Shell and pairing gaps: theory

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1 Shell and pairing gaps: theory Michael Bender Université Bordeaux 1; CNRS/IN2P3; Centre d Etudes Nucléaires de Bordeaux Gradignan, UMR5797 Chemin du Solarium, BP120, Gradignan, France Nuclear Structure and Astrophysical Applications ECT* Trento, July

2 Mass filters They are used 1. to identify and quantify shell closures (S 2n, S 2p, δ 2n, δ 2p, also Q α) 2. to quantify pairing correlations ( (3), (4), (5) ) 3. to quantify specific interactions ( matrix elements ) (δv pn,...) Masses are undoutably observables. The interpretation of the mass filters relies on simple models Atomic nuclei are complex systems. Simple models which might work for isolated systems, but will fail when describing the systematics over many randomly chosen) nuclei

3 Pairing Gaps ÅBohr, B. R. Mottelson, D. Pines, Phys. Rev. 110 (1958) 936

4 Pairing Gaps: Seniority model J. A. Maruhn, P.-G. Reinhard, E. Suraud, Simple Models of Many-Body Systems (Springer 2010) s s s s s s s s s s s s s s tr t r t2 r t t r t r rt r r t2 r t t r 2 tr r t r t t t r 2 3 r The excited states with seniority 2 have an excitation energy of two times the pairing gap 2 = GΩ Pairing interaction with Ĥ = G Â Â Â Ω α=1 â +αâ α in a single shell of degeneracy 2Ω. For even nuclei that can be described within the model space (3) = GΩ 2 =

5 Mass differences for pairing mass surfaces of even-even and odd nuclei E even even(z,n) = E 0(Z,N), E odd Z (Z,N) = E 0(Z,N)+ p(z,n), E odd N (Z,N) = E 0(Z,N)+ n(z,n). where q(z,n) is the energy lost when blocking a particle. The energy of any nucleus E(N) can be obtained by an expansion of the energy around a reference nucleus E(N 0) E(N) = n=0 1 n E 0 n! N n (N N 0) n +D(N 0) N0 where the gap D is given by 0 even proton and neutron number, D = n odd neutron number, odd proton number. p D. Madland, J. R. Nix, Nucl. Phys. A 476 (1988)1. M. B., K. Rutz, P.-G. Reinhard, J. A. Maruhn, EPJA8 (2000) 59.

6 1-nucleon separation energy S 1n S 1n(N 0) = E(N 0) E(N 0 1) = E0 N 1 2 E 0 N0 2 N E 0 N0 6 N E 0 N0 24 N N0 +D(N 0) D(N 0 1) mixes everything: variation of the smooth background and pairing gap. better use something else

7 3-point pairing gap (3) 3-point gap mass difference E(N 0 +1) 2E(N 0)+E(N 0 1) = 2 E 0 N E 0 N0 12 N 4 N0 + +D(N 0 +1) 2D(N 0)+D(N 0 1), provides the distance of the mass surfaces if E is a smooth function (which is not the case when not having pairing) D does not vary when going from N 0 1 to N E 0 N 2 N0 (and higher-order derivatives) is (are) negligible (3) (N 0) π [ ] N 0 E(N 0 1) 2E(N 0)+E(N 0 +1) 2 where π N0 = ( 1) N 0 is the number parity. (note that in the liquid drop model, 2 E 0 N 2 is non-zero due to the symmetry and Coulomb ( Z 2 ) energies) M. B., K. Rutz, P.-G. Reinhard, J. A. Maruhn, EPJA8 (2000) 59.

8 4-point pairing gap(s) (4) The 4-point gap mass difference E(N 0 2) 3E(N 0 1)+3E(N 0) E(N 0 +1) = 3 E 0 N 3 + +D(N 0 2) 3D(N 0 1)+3D(N 0) D(N 0 +1). N0 can be used to define (4) (N 0 = 1/2) := π [ N 0 E(N 0 2) 3E(N 0 1) 4 ] +3E(N 0) E(N 0 +1). M. B., K. Rutz, P.-G. Reinhard, J. A. Maruhn, EPJA8 (2000) 59.

9 5-point pairing gap(s) (4) The 5-point gap mass difference E(N 0 +2) 4E(N 0 +1)+6E(N 0) 4E(N 0 1)+E(N 0 2) = 4 E 0 N 4 + N0 = +D(N 0 2) 4D(N 0 1)+6D(N 0) 4D(N 0 +1) D(N 0 +2). can be used to define (5) (N 0) := π [ N 0 E(N 0 +2) 4E(N 0 +1)+6E(N 0) 8 ] 4E(N 0 1)+E(N 0 2). M. B., K. Rutz, P.-G. Reinhard, J. A. Maruhn, EPJA8 (2000) 59.

