Shell and pairing gaps from mass measurements: experiment

```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
2
Shell gaps
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
 In practice: small correlations (thus little deformation)
fp-shell
28
20
sd-shell
8
p-shell
2
s-shell
3
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
4
Binding energy
Net effect of all forces
 Parabolic behaviour
 Odd-even staggering
 Discontinuity at magic numbers
N
5
Separation energy
First mass derivative
(almost linear)
 Odd-even staggering
(larger for even-Z)
 Larger decrease at
magic numbers
N
6
2-nucleon separation energy
 Close-to-linear decrease
 No odd-even staggering
 Larger decrease at magic numbers
N
7
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
8
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
9
Two-proton separation energy
Z=28
Z=50
Decrease for smaller N
Z=82
N
10
Two-neutron separation energy
N=20
N=50
N=82
N=126
Z
11
Two-neutron separation energy
N
12
S2n – zoom1
N
13
S2n – zoom1
N=20
N=50
N=28
N=82
Decrease for smaller Z
14
Z
DS2N/2 [keV]
Shell gap-zoom1
1/2 x Empirical shell gap
DS2N/2:
1/2 x S2N(Z,N)-S2N(Z,N+2)]
15
S2n – zoom2
N
16
DS2N/2 [keV]
Shell gap-zoom2
17
Empirical shell gaps
D(S2n)/2 [keV]
Decrease for smaller Z
Z
18
Example: Ca
Binding energy
19
Separation energy
x
Pairing energy
20
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
21
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).
22
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).
23
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).
24
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
25
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?