Transcript Document

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
Symmetries
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
Symmetries in nuclear physics
p n
Isospin Symmetry: 1932 Heisenberg SU(2)
Spin-Isospin Symmetry: 1936 Wigner SU(4)
Seniority Pairing: 1943 Racah
J   0

j
Spherical Symmetry: 1949 Mayer


J   0
J   2
Nuclear Deformed Field (spontaneous symmetry breaking)
Restore symm. rotational spectra: 1952 Bohr-Mottelson
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
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
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)
051304(R)
• Theory on CED: A.P. Zuker, S.M. Lenzi, G. Martinez-Pinedo and A.
Poves, Phys. Rev. Lett. 89 (2002) 142502;
• 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) 311 - 318
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
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
t  1
2
proton : 
tz  
1
2
neutron : 
tz   1
2
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
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.
A
Tz   t z ,i 
i 1
N Z
2
NZ
NZ
T 
2
2
Isobaric analogue
multiplets:
States with the same J,T in
nuclei with the same A=N+Z
Nuclear interaction:
• Charge Symmetry: Vpp=Vnn
• Charge Independence: Vpp=Vnn=Vpn
π
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
ν
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
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.
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
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
IMME and Coulomb Displacement Energies
IMME:
Isobaric Multiplet
Mass Equation
M
BE((TT
TTz z))aabT
bTz z cT
cT
22
zz
a: isoscalar (~100 MeV)
b: isovector (~10 MeV)
c: isotensor (~300 keV)
-145
CDE  ,T ,Tz  M  ,T ,Tz  M  ,T ,Tz 1  M np
 b  c2Tz  1  M np
Isobaric Analogue States
-150
-BE(MeV)
For a set of isobaric analogue states,
the difference between the masses or
BE of two neighbours defines the CDE
Ground States
-155
CDE
-160
A=21 Isobars
-165
-170
8
9
10
Z
11
12
13
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
Excited analogue states
Normalize the
ground state energies
and look at the excited
analogue states…
Mirror nuclei with Tz = ±1/2
27
13
Al14
27
14
Si13
Tests isospin
symmetry
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
T=1 isobaric triplets
The nucleus can be characterized by isospin quantum numbers which restrict the
possible states in which the many-nucleon system can exist.
22
22
22
Mg
Na
12
10
11 11
10 Ne12
Look at the isobaric triplet:
We expect:
T=1 states low in energy in 22Mg and 22Ne
T=0 and T=1 states in 22Na (N=Z)
Tests isospin
independence
MeV
MeV
5
5
4
4
4+
3
4+
4+
2
3
2
1
2+
0
0+
22
12 Mg10
0.693
4+
3+
22
11 Na11
2+
2+
1
0+
0+
0
22
10 Ne12
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
T=3/2 Isobaric quadruplets: the spectra
21
9 F12
21
10 Ne11
21
11Na10
21
12 Mg 9
T=3/2
T=1/2
Small differences in excitation energy due (mainly?) to Coulomb effects
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Coulomb energy differences
