Mirror Symmetry

Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014

Mirror Symmetry

Silvia LenziUniversity of Padova and

INFN

Silvia M. LenziDipartimento di Fisica e Astronomia“Galileo Galilei”

Università di Padova and INFN

Isospin breaking in Coulomb energy

differences


Neutron-proton exchange symmetry

Charge symmetry : Vpp = Vnn

Charge independence: (Vpp + Vnn)/2= Vnp

Deviations are small

3+1+0+

4+

2+

0+

2+

MeV

0

1

2

3

4

5

0.693

4+

0+

4+

2+

0

1

2

3

4

5MeV

102212Mg

112211Na 12

2210Ne

T=1 T=1T=0 and T=1


TTJTTJJ zzxx ,, EEMED

Differences in analogue excited states

0,,, E2EETED zzz TJTTJTTJJ xxx

N=ZZ

N

Mirror Energy Differences (MED)

Test the charge symmetry of the interaction

Triplet Energy Differences (TED)

Test the charge independency of the interaction


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 properties.


Measuring MED and TED

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

We measure nuclear structure features:

They contain a richness of information about spin-dependent structural phenomena

Can we reproduce such small energy differences?What can we learn from them?


Coulomb effects

VCm Monopole Coulomb energy

R

ZZeECr

)1(

5

3 2

keVNA

NNllZE cs

Cll )2/3(

)]3()1(2[5.43/1

12/13

l.s

dr

dV

rcmggE C

NlsCls

1

4

1)(

22

radial effect: radius changes with J

change the single-particle

energies

CmCMC VVV

VCM Multipole Coulomb energy:

Between valence protons only

L2 term to account for shell effects

electromagnetic LS term


Are Coulomb corrections enough?

Another isospin symmetry breaking (ISB) term is needed and it has to be big!

VCM+VCm

Exp

25492424

4925 CrMn

VCM

VCm


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 :

VVV ,,

We can recast them in terms of isoscalar, isovector and isotensor contributions

VVVU

VVU

VVVU

2)2(

)1(

)0(

)1(,

)1(,

)1(,

4242

2/7)CaTi-( JBJCJfJ VVUMED Mirrors Isovector

)2(,

)2(,

)2(,

424242

2/7)Sc2-CaTi( JBJCJfJ VVUTED Triplet Isotensor

We assume that the configurations of these states are pure (f7/2)2

ππ πν νν


Looking for an empirical interaction

A. P. Zuker et al., PRL 89, 142502 (2002)

From the yrast spectra of the T=1 triplet 42Ti, 42Sc, 42Ca we deduce the interaction

J=0 J=2 J=4 J=6

VC

81 24 6 -11

MED-VC 5 93 5 -48

TED-VC 117 81 3 -42

estimate VB (1)

estimate VB (2)

Calculated

Simple ansatz for the application tonuclei in the pf shell:

keV100))(( 22)1(

2/7JBpf fV keV100))(( 0

2)2(

2/7JBpf fV

J=2 anomally


The “J=2 anomaly”

Calculation (using Harmonic Oscillator w.f)

Spatial correlation probability for two nucleons in f7/2

Co

ulo

mb

ma

trix

ele

me

nts

(M

eV)

Angular momentum J

Is this just a Coulomb two-body effect?

Two possibilities:1) Increase the J=2 term 2) Decrease the J=0 termWe choose 1) but there is not much difference


Calculating MED and TED

JBMJCMMJCmMtheoJ VVVMED )1(

JBTJCMTtheoJ VVTED )2(

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

ZEZEMED JJJ**exp

ZNEZEZETED JJJJ ***exp 2


Calculating the MED with SM

VCM: gives information on the nucleon alignment or recoupling

VCm: gives information on changes in the nuclear radius

Important contribution from the ISB VB term:

of the same order as the Coulomb contributions

VCM

VCm

Exp

VB

Theo49Mn-49Cr

JBMJCMMJCmMtheoJ VVVMED )1(

A. P. Zuker et al., PRL 89, 142502 (2002)

M.A. Bentley and SML, Prog. Part. Nucl. Phys. 59,

497-561 (2007)


MED in T=1/2 states

A = 45

A = 53

A = 51

A = 49

A = 4723452222

4523 TiV 24

472323

4724 VCr

25492424

4925 CrMn

26512525

5126 MnFe

27532626

5327 FeCo

Very good quantitative description of data without free parameters


497-561 (2007)


-40

-20

0

20

40

60

80

100

120

140

0 2 4 6

MED in T=1 states

A = 48

A = 50

A = 46

A = 54

A = 42 22422020

4222 CaTi 24

462222

4624 TiCr

25482323

4825 VMn

26502424

5026 CrFe

28542626

5428 FeNi

Same parameterizationfor the whole f7/2 shell!


497-561 (2007)


TED in the f7/2shell

Only multipole effects are relevant. The ISB term VB is of the same magnitude of the Multipole Coulomb term

TE

D (

keV

)T

ED

(ke

V)

TE

D (

keV

)T

ED

(ke

V)


Some questions arise…

Is the ISB term confined to the f7/2 shellor is a general feature?

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

If so the same prescription should work!


17

Necessary conditions for such studies:• good and enough available data

• good shell model description of the structure

Ideal case: the sd shellBut…few data at high spin and

no indications of “J=2 anomaly” in A=18

Looking for a systematic ISB term

Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014 18

A systematic analysis

of MED and TED in the sd shell


The method

JBMJCMMJCrMtheoJ VVlsllVMED )1(

JBTJCMTTheoJ VVTED )2(

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

VCr (radial term): looks at changes in occupation of the s1/2

keV100))(,)(( 22

2/322

2/5)1( JJBpf ddV

keV100))(,)(,)(( 02

2/102

2/302

2/5)2( JJJBpf sddV


MED: different contributions

T=1/2

A=29

T=1/2A=26

T=1


MED in the sd shellM

ED

(ke

V)


TED in the sd shell

The prescription applies successfully also in the sd shell!

TE

D (

keV

)

Silvia Lenzi – ARIS 2014, Tokyo, June 2-6, 2014 23

MED and TED in the upper pf shell


The method

JBMJCMMJCrMtheoJ VVlsllVMED )1(


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

VCr (radial term): looks at changes in occupation of both p orbits

keV100)),,,(( 02222)1(

2/52/12/32/7JBpf fppfV

keV100)),,,(( 02222)2(

2/52/12/32/7JBpf fppfV


MED in the upper pf shellM

ED

(ke

V)


TED in the upper pf and fpg shells


keV100)),(( 022

,2

,2)2(

2/12/52/32/7JBpf pfpfV keV100)),,,(( 0

2222)2(

2/92/12/52/3JBfpg gpfpV


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.

A.P. Zuker, A. Poves, F. Nowacki and SML, arXiv:1404.0224

MED are sensitive to shape changes and

therefore a full calculation is needed,which is not always

achievable with large scale SM calculations

s1/2

d5/2

g9/2 40

quasiSU3

pseudoSU3

f5/2

p

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


Proton-rich N~Z nuclei present several interesting properties and phenomena that can give information on specific terms of the nuclear interaction.

Conclusions

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.

N=ZZ

N

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.


In collaboration withMike Bentley

Rita LauAndres Zuker

Documents

Mirror Symmetry