1. Carpathian Summer School of Physics 2007
Sinaia, Romania, August 20th-31st, 2007
Mirror Nuclei: symmetry breaking and nuclear dynamics Silvia M. Lenzi Dipartimento di Fisica and INFN, Padova, Italy

2. Symmetries conservation laws good quantum numbers
Symmetries help to understand Nature
Examination of fundamental symmetries:
a key question in Physics
conservation laws
good quantum numbers
In nuclear physics, conserved quantities imply underlying symmetries of the interactions and help to interpret nuclear structure features
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

3. Symmetries in nuclear physics
Isospin Symmetry: 1932 Heisenberg SU(2)
Spin-Isospin Symmetry: 1936 Wigner SU(4)
Seniority Pairing: 1943 Racah
Spherical Symmetry: 1949 Mayer
Nuclear Deformed Field (spontaneous symmetry breaking)
Restore symm. rotational spectra: 1952 Bohr-Mottelson
have let to deep insights in describing many facets of the atomic nucleus
SU(3) Dynamical Symmetry: 1958 Elliott
Interacting Boson Model (IBM dynamical symmetry):
1974 Arima and Iachello
Critical point symm. E(5), X(5) ….
2000… F. Iachello
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

4. Outline What’s isospin symmetry? Why studying isospin symmetry?
How do we study it?
Experimental methods
Theoretical methods
What do we learn from the data?
Few illustrative recent examples
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

5. Bibliography
Isospin in Nuclear Physics, Ed. D.H. Wilkinson, North Holland, Amsterdam, 1969.
Review article on CDE: J.A. Nolen and J.P. Schafer, Ann. Rev. Nucl. Sci 19 (1969) 471; S. Shlomo, Rep. Prog. Phys.41 (1978) 66; N. Auerbach, Phys. Rep. 98 (1983) 273
Recent on CDE: J. Duflo and A.P. Zuker, Phys. Rev. C 66 (2002) (R)
Theory on CED: A.P. Zuker, S.M. Lenzi, G. Martinez-Pinedo and A. Poves, Phys. Rev. Lett. 89 (2002) ;
Theory on CED: J.A. Sheikh, D.D. Warner and P. Van Isacker, Phys. Lett. B 443 (1998) 16
Shell model reviews: B.A. Brown, Prog. Part. Nucl. Phys 47 (2001) 517; T. Otsuka, M. Honma, T. Mizusaki and N. Shimitzu, Prog. Part. Nucl. Phys 47 (2001) 319; E. Caurier, G. Martinez-Pinedo, F. Nowacki, A.Poves, and A.P. Zuker, Rev. Mod. Phys. 77 (2005) 427
Review article on CED: M.A. Bentley and S.M. Lenzi, Prog. Part. Nucl. Phys. (2006).
Review article on N~Z: D. D. Warner, M. A. Bentley, P. Van Isacker, Nature Physics 2, (2006)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

6. The nucleus: a unique quantum laboratory
Composed by two types of fermions differing only on its charge
Strong interaction: largely independent of the charge
Proton – Neutron exchange symmetry
Proton and neutron can be viewed as two alternative states of the
same particle: the nucleon.
The quantum number that distinguish the two charge states is the isospin
This is in analogy to the two intrinsic spin states of an electron
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

7. Charge invariance and isospin
1932 Heisenberg applies the Pauli matrices
to the new problem of labeling the two
alternative charge states of the nucleon.
1937 Wigner: isotopic spin is a good quantum
number to characterize isobaric multiplets.
Isobaric analogue
multiplets:
States with the same J,T in
nuclei with the same A=N+Z
We define a z-axis in space by establishing a magnetic field B. The electron is a tiny magnet that precesses about the magnetic field. When we try to measure the projection of the electrons spin along the z axis we discover that we can only measure two values, either along B or anti parallel to B. These spin projections, +1/2 and -1/2 are commonly called spin up or spin down.
The nucleon is a two state object. We define an abstract internal space called isospin. The spin one-half nucleon is said to have projections either along z ( the proton) or anti parallel to z
( the neutron). We must note that this picture of an abstract internal isospin space is a convenient mathematical device. Whatever we learn from the angular momentum algebra for electron spin we can apply to isospin.
This picture does not explain the dynamical origins of isospin symmetry. It was not until the discovery of quarks that the cause of isospin symmetry was understood.
It was discovered that the isospin properties of a pair of interacting nucleons influenced the reactions possible between them. The nucleus could be characterized by isospin quantum numbers which restricted the possible states in which the many nucleon system can exist. The strong nuclear force and the electromagnetic force "conserve" isospin. This means that the total isospin and its projection were constants of motion if these two forces were responsible for the dynamics of the system. The weak interaction does not conserve isospin.
Nuclear interaction: π ν
Charge Symmetry: Vpp=Vnn
Charge Independence: Vpp=Vnn=Vpn
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

