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Shell and pairing gaps from mass measurements: experiment. Magdalena Kowalska CERN, ISOLDE. Masses and nuclear structure. Atomic masses and nuclear binding energy show the net effect of all forces inside the nucleus
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Shell and pairing gaps from mass measurements: experiment Magdalena Kowalska CERN, ISOLDE
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
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)
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
Binding energy • Net effect of all forces • Parabolic behaviour • Odd-even staggering • Discontinuity at magic numbers N
Separation energy • First mass derivative • Steady decrease (almost linear) • Odd-even staggering (larger for even-Z) • Larger decrease at magic numbers N
2-nucleon separation energy • Close-to-linear decrease • No odd-even staggering • Larger decrease at magic numbers N
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
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
Two-proton separation energy Z=28 Z=50 Decrease for smaller N Z=82 N
Two-neutron separation energy N=20 N=50 N=82 N=126 Z
S2n– zoom1 N=20 N=50 N=28 N=82 Decrease for smaller Z Z
Shell gap-zoom1 DS2N/2 [keV] 1/2 x Empirical shell gap DS2N/2: 1/2 x S2N(Z,N)-S2N(Z,N+2)]
Shell gap-zoom2 DS2N/2 [keV]
Empirical shell gaps • D(S2n)/2 [keV] Decrease for smaller Z Z
Example: Ca Binding energy
Separation energy x Pairing energy
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
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).
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).
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).
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
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?