10 Mean-field models. Analysis I 81, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 OCTOBER 1998 Odd-Even Staggering of Nuclear Masses: Pairing or Shape Effect? W. Satu a, 1,2,3 J. Dobaczewski, 1,2,3 and W. Nazarewicz 2,3,4 1 Joint Institute for Heavy Ion Research, Oak Ridge, Tennessee Department of Physics, University of Tennessee, Knoxville, Tennessee Institute of Theoretical Physics, University of Warsaw, ul. Hoża 69, PL Warsaw, Poland 4 Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee (Received 20 April 1998) The odd-even staggering of nuclear masses was recognized in the early days of nuclear physics. Recently, a similar effect was discovered in other finite fermion systems, such as ultrasmall metallic grains and metal clusters. It is believed that the staggering in nuclei and grains is primarily due to pairing correlations (superconductivity), while in clusters it is caused by the Jahn-Teller effect. We find that, for light- and medium-mass nuclei, the staggering has two components. The first originates from pairing while the second, comparable in magnitude, has its roots in the deformed mean field. [S (98)07481-X] Values of (3) (N) calculated at odd N can be, roughly, associated with the pairing gap. the differences of (3) at adjacent even and odd values of N give information about the single-particle spectra e n+1 e n = 2[ (3) (N = 2n) (3) (N = 2n +1)] FIG. 2. Schematic illustration of various contributions to the OES. The odd-even energy difference, D o2e, is decomposed into the pairing part, D, and the mean-field part, de 2.

11 Mean-field models. Analysis II (3) FIG. 9. BCS (5), BCS, and as a function of A. From top to bottom, the pairing gap increases from 0 to 1200 kev. Left column: calculation with an equidistant doubly degenerate spectrum. Right column: calculation using the selfconsistent neutron HF spectrum of 152 Ce. In addition to, the lowest qp in odd nuclei is shown. Results are displayed between 148 Ce and 156 Ce. T. Duguet, P. Bonche, P.-H. Heenen, J. Meyer, PRC65 (2002)

12 Equi Spect denotes a schematic model: equidistant two-fold degenerate single-particle levels with level spacing δǫ = 400 kev constant gap pairing varied between = 0 and 1200 kev

13 Mean-field models. Analysis II BCSE: calculating odd nuclei as if they were even ones without blocking (3) FIG. 10. Energy differences BCSE (3) (5) BCS (5), BCSE,. Left and right panels: same as and BCS for Fig. 9. The results are displayed between 148 Ce and 156 Ce. T. Duguet, P. Bonche, P.-H. Heenen, J. Meyer, PRC65 (2002)

14 Mean-field models. Analysis II FIG. 11. Binding energy as a function of N for the even part squares joined by dashed line and for the full odd states circle. Left panel: no pairing. Right panel: realistic-pairing case. T. Duguet, P. Bonche, P.-H. Heenen, J. Meyer, PRC65 (2002)

15 Mean-field models. Analysis II some conclusions relevant for the present discussion: (3) (N = 2n 1) gives the pairing gap (+polarization effects) only in the no-pairing limit for realistic pairing, (5) gives the pairing gap (+polarization effects), while values for (3) alternate around it as the 2nd derivative of the smooth background energy is not zero

16 Mean-field models. Analysis IIa T. Duguet, T. Lesinski, Proc. of the Intl. Conf. on Nuclear Structure and Dynamics, Dubrovnik, Croatia, May , arxiv: LCS (even) 2.5 LCS (odd) [MeV] (3) th, unblocked (3) th (3) exp N FIGURE 1. Experimental three-point mass differences (crosses) for neutrons along the tin isotopic chain versus several theoretical measures of the OEMS: n LCS even (dashed-line), n LCS odd (dasheddotted line) and n 3 N (full line). Theoretical three-point mass differences are also shown for oddeven nuclei computed using an even-number-parity vacuum as areferencestate,i.e.ahfbstatewithout any quasi-particle blocking as if odd-even nuclei had the same structure as even-even ones (dotted line) [33, 34].

