Isospin symmetry and independence in analogue excited states

11th INTERNATIONAL SPRING SEMINAR ON NUCLEAR PHYSICS

Mirror Symmetry

Silvia M. Lenzi

Dipartimento di Fisica e Astronomia“Galileo Galilei” Università di Padova and INFN

University of Padova and INFN Silvia Lenzi – 11 th

Silvia Lenzi

Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Neutron-proton exchange symmetry

Charge symmetry : V pp = V nn p n Charge independence: (V pp + V nn )/2= V np Deviations are small Electromagnetic interactions lift the degeneracy of the analogue states, but do not generally affect the underlying symmetry.

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Analogue states in the A=22, T=1 triplet

3 2 1 0 MeV 5 4

T=1 4 + T=0 and T=1 4 + 22 12

Mg

10 2 + 0 +

0.693

4 + 1 + 3 + 22 11

Na

11 2 + 0 + T=1 22 10

Ne

12

5 4

4 +

3 2

2 +

1

0 +

0 MeV

LARGE differences in mass/binding energy mainly due to Coulomb effects SMALL differences in excitation energy Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Z

Differences in analogue excited states

Mirror Energy Differences (MED) N MED

J

 E

x J

,

T z

 

T

 E

x J

,

T z

 

T

Test the charge symmetry of the interaction Triplet Energy Differences (TED) TED

J

 E

x J

,

T z

 

T

 E

x J

,

T z

 

T

 2 E

x J

,

T z

 0 Test the charge independency of the interaction Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 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 properties .

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 – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

MED and nucleon spatial correlations

probability distribution for the relative distance of two like particles in the

f

7/2 shell

J

j

8

j

6

neutron

4

align.

2 0

j j A( N , Z ) 7 ΔE C

J 8

j

6

j

courtesy P. Van Isacker

A( Z , N )

4

j

2 0

proton align.

I=8 angular momentum j J=0 Shifts between the excitation energies of the mirror pair at the backbending indicate the type of nucleons that are aligning Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

0 3 100

MED and nucleon alignment

0

D.D. Warner, M.A. Bentley and P. Van Isacker, Nature Physics 2 (2006) 311

-50 6 -100

Energy (MeV) 51 Fe

-200 5 7 9 11 13 15

51 Mn

17 19 21 23 + Coulomb 25 27 25 21

MED

Alignment 17

2J

13 9 100 0 -100 MED (keV) Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014 5

Including monopole Coulomb effects Can we do better?

When we “normalize” to the g.s. energy, large Coulomb effects vanish, however… a small but important effect remains as a function of the angular momentum, and it is related to changes of the nuclear radius, or deformation and to single-particle effects .

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014 9

Improving the description of Coulomb effects

V C



V CM



V Cm

V CM Multipole part of the Coulomb energy: Between valence protons only

E Cr

 3 5

e

2

Z

(

Z R

 1 )

radial effect: radius changes with J

V Cm Monopole part of the Coulomb energy: L 2 term to account for shell effects

E Cll

  13 / 12 4 .

5

Z cs

[ 2

l

(

l A

1 / 3 (

N

 1 ) 

N

(

N

 3 / 2 )  3 )]

keV

electromagnetic LS term

change the single-particle energies

1

dV C E Cls

 (

g s



g l

) 1 4

m N

2

c

2

r dr

l.s

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

The radial term

The difference between the Coulomb energy of the ground states (CDE): 

E C



J

 0  

E C

 

E C

  3

n

 2

Z

 5

R C



n



e

2

T z

 

n

2 J If R C changes as a function of the angular momentum… 

E Cr

 

E C

 

E C



nC

 

R C

In f 7/2 nuclei the radial contribution can be calculated from the relative p 3/2 occupation number along the yrast band in the shell model framework Δ 

V Cr



J



na m



m p

3 / 2 

J



na m z p

3 / 2 

n p

3 / 2 2

J z and n are the number of protons and neutrons in the p 3/2 orbit relative to the g.s. (J=0) ,

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

The electromagnetic spin-orbit term

Analogous to the atomic case, the nuclear electromagnetic spin-orbit coupling has relativistic origin.

V Cls

 

g s



g l

 1 2

m N

2

c

2    1

r dV dr C

  

l

 

s



l



l

 

s

 

s

   

l



l

 1 

j



l j

 

s l



s

s ℓ f 7/2 j=l+ ½ ΔE p ~ 220 keV f 7/2 j=l+ ½ ℓ s d 3/2 j=l ½ d 3/2 j=l ½ π ν Its contribution to the MED becomes significant for configurations with a

pure

single-nucleon excitation to the f 7/2 shell: a proton excitation in one nucleus and a neutron excitation in its mirror Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Are Coulomb corrections enough?