8. Two-nucleon system For a two-nucleon system
four different isospin states
can exist:
Triplet T=1
Singlet T=0
The isospin quantum number T directly couples together the two effects of charge symmetry/independence and the Pauli principle
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

9. Isobaric spin (isospin)
In the absence of Coulomb interactions between the protons, a perfectly charge-symmetric and charge-independent nuclear force would result in the binding energies of all these isobaric analogue nuclei being identical; that is, they would be structurally identical.
We define a z-axis in space by establishing a magnetic field B. The electron is a tiny magnet that precesses about the magnetic field. When we try to measure the projection of the electrons spin along the z axis we discover that we can only measure two values, either along B or anti parallel to B. These spin projections, +1/2 and -1/2 are commonly called spin up or spin down.
The nucleon is a two state object. We define an abstract internal space called isospin. The spin one-half nucleon is said to have projections either along z ( the proton) or anti parallel to z
( the neutron). We must note that this picture of an abstract internal isospin space is a convenient mathematical device. Whatever we learn from the angular momentum algebra for electron spin we can apply to isospin.
This picture does not explain the dynamical origins of isospin symmetry. It was not until the discovery of quarks that the cause of isospin symmetry was understood.
The strong nuclear force and the electromagnetic force "conserve" isospin. This means that the total isospin and its projection were constants of motion if these two forces were responsible for the dynamics of the system. The weak interaction does not conserve isospin.
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

10. Symmetry breaking…
Isospin symmetry breakdown, mainly due to the Coulomb field, manifests when comparing mirror nuclei. This constitutes an efficient observatory for a direct insight into nuclear structure properties.
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

11. IMME and Coulomb Displacement Energies
Isobaric Multiplet
Mass Equation
a: isoscalar (~100 MeV)
b: isovector (~10 MeV)
c: isotensor (~300 keV)
T=3/2 Jπ= 5/2+
CDE
For a set of isobaric analogue states,
the difference between the masses or
BE of two neighbours defines the CDE
Nolen-Schiffer anomaly: calculated CDE underestimate the data by 7% (100’s keV)
Recent works show that this discrepancy can be reduced to the order of ~200 keV
The understanding of Coulomb effects at the level of less than
100 keV seemed likely to be very difficult…
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

12. Excited analogue states
Normalize the
ground state energies
and look at the excited
analogue states…
Mirror nuclei with Tz = ±1/2
This is generally not the case for any set of IAS,
where the isospin symmetry across a multiplet relies on both
charge symmetry {\sl and} charge independence.
Tests isospin
symmetry
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

13. T=1 isobaric triplets Look at the isobaric triplet:
The nucleus can be characterized by isospin quantum numbers which restrict the possible states in which the many-nucleon system can exist.
Look at the isobaric triplet:
T=1 states low in energy in 22Mg and 22Ne
T=0 and T=1 states in 22Na (N=Z)
Tests isospin
independence
We expect:
MeV
MeV 5 5 4 4 4+ 4+ 4+ 3 3 2 0+ 2+ 2 2+ 2+ 1 1 0+ 0+
0.693 4+ 3+
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

14. T=3/2 Isobaric quadruplets: the spectra
Small differences in excitation energy due (mainly?) to Coulomb effects
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

15. Coulomb energy differences
(differences of excitation energies) Z
N=Z
Mirror Energy Differences (MED) N
Tests the charge symmetry of the interaction
Triplet Energy Differences (TED)
Tests the charge independency of the interaction
MED and TED are of the order of 10’s of keV
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

16. A classical example: MED in T=1/2 states
- bulk energy (100’s of MeV)
- displacement energy (g.s.) CDE (10’s of MeV)
- differences between excited states (10’s of keV)
Coulomb effects in
isobaric multiplets:
25Mn24
24Cr25 49
Mirror Energy Differences
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

17. Measuring the Isospin Symmetry Breaking
Can we reproduce such small energy differences?
What can we learn from them?
Interestingly they contain a richness of information
about spin-dependent structural phenomena
We measure nuclear structure features:
How the nucleus generates its angular momentum
Evolution of the radii (deformation) along a rotational band
Learn about the configuration of the states
Isospin non-conserving terms in the nuclear interaction
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

