Isospin breaking in
Coulomb energy differences
Mirror Symmetry
Silvia M. Lenzi
Dipartimento di Fisica e Astronomia“Galileo Galilei”
Università di Padova and INFN
Silvia Lenzi
University
of– ARIS
Padova
and
INFN
Silvia Lenzi
2014, Tokyo, June
2-6, 2014
Neutron-proton exchange symmetry
Charge symmetry : Vpp = Vnn
Charge independence: (Vpp + Vnn)/2= Vnp
Deviations are small
MeV
5
T=0 and T=1
T=1
4
MeV
T=1
5
4
4+
4+
4+
1
2+
2+
2+
0
0+ 0.693 1+
0+
0+0
3
2
3
2
22
12 Mg 10
3+
22
11Na11
22
10Ne12
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
1
Differences in analogue excited states
Z
Mirror Energy Differences (MED)
N
MED J  ExJ ,Tz  T  ExJ ,Tz   T
Test the charge symmetry of the interaction
Triplet Energy Differences (TED)
TED J  ExJ ,Tz  T  ExJ ,Tz   T  2 ExJ ,Tz 0
Test the charge independency of the interaction
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
Mirror symmetry is (slightly) broken
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
Silvia Lenzi
– ARIS 2014,
Tokyo, Juneproperties.
2-6, 2014
Measuring MED and TED
Can we reproduce such small energy differences?
What can we learn from them?
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 radii (deformation) along a rotational band
Learn about the configuration of the states
Isospin non-conserving terms of the interaction
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
Coulomb effects
VC  VCM  VCm
VCM Multipole
Coulomb energy:
Between valence
protons only
3 e 2 Z ( Z  1)
ECr 
5
R
VCm Monopole
Coulomb energy
radial effect:
radius changes with J
L2 term to account for shell effects
ECll
 4.5Z cs13 /12 [2l (l  1)  N ( N  3)]

keV
A1/ 3 ( N  3 / 2)
electromagnetic LS term
ECls  ( g s  gl )
1  1 dVC 
 l.s
2 2 
4mN c  r dr 
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
change the
single-particle
energies
Are Coulomb corrections enough?
49
25
VCM+VCm
Mn24 49
24 Cr25
VCM
Exp
VCm
Another isospin symmetry breaking (ISB) term is needed
and it has to be big!
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
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 :
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
Triplet
)
(1)
MEDJ (42 Ti-42We
Ca)assume
 U (f17)/ 2 , J that
 VC(1the

V
configurations
,J
B, J
Isovector
of these states are pure (f7/2)2
TEDJ (42 Ti42 Ca - 242 Sc )  U (f72/)2 , J  VC(,2J)  VB(,2J) Isotensor
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
Looking for an empirical interaction
From the yrast spectra of the T=1 triplet 42Ti, 42Sc, 42Ca we deduce the interaction
J=0
J=2
J=4
J=6
81
24
6
-11
MED-VC
5
93
5
-48
TED-VC
117
81
3
-42
VC
Calculated
estimate VB (1)
estimate VB (2)
Simple ansatz for the application to
nuclei in the pf shell:
(1)
VBpf
( ( f 72/ 2 ) J 2 )  100 keV
(2)
VBpf
( ( f 72/ 2 ) J 0 )  100 keV
J=2 anomally
A. P. Zuker et al., PRL 89, 142502 (2002)
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
The “J=2 anomaly”
Coulomb matrix elements (MeV)
Is this just a Coulomb two-body effect?
Spatial correlation probability for two nucleons in f7/2
Calculation (using Harmonic Oscillator w.f)
Two possibilities:
1) Increase the J=2 term
2) Decrease the J=0 term
We choose 1) but there is not
much difference
Angular momentum J
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
Calculating MED and TED
We rely on isospin-conserving shell model wave functions and obtain the
energy differences in first order perturbation theory as sum of expectation
values of the Coulomb (VC) and isospin-breaking (VB) interactions
MED
exp
J
 E Z    E Z  
*
J
*
J
(1)
MED theo