(differences of excitation energies)
Z
Mirror Energy Differences (MED)
MED J  E *J ,Tz  1 / 2 E *J ,Tz  1 / 2  ΔbJ
N
Tests the charge symmetry of the interaction
Triplet Energy Differences (TED)
TED J  E *J ,Tz  1  E *J ,Tz  1 2E *J ,Tz 0  2Δc J
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
A classical example: MED in T=1/2 states
Coulomb effects in
isobaric multiplets:
- bulk energy (100’s of MeV)
- displacement energy (g.s.) CDE (10’s of MeV)
- differences between excited states (10’s of keV)
Mirror Energy Differences
MEDJ  EJ (Z  N )  EJ (Z  N )
49
25Mn24
49
24Cr25
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
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
Mirror energy differences and alignment
J
j
j
j
ΔEC
8
j
8
6
6 J=6
neutron
align. 4
4 proton
align.
2
0
2
0
A(N,Z)
j
A(Z,N)
j
J=0
j
j
j
MED
j
probability distribution for
the relative distance of two
like particles in the f7/2 shell
0
I=8
angular momentum
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
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
j
j
Experimental MED
49Cr
j
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
Coulomb energy differences:
Experimental methods
Studying the f7/2 shell
The 1f7/2 shell is isolated in energy from other major orbits
Wave functions dominated by (1f7/2)n configurations
N=Z
High-spin states experimentally reachable
f5/2
p1/2
p3/2
f7/2
28
28
proton
number
55Ni
56Ni
27
52Co 53Co 54Co 55Co
26
50Fe 51Fe 52Fe 53Fe 54Fe
25
48Mn 49Mn 50Mn 51Mn 52Mn 53Mn
24
20
54Ni
46Cr 47Cr 48Cr 49Cr 50Cr 51Cr
23
44V
45V
46V
47V
48V
43Ti
44Ti
45Ti
46Ti
47Ti
d3/2
s1/2
22
42Ti
d5/2
21
41Sc 42Sc 43Sc 44Sc 45Sc
20
40Ca 41Ca 42Ca 43Ca 44Ca
20
21 22 23 24 25
49V
neutron
number
26
Experimental issue : proton-rich Tz< 0 isobars very weakly populated
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
27 28
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
Gamma spectroscopy
Anti-Compton
gamma ray
Constructing
a level scheme
γ1
Gamma array
Ge crystal
156Dy
γ2
γ3
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Gamma-ray spectrometers
Present
GASP
GAMMASPHERE
e ~ 10 — 5 %
( Mg=1 — Mg=30)
Next future
AGATA
GRETA
e ~ 40 — 20 %
( Mg=1 — Mg=30)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Techniques for proton-rich spectroscopy
Three basic techniques for selecting proton-rich systems
1. High efficiency & high granularity gamma-ray spectrometer
(e.g. Euroball, Gammasphere) - high fold gn (n  3)
coincidence spectroscopy
2. Gamma-ray array + 0o recoil mass spectrometer + focal
plane detectors - identify A,Z of recoiling nucleus  tag
emitted gamma-rays
3. Identify cleanly all emitted particles from reaction - needs a
charged-particle detector (p, ) + high-efficiency & high
granularity neutron detector array
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
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
  260 ms
Double-coincidence spectra after
gating on 2 analogue transitions
S.J. Williams et al., Phys. Rev. C 68 (2003) 011301
32