17 Mean-field models. Analysis III M. B., K. Rutz, P.-G. Reinhard, J. A. Maruhn, EPJA8 (2000) 59.

18 Mean-field models. Analysis III M. B., K. Rutz, P.-G. Reinhard, J. A. Maruhn, EPJA8 (2000) 59.

19 Mean-field models. Analysis III M. B., K. Rutz, P.-G. Reinhard, J. A. Maruhn, EPJA8 (2000) 59.

20 Mean-field models. Analysis III (5) provides better measure for the distance between mass surfaces of even and odd nuclei than (3) as all contributions from the smooth surface are suppresed up to terms that go as the power N 4 or Z provided there is no discontinuity in the mass surface as it happens at shell closures t t t t t t t t tt t t t t t t t t t t Þ t t t t t t t M. B., K. Rutz, P.-G. Reinhard, J. A. Maruhn, EPJA8 (2000) 59.

21 How is the evolution of shell structure related to systematics of masses? D. J. Rowe and J. L. Wood, Fundamentals of Nuclear Models (World Scientific, 2010)

22 1-nucleon separation energy S 1n S 1n(N 0) = E(N 0) E(N 0 1) = E0 N 1 2 E 0 N0 2 N E 0 N0 6 N E 0 N0 24 N N0 +D(N 0) D(N 0 1) mixes everything: variation of the smooth background and pairing gap. better use something else

23 2-nucleon separation energies and shell gaps two-nucleon separation energy S 2n(N 0) = E(N 0) E(N 0 2) = 2 E0 N 2 2 E 0 N0 N E 0 N0 3 N E 0 N0 3 N 4 shell gap δ 2n(N 0) = S 2n(N 0) S 2n(N 0 +2) = E(N 0 2) 2E(N 0)+E(N 0 +2) = 4 2 E 0 N E 0 N0 3 N N0 N0

24 How is the evolution of shell structure related to systematics of masses? PRL 101, (2008) P H Y S I C A L R E V I E W L E T T E R S week ending 1 AUGUST 2008 Evolution of then 50 Shell Gap Energy towards 78 Ni J. Hakala, S. Rahaman, V.-V. Elomaa, T. Eronen, U. Hager,* A. Jokinen, A. Kankainen, I. D. Moore, H. Penttilä, S. Rinta-Antila, J. Rissanen, A. Saastamoinen, T. Sonoda, C. Weber, and J. Äystö x Department of Physics, P.O. Box 35 (YFL), FI University of Jyväskylä, Finland (Received 20 March 2008; published 31 July 2008) exp FRDM95 SLy4 SkP SLy4+GCM GT3 D1S DuZu 9 8 Shell gap (MeV) Proton number δ 2n (N,Z) = S 2n (Z,N) S 2n (Z,N +2) = E(N 2,Z) 2E(N,Z)+E(N +2,Z)

25 Eigenvalues of the single-particle Hamiltonian vs. S 2q lower panel: S 2p (Z =50,N)/2 The global linear trend is taken out subtracting N 82 [S 2 2p (Z=50,N=50) S 2p (Z=50,N=82)] using the spherical mean-field S 2p M. B., G. F. Bertsch, P.-H. Heenen, PRC 78 (2008) lower panel: S 2n (Z,N=50)/2 The global linear trend is taken out subtracting N 50 2 [S 2n (Z=28,N=50) S 2n (Z=50,N=50)] using the spherical mean-field S 2n

26 Eigenvalues of the single-particle Hamiltonian vs. S 2q lower panel: S 2p (Z =50,N)/2 The global linear trend is taken out subtracting N 82 [S 2 2p (Z=50,N=50) S 2p (Z=50,N=82)] using the spherical mean-field S 2p M. B., G. F. Bertsch, P.-H. Heenen, PRC 78 (2008) lower panel: S 2n (Z,N=50)/2 The global linear trend is taken out subtracting N 50 2 [S 2n (Z=28,N=50) S 2n (Z=50,N=50)] using the spherical mean-field S 2n

27 Eigenvalues of the single-particle Hamiltonian vs. S 2q lower panel: S 2p (Z =50,N)/2 The global linear trend is taken out subtracting N 82 [S 2 2p (Z=50,N=50) S 2p (Z=50,N=82)] using the spherical mean-field S 2p M. B., G. F. Bertsch, P.-H. Heenen, PRC 78 (2008) lower panel: S 2n (Z,N=50)/2 The global linear trend is taken out subtracting N 50 2 [S 2n (Z=28,N=50) S 2n (Z=50,N=50)] using the spherical mean-field S 2n