18. Mirror energy differences and alignmentj J
probability distribution for the relative distance of two like particles in the f7/2 shell j
J=6
ΔEC 8 2 4 6 8
MED
angular momentum
I=8 6
neutron
align. 4
proton
align. 2 j
A(N,Z)
A(Z,N) j
J=0
Shifts between the excitation energies
of the mirror pair at the back-bend
indicate the type of nucleons that are
aligning
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

19. Nucleon alignment at the backbending
J.A. Cameron et al., Phys. Lett. B 235, 239 (1990)
C.D. O'Leary et al., Phys. Rev. Lett. 79, 4349 (1997)
49Mn
49Cr
Experimental MED j
J=6
Alignment
MED are a probe of nuclear structure:
reflect the way the nucleus generates its angular momentum
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

20. Coulomb energy differences: Experimental methods

21. Studying the f7/2 shell f7/2
The 1f7/2 shell is isolated in energy from other major orbits
Wave functions dominated by (1f7/2)n configurations
41Ca
40Ca
43Sc
42Sc
41Sc
45Ti
44Ti
43Ti
42Ti
47V
46V
45V
44V
49Cr
48Cr
47Cr
46Cr
51Mn
50Mn
49Mn
48Mn
53Fe
52Fe
51Fe
50Fe
55Co
54Co
53Co
52Co
56Ni
55Ni
54Ni
44Ca
43Ca
42Ca
45Sc
44Sc
47Ti
46Ti
49V
48V
51Cr
50Cr
54Fe
53Mn
52Mn
N=Z 20 21 22 23 24 25 26 27 28
proton
number
neutron
High-spin states experimentally reachable
f7/2
f5/2
d5/2
d3/2
s1/2
p3/2
p1/2 28 20
Experimental issue : proton-rich Tz< 0 isobars very weakly populated
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

22. Experimental requirements
High efficiency for γ detection
Low cross section at high spin (small masses)
High energy transitions
High selectivity: particle detectors
Many channels opened: high efficient charged-particle det.
Kinematics reconstruction
for Doppler broadening
Mass spectrometers
Neutron detectors to select proton rich channels
Polarimeters and granularity (J, π, δ)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

23. Gamma spectroscopy γ1 γ2 γ3 Anti-Compton gamma ray Ge crystal
Constructing
a level scheme
Gamma array γ1 γ2 γ3
156Dy
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

24. Gamma-ray spectrometers
Present
Next future
GASP
GAMMASPHERE
AGATA
GRETA
e ~ 10 — 5 %
( Mg=1 — Mg=30)
e ~ 40 — 20 %
( Mg=1 — Mg=30)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

25. Techniques for proton-rich spectroscopy
Three basic techniques for selecting proton-rich systems
High efficiency & high granularity gamma-ray spectrometer (e.g. Euroball, Gammasphere) - high fold gn (n  3) coincidence spectroscopy
Gamma-ray array + 0o recoil mass spectrometer + focal plane detectors - identify A,Z of recoiling nucleus  tag emitted gamma-rays
Identify cleanly all emitted particles from reaction - needs a charged-particle detector (p, a) + high-efficiency & high granularity neutron detector array
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

26. 1. High fold gn (n > 3) coincidence spectroscopy
Rely on the power of the array:
high-fold gamma ray coincidences
high granularity…
and on the similarity between the
energy of the transitions with those
of the known mirror nucleus
Double-coincidence spectra after gating on 2 analogue transitions
S.J. Williams et al., Phys. Rev. C 68 (2003)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

27. 2. Identification of A,Z of the recoiling nucleusGS
E.g.: Fragment Mass Analyser (FMA) at Argonne National Lab
Combined electric and magnetic dipoles
 beam rejection & A/q separation
A/q identified by x-position at focal plane
Z identified by energy loss (E-DE) in
gas-filled ionisation chamber
FMA information used to “tag” coincident
gamma-rays at target
Efficiency - up to ~ 15%
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

28. Z identification Example: No excited states known in 48Mn
 Z identification essential… E DE
Ionisation Chamber Fe Mn Cr V Ti
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

29. γ-γ coincidence analysis
(A/q = 3, Z=25)-gated
g-g coincidence analysis…
M.A. Bentley et al.,
Phys.Rev. Lett. 97, (2006)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

30. 3. Measuring the evaporated particles
With this method we do not measure directly the final residue but
the particles emitted from the compound nucleus
neutrons
charged particles
We need high efficiency
detectors for:
Advantage: more flexible than recoil mass spectrometry
 more channels can be measured!
Disadvantage: not as clean as RMS and, if neutrons are needed,
it can be much less efficient
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

