N=Z nuclei original HF result for pph excitation G.Stoitcheva, WS, W.Nazarewicz, D.J.Dean, M.Zalewski, H.Zduńczuk, PRC73, 061304(R) (2006) SkO Shifted by 480keV reduced s-o**-1**d3/2 g9/2 48Cr 4 4 [nf7/2pf7/2] 16+ HFB calculations including T=0 and T=1 pairing J. Terasaki, R. Wyss, and P.H. Heenen PLB437, 1 (1998) • Skyrme interaction in p-h • DDDI in p-p channel • fully self-consistent theory • no spherical symmetry isoscalar pairing no T=0 at low spins Non-collective (oblate) rotation T=1 collapses Collective (prolate) rotation (termination) data**Isovector dependence of shell-model**matrix elements BFZ mechanism: „... It is a grave error to assume that the p-h intraction is independent of isotopic spin...” ... isotopic dependence of p-h interaction can be approximated by a monopole poten- tial vT~bt1.t2 Bansal & French, Phys.Lett. 11, 145 (1964) modified SM SM Zamick, Phys. Lett. 19, 580 (1965) DE=1/2b[T(T+1)- -Tp(Tp+1)-Th(Th+1)] 0.4 particle contribution Tp=T 1/2 single-hole contribution: Th=1/2 0.2 DESM-DEEXP [MeV] N = Z (Tp=1/2): + 0 Single j-shell J-T SM phenomenology: DE=-3b/4 VT=0+3d N = Z (Tp=T-1/2): bsym = bsym- 4d -0.2 Vphph(JT=0)+3d Vphph(JT=1)-d DE=b(T-1/2)/2 d=175keV -0.4 The SM overestimates b by ~700keV!!! 42Ca 43Sc 45Sc 46Ti 44Ca 44Sc 45Ti 47V 42Sc 46V 40Ca 44Ti (2j+3)VT=1,n=2-(2j+1)VT=0,n=2-2VT=1,J=0 N=Z 2(2j+1) VT=1-d N. Zeldes, Handbook of Nuclear Properties, Clarendon Press, Oxford, 1996, p.13**modified SM**SM 3/2+ 3- 0.8 3/2+ 0.6 3- 3- ESM-EEXP [MeV] 0.4 3- 3- 3- 0.2 3/2+ 3- 3- 3- 3- 3- 3- 3/2+ 3/2+ 2- 3/2+ 0 3/2+ 42Ca 44Sc 44Ti 46Ti 42Sc 47V 40Ca 44Ca 43Sc 45Sc 45Ti 46V Low-spin particle-hole intruder states SM versus modified-SM**OES in high-spin isomeric states:**• new opportunities to study blocking, TO terms, • residual-pn, and mean-field (sp-splitting) effects Volume-, mixed- or surface-like local pairing • selective nuclear data exist but must be systematically • identified (and understood) thrughout the nuclear chart Termination in N~Z, A~50 nuclei: 73Kr – possible fingerprint of enhanced T=0 pairing Concluding remarks Consistent superfluid local density approximation is just behind our doors! - excellent laboratory for fine-tuning of ph MF interaction and SM interaction**(1) 73Kr - a fingerprint of T=0 pairing?**3qp 2.5 30 2.0 3qp 1.5 25 1.0 20 0.5 15 0.0 -0.5 10 5 g 40 0.5 0.5 0.5 1.0 1.0 1.0 1.5 1.5 1.5 fp R.Wyss, P.J. Davis, WS, R. Wadsworth Conventional TRS calculations involving only T=1 pairing: negative parity negative parity positive parity Ix (-,-) (+,+) 73Kr 73Kr (-,-) Ew [MeV] 5qp 1qp 1qp 73Kr: Kelsall et al., Phys. Rev. C65 044331 (2005) hw[MeV] hw[MeV] |1qp> = a+n(fp)|0> |3qp> = a+ng a+pg a+p(fp)|0> <1qp|E2|3qp> ~ 0 (one-body operator)**(2) 73Kr a fingerprint of T=0 pairing?**TRS involving T=0 and T=1 pairing Dn 1.0 Dp D [MeV] n(fp) p(fp) p(fp) DT=0 0.5 ng9/2 pg9/2 pg9/2 0 p(fp) n(fp) p(fp) 73Kr 30 pg9/2 pg9/2 ng9/2 25 Ix 20 theory 15 exp 10 5 0 hw [MeV] 1.4 0.4 0.8 1.0 1.2 1.6 0.2 0.6 What makes the 1qp and 3qp configurations alike? Scattering of a T=0 np pair in 73Kr n(fp)(-) vacuum 1qp configuration n(fp) ng9/2 n(fp) ng9/2 ng9/2(+) pg9/2 p(fp)(-) 3qp configuration**Ix**(-,+) 3qp 75Rb 75Rb (+,+) 2.0 30 1.5 Ew [MeV] 25 1qp 3qp 1.0 20 0.5 1qp 15 0.0 10 hw[MeV] hw[MeV] -0.5 5 0.5 1.0 1.5 0.5 0.5 1.0 1.0 1.5 1.5 (3) 73Kr a fingerprint of T=0 pairing? Conventional TRS calculations involving only T=1 pairing in neighbouring nuclei: negative parity positive parity all bands Excellent agreement was obtained in: Tz=1 : 74Kr,76Rb, D. Rudolph et al. Phys. Rev. C56, 98 (1997) Tz=1/2: 75Rb, C. Gross et al. Phys. Rev. C56, R591 (1997) Tz=1/2: 79Y, S.D. Paul et al. Phys. Rev. C58, R3037 (1998)