28 Two-nucleon gaps δ 2p(N,Z) = S 2p(Z,N) S 2p(Z 2,N) δ 2n(N,Z) = S 2n(Z,N) S 2n(Z,N 2) In spite of their name, the two-nucleon shell gaps do not measure the gap in the single-particle spectrum in self-consistent mean-field model M. B., G. F. Bertsch, P.-H. Heenen, Phys. Rev. Lett. 94 (2005) M. B., G. F. Bertsch, P.-H. Heenen, Phys. Rev. C 73 (2006) experimental values shown here include more recent data than the plots in the papers

29 Collectivity-enhanced quenching of signatures of shell closures M. B., G. F. Bertsch, P.-H. Heenen, PRC 78 (2008)

30 Collectivity-enhanced quenching of signatures of shell closures M. B., G. F. Bertsch, P.-H. Heenen, PRC 78 (2008)

31 Collectivity-enhanced quenching of signatures of shell closures same effect seen in 5-dimensional collective Hamiltonian from Gogny D1s S 2N (MeV) (a) HFB 162 S 2P (MeV) (b) 82 HFB S 2N (MeV) DCH S 2P (MeV) DCH EXP 40 EXP S 2N (MeV) 20 S 2P (MeV) Z N J.-P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. Péru, N. Pillet, G. F. Bertsch, PRC81 (2010)

32 Empirical single-particle energies vs. eigenvalues of the single-particle Hamiltonian It is customary to discuss shell structure in terms of the spectrum of eigenvalues of the single-particle Hamiltonian ǫ µ in even-even nuclei K. Rutz, M. B., P.-G. Reinhard, J. A. Maruhn and W. Greiner, Nucl. Phys. A634 (1998) 67 ĥψ µ = ǫ µψ µ Koopman s theorem states that ǫ µ is equal to the one-nucleon separation energy if The nucleus is perfectly described by a HF state (i.e. that there are no correlations of any kind whatsoever) rearrangement and polarization effects changing the single-particle wave functions when adding or removing a particle are negligible The structure of the mean-field state of an even- even and an odd-a nucleus is different (blocking, additional mean fields that originate from interactions involving currents and spin densities in the odd-a nucleus,...), there always are correlations and they will give a different contribution to the binding in even-even and odd-a nuclei, etc.

33 Mutually enhanced magicity effect earlier seen by K. H. Schmidt et al. Nucl. Phys. A318 (1979) 253

34 Q α values of even-even nuclei M. B.and P.-H. Heenen, to be published Q α values calculated from spherical liquid drop with parameters derived from SLy4 (a) spherical self-consistent mean field with SLy4 (b) deformed self-consistent mean field with SLy4 (c) J = 0, N and Z projected GCM of axial states with SLy4 (d) isotopic chains Z = 90, 100, 110, 120 shown in red

35 What is represented by δv pn? δv pn(n,z) = 1 4 [ E(Z,N) E(Z,N 2) E(Z 2,N)+E(Z 2,N 2) ] δv pn = 1 4 ( ) M. B., P.-H. Heenen, PRC 83 (2011)

36 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011)

37 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011)

38 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011)

39 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011)

40 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011)

41 What is represented by δv pn? N Z = 32 chain across the rare-earth region deformation well reproduced contribution from deformation is the key ingredient to describe fine structure of δv pn contribution from fluctuations in deformation is important in transitional regions M. B., P.-H. Heenen, PRC 83 (2011)

42 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011)

43 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011)

44 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011)

45 What is represented by δv pn? ( ) δv pn = 1 4 M. B., P.-H. Heenen, PRC 83 (2011)

46 What is represented by δv pn? M. B., P.-H. Heenen, PRC 83 (2011) δv pn = 1 (red + green - blue) 4 Comparison of frozen HF: calculation without pairing using the same single-particle states of the HF solution of 208 Pb for all 4 nuclei HF: calculation without pairing using self-consistent states HF+BCS+LN: calculation with pairing using self-consistent states J=0 GCM: calculation with pairing using the mixing of symmetry-restored self-consistent states with different quadrupole deformation

47 Summary The interpretation of observables in terms of simple models leads to a model-dependent interpretation of observables. This is a constant source of confusion and misunderstanding when comparing different models. the simplicity of the model assumptions made when interpreting mass filters as gaps of some kind clashes with the realities of complex systems It is recommended to compare only calculated mass filters with experimental mass filters, not some model-dependent ingredient of the model used (spectral gaps,...)

Shell structure of superheavy elements

Shell structure of superheavy elements Michael Bender Université Bordeaux 1; CNRS/IN2P3; Centre d Etudes Nucléaires de Bordeaux Gradignan, UMR5797 Chemin du Solarium, BP120, 33175 Gradignan, France Workshop