V CM +V Cm 49 25 Mn 24  49 24 Cr 25 Exp

V CM V Cm

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

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Looking for an empirical interaction

In the single

f

7/2 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



U

( 1 )

U

( 2 ) 

V

 

V

 

V

 

V

 

V

 

V

  2

V

 Mirrors Triplet

TED J MED J

( 42 Ti ( 42 Ti  42 Ca 42 Ca ) 

U

(

f

1 7 ) / 2 ,

J



V C

( 1 )

J



V B

( 1 ) Isovector 2  of these states are pure (f 7/2 ) 2 42 Sc ) 

U

( 2 )

f

7 / 2 ,

J



V C

( 2 ) ,

J



V B

, ( 2 )

J

Isotensor Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Looking for an empirical interaction

V C

V C is calculated for every J state in the f 7/2 shell and then subtracted to MED and TED to estimate V B

MED J

( 42 Ti 42 Ca ) 

V C

( 1 ) ,

J



V B

( 1 ) ,

J

MED-V C = V (1) B J=0

81

J=2 J=4 J=6

24 6 -11 5

93

5 -48 150 100 50 0 -50 -100 0 2 Coul MED-Coul 4 6

TED-V C =V (2) B 117

81 3 -42

spin

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

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

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

The “J=2 anomaly”

Is this just a Coulomb two-body effect?

Spatial correlation probability for two nuclons in

f 7/2

1) Increase the J=2 term 2) Decrease the J=0 term We choose 1) but there is not much difference See talk by M. Bentley Angular momentum J Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Looking for an empirical interaction

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

V C MED-V C J=0

81

J=2

24

J=4

6

J=6

-11 5

93

5 -48

TED-V C 117

81 3 -42 Calculated estimate V Bf7/2 (1) estimate V Bf7/2 (2) Simple ansatz for the application to nuclei in the pf shell: ( 1 )

V Bpf

(  ( 2

f

7 / 2 )

J

 2 )   100 keV (

V Bpf

2 ) (  ( 2

f

7 / 2 )

J

 0 )   100 keV

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

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 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 (V C ) and isospin-breaking (V B ) interactions

MED

exp

J



E J

* 

E J

*

MED J theo

 

M



V Cm



J

 

M



V CM



J

 

M



V B

( 1 ) 

J TED J

exp 

E J

* 

E J

*  2

E J

* 

N TED J Theo

 

T



V CM



J

 

T



V B

( 2 ) 

J



Z

 Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Calculating the MED with SM

MED J theo

 

M



V Cm



J

 

M



V CM



J

 

M



V B

( 1 ) 

J

Theo VCM : gives information on the nucleon alignment or recoupling 49 Mn 49 Cr V CM VCm : gives information on changes in the nuclear radius Exp V Cm Important contribution from the ISB VB term : of the same order as the Coulomb contributions V B

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

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

MED in T=1/2 states

Very good quantitative description of data without free parameters

A = 45

45 23 V 22  45 22 Ti 23

A = 47

47 24 Cr 23  47 23 V 24

A = 49

49 25 Mn 24  49 24 Cr 25

A = 51

51 26 Fe 25  51 25 Mn 26

A = 53

53 27 Co 26  53 26 Fe 27 M.A. Bentley and S.M.L., Prog. Part. Nucl. Phys. 59, 497-561 (2007) Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

A = 42

42 22 Ti 20  42 20 Ca 22

MED in T=1 states

A = 46

46 24 Cr 22  46 22 Ti 24

A = 48

48 25 Mn 23  48 23 V 25

A = 50

50 26 Fe 24  50 24 Cr 26 M.A. Bentley and SML, Prog. Part. Nucl. Phys. 59, 497-561 (2007) 140 120 100 80 60 40 20 0 Same parameterization -20 -40 0 for the whole f 7/2 shell!

