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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

Shell and pairing gaps from mass measurements: experiment

Magdalena Kowalska

CERN, ISOLDE


Masses and nuclear structure
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
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
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
Binding energy

  • Net effect of all forces

    • Parabolic behaviour

    • Odd-even staggering

    • Discontinuity at magic numbers

N


Separation energy
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
2-nucleon separation energy

  • Close-to-linear decrease

  • No odd-even staggering

  • Larger decrease at magic numbers

N


3 point mass difference
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
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
Two-proton separation energy

Z=28

Z=50

Decrease for smaller N

Z=82

N


Two neutron separation energy
Two-neutron separation energy

N=20

N=50

N=82

N=126

Z



S 2n zoom1
S2n – zoom1

N


S 2n zoom11
S2n– zoom1

N=20

N=50

N=28

N=82

Decrease for smaller Z

Z


Shell gap zoom1
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 and pairing gaps from mass measurements experiment

Shell gap-zoom2

DS2N/2 [keV]


Empirical shell gaps
Empirical shell gaps

  • D(S2n)/2 [keV]

Decrease for smaller Z

Z


Example ca
Example: Ca

Binding energy


Separation energy1
Separation energy

x

Pairing energy


Pairing gap
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
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
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
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
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
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?


Shell and pairing gaps from mass measurements experiment

S. Naimi, ISOLTRAP PhD thesis 2010