S
24
Mg,1p2n

53
Co
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
2. Identification of A,Z of the recoiling nucleus
E.g.: Fragment Mass Analyser (FMA)
at 0o @ Argonne National Lab
GS
• 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
Z identification
Example:
Ca 20  B5  Mn
10
5
50
25
*
25
48
25
No excited states known in 48Mn
DE
40
20
V25  2 p
48
23
Mn 23  2n


 48 Mn
4
~
10
 48 V
 
 Z identification essential…
Ionisation Chamber
Fe
Mn
Cr
E
V
Ti
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
γ-γ coincidence analysis
(A/q = 3, Z=25)-gated
gg coincidence analysis…
48
25
Mn 23
M.A. Bentley et al.,
Phys.Rev. Lett. 97, 132501 (2006)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
48
23
V25
3. Measuring the evaporated particles
With this method we do not measure directly the final residue but
the particles emitted from the compound nucleus
charged particles
neutrons
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
Observation of excited states in
50Fe
sum
of gates (*)
counts
50Fe
50Cr
sum
of gates (*)
50
26
Fe
50
24
24
Cr
26
Eg (keV)
σ(Fe)/ σ(Cr) ≈ 10-4
S. M. Lenzi et al., Phys. Rev. Lett. 87, 122501 (2001)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Coulomb energy differences:
Theoretical methods
Cranked shell model and alignment
alignment
Cranked shell-model
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
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
the valence
space
The choice is determined by
the limits in computing time
and memory: large dimension
of the matrices to be
diagonalised.
28
f5/2
p1/2
p3/2
f7/2
20
d3/2
s1/2
inert core
d5/2
8
Current programs
diagonalise matrices
of dimension ~109
p1/2
p3/2
2
s1/2
N or Z
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
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
-50
D.D. Warner et al., Nature Physics 2 (2006) 311
-100
51Fe
Energy
-150
(MeV)
6 -200
5
7
51Mn
9
Experiment
Experiment
fp-shell
Model
Shell Model
11 13 15 17 19 21 23 25 27
Alignment in A=51
Experiment
0
fp-shell Model
50
25
11 13 15 17 19 21 23 25 27
MED
21
Alignment
17
2J
3
9
9
5
MED (keV)
-200
-100
-150
-50
0
-100
M.A. Bentley et al, PRC 62 (2000) 051303
100
100
0
0
50
7
100 keV
5
13
J.Ekman et al, EPJ A9 (2000) 13
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Improving the description of Coulomb effects
Can we do better?
Multipole term
of the Coulomb
energy VCM:
Monopole term
of the Coulomb
energy Vcm
interaction
with the core
VC  VCM  VCm
What is missing?
Between valence protons only
3 e 2 Z ( Z  1)
radial effect:
ECr 
radius changes with J
5
R
ECll
 4.5Z cs13/12[2l (l  1)  N ( N  3)]

keV
A1/ 3 ( N  3 / 2)
ECls  ( g s  gl )
1  1 dVC 

 l.s
4mN2 c 2  r dr 
change the
single-particle
energies
A.P. Zuker
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
The radial term
Coulomb energy of a charged sphere:
3Z ( Z  1 )e 2
EC 
5 RC
The difference between the energy of the ground states (CDE):
3n2 Z  n e 2
EC  J  0  EC Z    EC Z   
5RC
If RC changes as a function of the angular momentum…
n
Tz  
2
J
 1
3
1 

ECr J   EC J   EC 0  n2Z  n e 2 

5
 RC J  RC 0 
3
2  RC 0   RC  J  

 n2Z  n e 
2
5
RC


Radial contribution
to the MED
RC J 
3
  n2Z  n e 2
5
RC2
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Evidencing the monopole radial effect
Multipole (alignment) effects
are cancelled out
48
25
Mn 23
 f 7/32  f 73/ 2
48
23
V25
 f 73/ 2  f 7/32
radial term
VCr
Most important contribution
M.A. Bentley et al.,
PRL 97, 132501 (2006)
The nucleus changes shape
towards band termination
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Electromagnetic single-particle effects:
the ℓ·s term
ECls
1
 g s  gl  2 2
2m N c
 Ze 2   

 l s

R 

50 times smaller than the nuclear spin-orbit term!!!
Acts differently on protons and neutrons
s
ℓ
f7/2
ΔEp ~ 220 keV
j=l+½
ℓ s d3/2
j=l-½
f7/2
j=l+½
d3/2
j=l-½