31. Observation of excited states in 50Fe
50Fe sum
of gates (*)
counts
50Cr sum
of gates (*) Fe 50 26 24 Cr 50 24 26
Eg (keV)
σ(Fe)/ σ(Cr) ≈ 10-4
S. M. Lenzi et al., Phys. Rev. Lett. 87, (2001)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

32. Coulomb energy differences: Theoretical methods

33. Cranked shell model and alignment
Approximations:
one shell only
fixed deformation
no p-n pairing
CSM: good qualitative
description of the data
J.A. Sheikh, P. Van Isacker, D.D. Warner and J.A. Cameron,
Phys. Lett. B 252 (1990) 314
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

34. Ingredients for the Shell Model calculations
1) an inert core
2) a valence space
3) an effective interaction that mocks up the general
hamiltonian in the restricted basis
s1/2
p1/2
p3/2
d3/2
d5/2
f7/2
f5/2 8 20 28 2
N or Z
the valence
space
The choice is determined by the limits in computing time and memory: large dimension of the matrices to be diagonalised.
inert core
Current programs
diagonalise matrices
of dimension ~109
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

35. Shell model and collective phenomena
Shell model calculations in the full fp
shell give an excellent description
of the structure of collective rotations
in nuclei of the f7/2 shell
Excitation energies
Transition probabilities
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

36. D.D. Warner et al., Nature Physics 2 (2006) 311
Alignment in A=51
D.D. Warner et al., Nature Physics 2 (2006) 311
51Fe
51Mn
Energy (MeV)
Experiment
Shell Model 6 25 21 17 13 9 5
MED
Alignment 2J 3
100 keV
MED (keV)
M.A. Bentley et al, PRC 62 (2000)
J.Ekman et al, EPJ A9 (2000) 13
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

37. Improving the description of Coulomb effects
Can we do better?
What is missing?
Multipole term
of the Coulomb energy VCM:
Between valence protons only
radial effect:
radius changes with J
Monopole term
of the Coulomb energy Vcm
interaction
with the core
change the
single-particle
energies
A.P. Zuker
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

38. The radial term Coulomb energy of a charged sphere:
The difference between the energy of the ground states (CDE): J
If RC changes as a function of the angular momentum…
Radial contribution
to the MED
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

39. Evidencing the monopole radial effect
Multipole (alignment) effects
are cancelled out
VCr
radial term
Most important contribution
M.A. Bentley et al.,
PRL 97, (2006)
The nucleus changes shape
towards band termination
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

40. Electromagnetic single-particle effects: the ℓ·s term
50 times smaller than the nuclear spin-orbit term!!!
Acts differently on protons and neutrons
ΔEp ~ 220 keV s
f7/2
j=l+½
f7/2
j=l+½ ℓ
d3/2
j=l-½
d3/2
j=l-½ ℓ s p n
Its contribution to the MED becomes significant for configurations with a pure single-nucleon
excitation to the f7/2 shell: a proton excitation in one nucleus and a neutron excitation in its mirror
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

41. The T=1/2 mirror nuclei A=35
Large MED for the 13/2-
state is found in A=35
35Ar
35Cl
23/2-
19/2-
19/2-
F. Della Vedova et al.,
PRC 75, (2007)
15/2-2
15/2-1
15/2-2
15/2-1
>300 keV!!!
13/2-
11/2-2
13/2-
11/2-2
J. Ekman et al.,
PRL 92 (2004)
Negative parity
The two 11/2 states have
very different configurations
Measured very large MED
values for all high spin states!
13/2-
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

42. MED in the sd shell: the EMSO
The mirror pair
35Ar – 35Cl
Calculations in the sdfp space
Large and similar contributions
from the multipole Coulomb
and the electromagnetic
spin-orbit terms
Exp. data: Th. Andersson et al., EPJA 6, 5 (1999)
F. Della Vedova et al.,
PRC 75, (2007)
The mirror pair
39Ca – 39K
Small effects due to the
orbital term
Puzzling results…
deformation effects?
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

43. EMSO effect in the fp and f7/2 shell
The mirror pair 61Ga – 61Zn
VCM
VCM+ls
VCM+ls+ll
VCM+ls+ll+VCr
Exp
Data: L-L. Andersson et al., PRC (2005)
The mirror pair 48Mn – 48V
M.A. Bentley et al.,
PRL 97, (2006)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

44. Are Coulomb corrections enough?
sum of Coulomb
terms
Another term of nuclear nature is needed, but it has to be big!
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