V




V




V
J
M
Cm J
M
CM J
M
B J
*
*
*




TED exp

E
Z

E
Z

2
E
J
J

J

J N  Z 
(2)
TED theo



V




V
J
T
CM J
T
B J
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
Calculating the MED with SM
MEDJtheo  M VCm  J  M VCM  J  M VB(1)  J
Theo
VCM: gives information on the
nucleon alignment or recoupling
49Mn-49Cr
VCM
Exp
VCm: gives information on
changes in the nuclear radius
VCm
Important contribution from the
ISB VB term:
of the same order as the
Coulomb contributions
M.A. Bentley and SML,
Prog. Part. Nucl. Phys. 59,
497-561 (2007)
VB
A. P. Zuker et al., PRL 89, 142502 (2002)
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
MED in T=1/2 states
Very good quantitative description of data without free parameters
A = 45
V22 4522T i23
45
23
A = 47
A = 49
A = 51
51
26
49
25
47
24
Cr23 4723V24
Mn24 49
24 Cr25
51
Fe25 25
Mn26
53
27
A = 53
M.A. Bentley and SML,
Prog. Part. Nucl. Phys. 59,
497-561 (2007)
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
53
Co 26 26
Fe27
MED in T=1 states
A = 42
T i20 42
20 Ca 22
42
22
A = 46
A = 48
50
26
48
25
46
24
Cr22 4622T i24
Mn23 4823V25
Fe24 50
24 Cr26
A = 50
140
120
A = 54
100
M.A. Bentley and SML,
Prog. Part. Nucl. Phys. 59,
497-561 (2007)
Same parameterization
for the whole f7/2 shell!
54
28
Ni26 54
26 Fe28
80
60
40
20
0
-20
-40
0
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
2
4
6
TED (keV)
TED (keV)
TED (keV)
TED (keV)
TED in the f7/2shell
Only multipole effects are relevant.
The ISB term VB is of the same magnitude of the Multipole Coulomb term
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
Some questions arise…
What happens farther from stability
or at larger T in the f7/2 shell?
The same prescription applies (poster by T. Henry)
Can we understand the origin of this term?
Work in progress
Is the ISB term confined to the f7/2 shell
or is a general feature?
If so the same prescription should work!
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
Looking for a systematic ISB term
Necessary conditions for such studies:
• good and enough available data
• good shell model description of the structure
Ideal case: the sd shell
But…few data at high spin and
no indications of “J=2 anomaly” in A=18
17
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
A systematic analysis
of MED and TED
in the sd shell
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
18
The method
We apply the same method as in the f7/2 shell
However, here the three orbitals, d5/2, s1/2 and d3/2 play an important role
MEDJtheo   M VCr  ll  ls J   M VCM  J   M VB(1)  J
VCr (radial term): looks at changes in occupation of the s1/2
(1)
VBpf
( ( d 52/ 2 ) J 2 ,  ( d 32/ 2 ) J 2 )  100 keV
TEDJTheo  T VCM  J  T VB( 2 )  J
(2)
VBpf
( ( d 52/ 2 ) J 0 ,  ( d 32/ 2 ) J 0 ,  ( s12/ 2 ) J 0 )  100 keV
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
MED: different contributions
A=29
T=1/2
T=1/2
A=26
T=1
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
MED (keV)
MED in the sd shell
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
TED (keV)
TED in the sd shell
The prescription applies successfully also in the sd shell!
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
MED and TED in the
upper pf shell
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
23
The method
We apply the same method as in the f7/2 shell
However, here the three orbitals, p3/2, f5/2 and p1/2 play an important role
MEDJtheo   M VCr  ll  ls J   M VCM  J   M VB(1)  J
VCr (radial term): looks at changes in occupation of both p orbits
(1)
VBpf
( ( f 72/ 2 , p 32/ 2 , p12/ 2 , f 52/ 2 ) J 0 )  100 keV
TEDJTheo  T VCM  J  T VB( 2 )  J
(2)
VBpf
( ( f 72/ 2 , p 32/ 2 , p12/ 2 , f 52/ 2 ) J 0 )  100 keV
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
MED (keV)
MED in the upper pf shell
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
TED in the upper pf and fpg shells
TEDJTheo  T VCM  J  T VB( 2 )  J
(2)
(2)
( ( p 32/ 2 , f 52/ 2 , p12/ 2 , g 92/ 2 ) J 0 )  100 keV
VBpf
( ( f 72/ 2 , p 32/ 2 , f 52/ 2 , p12/ 2 ) J 0 )  100 keV VBfpg
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
N~Z nuclei in the A~68-84 region
Around N=Z quadrupole correlations are dominant.
Prolate and oblate shapes coexist.
The fpg space is not able to reproduce this behaviour, the fpgds space is needed.
s1/2
d5/2
g9/2
f5/2
p
quasi
SU3
40
pseudo
SU3
MED are sensitive to
shape changes and
therefore a full
calculation is needed,
which is not always
achievable with large
scale SM calculations
A.P. Zuker, A. Poves, F. Nowacki and SML, arXiv:1404.0224
Experimentally may be not clear if what we measure are energy
differences between analogue states, as ISB effects may
exchange the order of nearby states of the same J
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
Conclusions
Z
Proton-rich N~Z nuclei present several interesting
properties and phenomena that can give information
on specific terms of the nuclear interaction.
N
The investigation of MED and TED allows to have
an insight on nuclear structural properties and their
evolution as a function of angular momentum such
as: alignments, changes of deformation, particular
s.p. configurations.
The need of including an additional ISB term VB in MED and TED
shows up all along the N=Z line from the sd to the upper fp shell,
therefore revealing as a general feature.
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014
In collaboration with
Mike Bentley
Rita Lau
Andres Zuker
Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014