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
The T=1/2 mirror nuclei A=35
35Ar
35Cl
23/2-
19/2-
19/2-
Large MED for the 13/2is found
A=35
15/2-in
15/2state
2
F. Della Vedova et al.,
PRC 75, 034317 (2007)
1
15/2-1
15/2-2
>300 keV!!!
13/2-
11/2-2
13/2-
11/2-2
J. Ekman et al.,
PRL 92 (2004) 132502
Negative parity
The two 11/2 states have
very different configurations
13/2-
Measured very large MED
values for all high spin states!
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
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
F. Della Vedova et al.,
PRC 75, 034317 (2007)
The mirror pair
39Ca – 39K
Exp. data: Th. Andersson et al., EPJA 6, 5 (1999)
Small effects due to the
orbital term
Puzzling results…
deformation effects?
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
EMSO effect in the fp and f7/2 shell
The mirror pair 61Ga – 61Zn
VCM+ls+ll+VCr
f5 / 2
61
31
Exp
Ga 30
VCM+ls+ll
VCM+ls
VCM
p3 / 2
Data: L-L. Andersson et al., PRC (2005)
The mirror pair 48Mn – 48V
48
25
Mn 23
48
25
48
23
V25
Mn 23
f7 / 2
d3 / 2
M.A. Bentley et al.,
PRL 97, 132501 (2006)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Are Coulomb corrections enough?
49
25
Mn 24  49
24 Cr25
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
Looking for an empirical interaction
In the single f7/2 shell, an interaction V can be defined by twobody matrix elements written in the proton-neutron formalism :
V  , V  , V 
We can recast them in terms of isoscalar, isovector and isotensor contributions
ππ
πν
νν
U (0)  V   V   V 
U (1)  V   V 
U (2)  V   V   2V 
Mirrors
)
(1)
configurations
MEDJ (42 Ti- 42We
Ca)assume
 U (f17)/ 2 , J that
 VC(1the

V
Isovector
,J
B, J
of these states are pure (f7/2)2
Triplet
TEDJ (42 Ti  42 Ca - 242 Sc)  U (f72/)2 , J  VC(,2J)  VB(,2J)
Isotensor
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, 142502 (2002)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Learning from MED and TED in A=42
From the yrast spectra of the T=1 triplet 42Ti, 42Sc, 42Ca we deduce:
VC  VCh.o. ( f 7 / 2 )
J=0
J=2
J=4
J=6
81
24
6
-11
MED-VC
5
93
5
-48
estimate VBf7/2 (1)
TED-VC
117
81
3
-42
estimate VBf7/2 (2)
VC
Calculated
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:
(1)
VBpf
 100 keV ( J  2)
( 2)
VBpf
 100 keV ( J  0)
A. P. Zuker et al., Phys. Rev. Lett. 89, 142502 (2002)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
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)
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, 142502 (2002)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
A=49
Coulomb energy differences
(CED):
Results
MED and TED in the shell model framework
Good quantitative
description of data
without free parameters
A = 47
47
24
A = 48
48
25
A = 49
Mn 23  4823V25
49
25
Mn 24  49
24 Cr25
54
28
Ni26  54
26 Fe 28
Cr23  4723V24
140
A = 54
120
100
A = 51
51
26
51
Fe25  25
Mn 26
A = 53
53
27
53
Co 26  26
Fe 27
80
60
40
20
0
-20
-40
0
2
4
6
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
Rotational T=1/2 analogue states
A=47/49
7 particles/holes in the f7/2 shell
Band termination state:
  72  52  32  12  8
  72  52  32  152
J

max
31

2

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
T=1 A=54/42 triplet: MED and TED
A=54
A=42
J=2 anomaly
no collectivity: only multipole effects:
smooth recoupling and J=2 anomaly
2 particles / holes
A=54
A=42
A.Gadea et al.,
PRL 97, 152501 (2006)
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
More results of MED measurements
P.E. Garrett et al.,
PRC 75, 014307 (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
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
f5 / 2
p3 / 2
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
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
Monopole Coulomb single-particle effects:
2) the ℓ2 (orbital) term
J. Duflo and A.P. Zuker, Phys. Rev. C 66 (2002) 051304(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:
ECll
13 / 12
2  1  N N  3 keV
 4.5Z cs

A1 / 3 N  32 
Eg. in the fp shell:
Z cs  20
N 3
150 keV
p3/2

f7/2

proton s.p. relative
energy is increased
by 150 keV
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007
Alignment and shell model
Define the operator

A   a a

 a a 
  J 6
51Fe-51Mn
J 6  0

“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
ΔA,J   J A Z    J  ' J A Z  ' J
7/2
5/2
3/2
1/2
51Fe
p
51Mn
n
p
n
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) 051303
Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007