A = 54

2 54 28 Ni 26  54 26 Fe 28 4 Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014 6

Some illustrative examples

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014 22

Evidence of the monopole radial effect

Multipole (alignment) effects are cancelled out 48 25 Mn 23 48 23 V 25 

f

7  3

/

2 

f

7 3

/

2 

f

7

/

3 2 

f

7  3

/

2 radial term Most important contribution M.A. Bentley et al., PRL 97, 132501 (2006) The nucleus changes shape towards band termination Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

The electromagnetic spin-orbit effect: disentangling configurations

23/2 -

35 Ar

23/2 -

35 Cl

19/2 19/2 15/2 2 15/2 1 15/2 2 15/2 1 11/2 1 11/2 2 11/2 2 11/2 2 11/2 1 11/2 1

Negative parity

MED > 300 keV From the MED experimental values we can identify those states with configurations of pure proton (neutron) excitation to the f 7/2 shell.

F. Della Vedova et al., Phys.Rev. C 75, 034317 (2007) Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

T=1 A=54/42 MED: the VB term

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, 152501 (2006)

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

TED in the f

7/2

shell

Only multipole effects are relevant. The ISB term VB is of the same magnitude of the Multipole Coulomb term Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Some questions arise…

What happens farther from stability or at larger T in the f 7/2 shell?

The same prescription applies (see M. Bentley’s talk) Can we understand the origin of this term?

Work in progress with A. Zuker Is the ISB term confined to the f 7/2 or is a general feature?

shell If so the same prescription should work Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 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 28 Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

A systematic analysis of MED and TED

Silvia Lenzi – 11 th

in the sd shell

Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014 29

The method

We apply the same method as in the

f 7/2

shell However, here the three orbitals, d 5/2 , s 1/2 and d 3/2 play an important role

MED J theo

 

M



V Cr



ll



ls



J

 

M



V CM



J

 

M



V B

( 1 ) 

J

VCr (radial term): looks at changes in occupation of the s 1/2 ( 1 )

V Bpf

(  2 (

d

5 / 2 )

J

 2 ,  2 (

d

3 / 2 )

J

 2 )   100 keV

TED J Theo

 

T



V CM



J

 

T



V B

( 2 ) 

J

(

V Bpf

2 ) (  (

d

5 2 / 2 )

J

 0 ,  (

d

3 2 / 2 )

J

 0 ,  (

s

1 2 / 2 )

J

 0 )   100 keV Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

MED: different contributions

A=29 T=1/2 T=1/2 A=26 T=1 Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

MED in the sd shell

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

TED in the sd shell

The prescription applies successfully also in the sd shell!

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

What about other mass regions?

The next mass region is the upper pf and fpg shells but… not much data to perform a systematic analysis The shell model description is not that good.

The development of deformation and shape coexistence enter into play Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

MED and TED in the upper pf shell

35 Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

The method

We apply the same method as in the

f 7/2

shell However, here the three orbitals, p 3/2 , f 5/2 and p 1/2 play an important role

MED J theo

 

M



V Cr



ll



ls



J

 

M



V CM



J

 

M



V B

( 1 ) 

J

VCr (radial term): looks at changes in occupation of both p orbits ( 1 )

V Bpf

(  ( 2

f

7 / 2 ,

p

2 3 / 2 ,

p

1 / 2 2 , 2

f

5 / 2 )

J

 0 )   100 keV

TED J Theo

 

T



V CM



J

 

T



V B

( 2 ) 

J

( 2

V Bpf

) (  ( 2

f

7 / 2 ,

p

2 3 / 2 ,

p

1 / 2 2 , 2

f

5 / 2 )

J

 0 )   100 keV Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

MED in the upper pf shell

Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

TED in the upper pf and fpg shells

TED J Theo

 

T



V CM



J

 

T



V B

( 2 ) 

J

( 2

V Bpf

) (  (

f

7 2 / 2 ,

p

3 2 / 2 , 2

f

5 / 2 , 2

p

1 / 2 )

J

 0 )   100 keV

V

( 2 )

Bfpg

(  (

p

3 2 / 2 ,

f

2 5 / 2 ,

p

1 / 2 2 ,

g

2 9 / 2 )

J

 0 )   100 keV Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

N~Z nuclei in the A~68-84 region

Around N=Z quadrupole correlations are dominant. The

fpg

Prolate and oblate shapes coexist.

space is not able to reproduce this behaviour, the

fpgds

space is needed .

f s

1/2

d

5/2

g

9/2 5/2

p

40 quasi SU3 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 it is 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 – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

Conclusions

Z 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 shows up not only in the f 7/2 shell but also in other mass regions (sd, upper pf and fpg). Investigation of its origin is in progress. Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014

In collaboration with

Mike Bentley Rita Lau Andres Zuker Silvia Lenzi – 11 th Int. Spring Seminar on Nuclear Physics, Ischia, May 12-16, 2014