Shell and pairing gaps from mass measurements: experiment

Magdalena KowalskaCERN, ISOLDE

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Masses and nuclear structure

Atomic masses and nuclear binding energy show the net effect of all forces inside the nucleusMass filters (i.e. various mass differences) “enhance” specific effects, compared to othersBest 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

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Shell gapsObservable: 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 gapAssumptions Single-particle picture: no correlations No rearrangement when adding the additional nucleons In practice: small correlations (thus little deformation)

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2s-shell

p-shell

sd-shell

fp-shell

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Pairing gapsObservable odd-even staggering in binding energy 3-, 4-, or 5-point mass-difference formula

“indirect observable” – pairing gapAssumptions No rearrangement (polarization) The same shell filled

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Binding energy

Net effect of all forces Parabolic behaviour Odd-even staggering Discontinuity at magic numbers

N

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Separation energyFirst mass derivative Steady decrease (almost

linear) Odd-even staggering

(larger for even-Z) Larger decrease at magic

numbers

N

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2-nucleon separation energy

Close-to-linear decrease No odd-even staggering Larger decrease at magic numbers

N

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3-point mass differenceSecond 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

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4-point mass differenceSecond 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

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Two-proton separation energy

N

Z=50

Z=82

Z=28

Decrease for smaller N

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Two-neutron separation energy

Z

N=20

N=50

N=82

N=126

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Two-neutron separation energy

N

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S2n – zoom1

N

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S2n – zoom1

Z

N=82

N=50

N=28

N=20

Decrease for smaller Z

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Shell gap-zoom1

1/2 x Empirical shell gapDS2N/2:1/2 x S2N(Z,N)-S2N(Z,N+2)]

DS2N

/2 [k

eV]

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S2n – zoom2

N

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Shell gap-zoom2DS

2N/2

[keV

]

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Empirical shell gapsD(S2n)/2 [keV]

Z

Decrease for smaller Z

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Example: Ca

Binding energy

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Separation energy

x

Pairing energy

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Pairing gapExample:

Neutron pairing gap in Ca

Smoother than D3, butCentred at N+1/2 or N-1/2

For even N – shell effects visible

D3(N) = B(N-1)-2B(N)+B(N+1)D4(N) = B(N-2)-3B(N-1)+3B(N)-B(N+1)

D(3)

D(4)

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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).

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Shell gap at N=50Empirical shell gap

Decrease for smaller Z

Theory: M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys. Rev. C 78, 054312 (2008).

Decrease also in spherical mean-filed -> shell gap indeed decreases

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Shell gap at Z=50Empirical shell gap

Decrease for smaller Z

Theory: M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys. Rev. C 78, 054312 (2008).

No decrease in spherical mean-filed -> shell gap doesn’t decrease; experimental value changes due to correlations

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N-pairing gap for odd and even Z

even-Z

odd-Z

even-Z

odd-Zp-n interaction?

Pairing gap difference: can we call it p-n pairing?

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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?

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30S. Naimi, ISOLTRAP PhD thesis 2010

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