45. Looking for an empirical interaction
In the single f7/2 shell, an interaction V can be defined by two-body matrix elements written in the proton-neutron formalism :
We can recast them in terms of isoscalar, isovector and isotensor contributions
ππ πν νν
We assume that the configurations
of these states are pure (f7/2)2
Mirrors
Isovector
Isotensor
Triplet
If the energy differences are due only to VC one expects
very small numbers for all J couplings for VB
42Ti
42Sc
42Ca
A. P. Zuker et al., Phys. Rev. Lett. 89, (2002)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

46. Learning from MED and TED in A=42
From the yrast spectra of the T=1 triplet 42Ti, 42Sc, 42Ca we deduce:
J=0
J=2
J=4
J=6 VC 81 24 6
-11
MED-VC 5 93
-48
TED-VC
117 3
-42
Calculated
estimate VBf7/2 (1)
estimate VBf7/2 (2)
This suggests that the role of the isospin non conserving nuclear force is at
least as important as the Coulomb potential in the observed MED and TED
J=2 anomaly
Simple ansatz
for the application to
nuclei in the pf shell:
A. P. Zuker et al., Phys. Rev. Lett. 89, (2002)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

47. Evidence of ISB of the nuclear interaction
Very good quantitative agreement between theory and data
The multipole Coulomb contribution
gives information on the nucleon alignment
MED (keV)
A=49
The monopole Coulomb contribution
gives information on changes in the nuclear radius (deformation)
Important contribution from the “nuclear” ISB term,
of the same order as the Coulomb contributions!!!!!
Now, without changing the parametrization, see how the rest of the
MED for nuclei along the f7/2 shell are described by the calculations…
A. P. Zuker et al., PRL 89, (2002)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

48. Coulomb energy differences (CED): Results

49. MED and TED in the shell model framework
Good quantitative
description of data
without free parameters
A = 48
A = 47
A = 54
A = 53
A = 51
M.A. Bentley and S.M. Lenzi,
Prog. Part. Nucl. Phys. (2006) in press
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

50. Rotational T=1/2 analogue states A=47/49
7 particles/holes in the f7/2 shell
Band termination state:
Deformed nuclei  Rotational bands
All terms contribute
significantly to the MED
Monopole effects: Cr
Multipole effects: CM and VB
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

51. T=1 A=54/42 triplet: MED and TED
J=2 anomaly
A=54
A=42
no collectivity: only multipole effects: smooth recoupling and J=2 anomaly
2 particles / holes
A=54
A=42
A.Gadea et al.,
PRL 97, (2006)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

52. More results of MED measurements
P.E. Garrett et al.,
PRC 75, (2007)
Rising and GANIL: T=2 mirror in A=36
P. Doorneball et al., Phys.Lett. B 647, 237 (2007)
F. Azaiez et al., to be published
Rising stopped-beam campaign: J=8,10 in A=54
D. Rudolph et al., to be published
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

53. along a rotational band
Summary and outlook
These studies allow to learn about:
Mechanism of nucleon alignment
at the backbending
Evolution of the radii
along a rotational band
Importance of the single-particle effects:
- test interactions and basis
- information on the configurations
Evidence of isospin-non-conserving
terms in the nuclear interaction
J=2 anomaly
Other interesting facets can be, and are being, studied in isobaric multiplets:
- lifetimes and decay probabilities
- magnetic and giromagnetic moments
- isospin mixing …..
These investigations will improve with the advent of intense stable and
radioactive beams and the next generation gamma-arrays and ancilaries
Much effort has to be put in the development of theoretical methods
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

54. Thanks to… Theory: A.P. Zuker, F. Nowacki (Strasbourg) Experiments:
M. A. Bentley (York)
N. Marginean, A. Gadea, F. Della Vedova, (LNL and Padova)
J. Ekman, D. Rudolph (Lund)
P.E. Garrett (Guelph)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

55. Monopole Coulomb single-particle effects: 2) the ℓ2 (orbital) term
J. Duflo and A.P. Zuker, Phys. Rev. C 66 (2002) (R)
The monopole Coulomb term accounts for
shell effects.
It changes the single-particle energy of the protons proportionally to the square of the orbital angular momentum.
For a proton in a main shell N above a closed shell Zcs is:
Eg. in the fp shell:
proton s.p. relative
energy is increased
by 150 keV
p3/2
150 keV
f7/2
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

56. Alignment and shell model
Define the operator
51Fe-51Mn
“Counts” the number of protons coupled to J=6
Calculate the difference of the
expectation value in both mirror
as a function of the angular momentum
1/2
3/2
5/2
7/2 p n
51Fe
51Mn
In 51Fe (51Mn) a pair of protons (neutrons)
align first and at higher frequency align
the neutrons (protons)
M.A.Bentley et al. Phys Rev. C62 (2000)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007