Shell and pairing gaps from mass measurements: experiment

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Shell and pairing gaps from mass measurements: experiment

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Shell and pairing gaps from mass measurements: experiment

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Shell and pairing gaps from mass measurements: experiment

Magdalena Kowalska

CERN, ISOLDE

- 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

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)

- 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

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

N

- First mass derivative
- Steady decrease (almost linear)
- Odd-even staggering (larger for even-Z)
- Larger decrease at magic numbers

N

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

N

- 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

- 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

Z=28

Z=50

Decrease for smaller N

Z=82

N

N=20

N=50

N=82

N=126

Z

N

N

N=20

N=50

N=28

N=82

Decrease for smaller Z

Z

DS2N/2 [keV]

1/2 x Empirical shell gap

DS2N/2:

1/2 x S2N(Z,N)-S2N(Z,N+2)]

N

Shell gap-zoom2

DS2N/2 [keV]

- D(S2n)/2 [keV]

Decrease for smaller Z

Z

Binding energy

x

Pairing energy

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

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).

- 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).

- 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).

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

even-Z

even-Z

p-n interaction?

odd-Z

odd-Z

- 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?

S. Naimi, ISOLTRAP PhD thesis 2010