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Shell and pairing gaps from mass measurements: experiment

This article discusses the use of mass measurements to determine shell and pairing gaps in nuclear structure. It explores the observable and assumptions associated with shell gaps and pairing gaps, and compares experimental values to theoretical models. The article also addresses open questions related to pairing gaps.

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Shell and pairing gaps from mass measurements: experiment

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  1. Shell and pairing gaps from mass measurements: experiment Magdalena Kowalska CERN, ISOLDE

  2. Masses and nuclear structure • Atomic masses and nuclear binding energy show the net effect of all forces inside the nucleus • Mass filters (i.e. various mass differences) “enhance” specific effects, compared to others • Best comparison to nuclear structure models: use models to calculate mass differences (i.e. compare the observables) • Easier in mean-field models than in shell model • Problems start when comparing to non-observables

  3. Shell gaps fp-shell 28 20 sd-shell 8 p-shell 2 s-shell • Observable: • Two-nucleon separation energy; how strongly bound are the 2 additional neutrons (protons) • “empirical shell gap”: Difference in two-nucleon separation energy • “indirect observable”: (single-particle) shell gap • Assumptions • Single-particle picture: no correlations • No rearrangement when adding the additional nucleons • In practice: small correlations (thus little deformation)

  4. Pairing gaps • Observable • odd-even staggering in binding energy • 3-, 4-, or 5-point mass-difference formula • “indirect observable” – pairing gap • Assumptions • No rearrangement (polarization) • The same shell filled

  5. Binding energy • Net effect of all forces • Parabolic behaviour • Odd-even staggering • Discontinuity at magic numbers N

  6. Separation energy • First mass derivative • Steady decrease (almost linear) • Odd-even staggering (larger for even-Z) • Larger decrease at magic numbers N

  7. 2-nucleon separation energy • Close-to-linear decrease • No odd-even staggering • Larger decrease at magic numbers N

  8. 3-point mass difference • Second mass derivative • Linear trend taken away • Showing the size of odd-even staggering (larger for even-Z) • Small residual odd-even staggering • Larger at magic numbers N

  9. 4-point mass difference • Second mass derivative • Linear trend taken away • Showing the size of odd-even staggering (larger for even-Z) • No residual odd-even staggering • Larger at magic numbers N

  10. Two-proton separation energy Z=28 Z=50 Decrease for smaller N Z=82 N

  11. Two-neutron separation energy N=20 N=50 N=82 N=126 Z

  12. Two-neutron separation energy N

  13. S2n – zoom1 N

  14. S2n– zoom1 N=20 N=50 N=28 N=82 Decrease for smaller Z Z

  15. Shell gap-zoom1 DS2N/2 [keV] 1/2 x Empirical shell gap DS2N/2: 1/2 x S2N(Z,N)-S2N(Z,N+2)]

  16. S2n – zoom2 N

  17. Shell gap-zoom2 DS2N/2 [keV]

  18. Empirical shell gaps • D(S2n)/2 [keV] Decrease for smaller Z Z

  19. Example: Ca Binding energy

  20. Separation energy x Pairing energy

  21. Pairing gap D(3) Example: • Neutron pairing gap in Ca For even N – shell effects visible D(4) D3(N) = B(N-1)-2B(N)+B(N+1) D4(N) = B(N-2)-3B(N-1)+3B(N)-B(N+1) Smoother than D3, but Centred at N+1/2 or N-1/2

  22. N=40 and 68Ni region From S. Naimi et al, Phys. Rev. C 86, 014325 (2012) Theory: M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys. Rev. C 78, 054312 (2008).

  23. Shell gap at N=50 • Empirical shell gap Decrease for smaller Z Decrease also in spherical mean-filed -> shell gap indeed decreases Theory: M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys. Rev. C 78, 054312 (2008).

  24. Shell gap at Z=50 • Empirical shell gap Decrease for smaller Z No decrease in spherical mean-filed -> shell gap doesn’t decrease; experimental value changes due to correlations Theory: M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys. Rev. C 78, 054312 (2008).

  25. N-pairing gap for odd and even Z Pairing gap difference: can we call it p-n pairing? even-Z even-Z p-n interaction? odd-Z odd-Z

  26. Summary • Mass differences can be used to obtain empirical • shell gaps – 2-nucleon separation energies • pairing gaps – odd-even mass staggering • To give them quantitative value, other effects should be small in a given region: • Shells: small deformations • Pairing: the same shell filled, similar deformation • Comparison to theoretical models: • Safest: compare to theoretical mass differences • Problems start when interpreting the values as shell or pairing gaps • Open questions mainly for pairing • Which formula to use? • What about p-n interaction?

  27. S. Naimi, ISOLTRAP PhD thesis 2010

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