Carpathian Summer School of Physics 2007 Sinaia, Romania, August 20th-31st, 2007 Mirror Nuclei: symmetry breaking and nuclear dynamics Silvia M. Lenzi

Carpathian Summer School of Physics 2007Sinaia, Romania, August 20th-31st, 2007

Mirror Nuclei: symmetry breaking and nuclear

dynamics

Silvia M. LenziDipartimento di Fisica

and INFN, Padova, Italy

Silvia Lenzi, Carpathian Summer School of Physics 2007, Sinaia, Romania, 20 August 2007

Symmetries

Examination of fundamental symmetries: a key question in Physics

Symmetries help to understand Nature

conservation lawsconservation laws

good quantum numbersgood quantum numbers

In nuclear physics, conserved quantities imply underlying symmetries of the interactions and help to interpret nuclear structure features


Critical point symm. E(5), X(5) ….2000… F. Iachello

p n

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 SU(3) Dynamical Symmetry: 1958 Elliott

0J

0J 2JInteracting Boson Model (IBM dynamical symmetry): 1974 Arima and Iachello

j


Outline

What’s isospin symmetry? Why studying isospin symmetry?

How do we study it?Experimental methodsTheoretical methods

What do we learn from the data?Few illustrative recent examples


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


The nucleus: a unique quantum laboratory

Composed by two types of fermions differing only on its charge

Strong interaction: largely independent of the chargeStrong 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

21t

21zt:proton

21zt:neutron

This is in analogy to the two intrinsic spin states of an electron


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.

Nuclear interaction:Nuclear interaction:

• Charge Symmetry: Vpp=Vnn

• Charge Independence: Vpp=Vnn=Vpn

π ν

21

ZNtT

A

ii,zz

22

ZNT

ZN

Isobaric analoguemultiplets:

States with the same J,T in nuclei with the same A=N+Z


Two-nucleon system

For a two-nucleon systemfour 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


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.


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.


IMME and Coulomb Displacement Energies

A=21 Isobars

-170

-165

-160

-155

-150

-145

8 9 10 11 12 13Z-B

E(M

eV)

Isobaric Analogue States

Ground States

T=3/2 Jπ = 5/2

+

CDE

IMME: Isobaric Multiplet

Mass Equation

For a set of isobaric analogue states,the difference between the masses or BE of two neighbours defines the CDE

npz

npT,T,T,T,T,T,

MTcb

MMMzzz

12

CDE 1

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…

a: isoscalar (~100 MeV)b: isovector (~10 MeV)c: isotensor (~300 keV)

2zzz cTbTa)TT(M 2

zzz cTbTa)TT(BE


Excited analogue states

Mirror nuclei with Tz = ±1/2

132714Si14

2713 Al

Tests isospin symmetry

Normalize the ground state energiesand look at the excitedanalogue states…


T=1 isobaric triplets

T=1 states low in energy in 22Mg and 22Ne

T=0 and T=1 states in 22Na (N=Z)

The nucleus can be characterized by isospin quantum numbers which restrict the possible states in which the many-nucleon system can exist.

We expect:

Look at the isobaric triplet: 12221011

221110

2212 NeNaMg

3+

4+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

Tests isospin independence


T=3/2 Isobaric quadruplets: the spectra

Small differences in excitation energy due (mainly?) to Coulomb effects

9211210

211111

211012

219 Mg Na Ne F

T=1/2

T=3/2


Coulomb energy differences

JT,JT,JT,JJ c**zzz

Δ2E2E*ETED 011

N=ZZ

N

Mirror Energy Differences (MED)

J/T,J/T,JJ bzz

Δ*E*EMED 2121

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

(differences of excitation energies)


A classical example: MED in T=1/2 states

Coulomb effects inisobaric multiplets:

- bulk energy (100’s of MeV) - displacement energy (g.s.) CDE (10’s of MeV) - differences between excited states (10’s of keV)

)()( NZENZEMED JJJ Mirror Energy Differences

25Mn24 24Cr2549 49


Measuring the Isospin Symmetry Breaking

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

We measure nuclear structure features:

Interestingly they contain a richness of information about spin-dependent structural phenomena

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


Mirror energy differences and alignment

Shifts between the excitation energies of the mirror pair at the back-bend indicate the type of nucleons that are aligning

02

4

6

8

02

4

6

8

j

jJ=0

j

j

jjj

j

J=6

ΔEC

probability distribution for the relative distance of two like particles in the f7/2 shelljj

J

MED

angular momentumI=8

0

A(N,Z) A(Z,N)

proton align.

neutron align.


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

MED are a probe of nuclear structure:reflect the way the nucleus generates its angular momentum

Alignment

jjj

j

J=6

Coulomb energy differences: Experimental methods


Studying the f7/2 shell

41Ca40Ca

43Sc42Sc41Sc

45Ti44Ti43Ti42Ti

47V46V45V44V

49Cr48Cr47Cr46Cr

51Mn50Mn49Mn48Mn

53Fe52Fe51Fe50Fe

55Co54Co53Co52Co

56Ni55Ni54Ni

44Ca43Ca42Ca

45Sc44Sc

47Ti46Ti

49V48V

51Cr50Cr

54Fe

53Mn52Mn

N=Z

20

21

22

23

24

25

26

27

28

20 21 22 23 24 25 26 27 28

proton number

neutron number

f7/2

f5/2

d5/2

d3/2s1/2

p3/2

p1/2

28

20

The 1f7/2 shell is isolated in energy from other major orbits

Wave functions dominated by (1f7/2)n configurations

High-spin states experimentally reachable

Experimental issue : proton-rich Tz< 0 isobars very weakly populated


Experimental requirements

High efficiency for γ detection

Low cross section at high spin (small masses)

High energy transitions

Many channels opened: high efficient charged-particle det.

High selectivity: particle detectors

Kinematics reconstruction for Doppler broadening

Neutron detectors to select proton rich channels

Polarimeters and granularity (J, π, δ)

Mass spectrometers


Gamma spectroscopy

γ1

γ2

γ3

Ge crystal

Anti-Comptongamma ray

Constructing a level scheme Gamma array

156Dy


Gamma-ray spectrometers

Next futurePresent

40 — 20 % ( M=1 — M=30)

10 — 5 % ( M=1 — M=30)

GAMMASPHEREGASP GRETAAGATA


Techniques for proton-rich spectroscopy

1. High efficiency & high granularity gamma-ray spectrometer (e.g. Euroball, Gammasphere) - high fold n (n 3) coincidence spectroscopy

3. Identify cleanly all emitted particles from reaction - needs a charged-particle detector (p, ) + high-efficiency & high granularity neutron detector array

2. Gamma-ray array + 0o recoil mass spectrometer + focal plane detectors - identify A,Z of recoiling nucleus tag emitted gamma-rays

Three basic techniques for selecting proton-rich systems


1. High fold n (n > 3) coincidence spectroscopy

S.J. Williams et al., Phys. Rev. C 68 (2003) 011301

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

CoMg,1p2nS 532432

Double-coincidence spectra after gating on 2 analogue transitions

ms 260


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%


Z identification

E

DE

Ionisation ChamberFe

Mn

Cr

V

Ti

Example: *25

50255

10520

4020 MnBCa

n

p

2Mn

2V

234825

254823

No excited states known in 48Mn

448

48

10V

Mn

~

Z identification essential…


γ-γ coincidence analysis

(A/q = 3, Z=25)-gated

coincidence analysis…

234825 Mn 25

4823V

M.A. Bentley et al., Phys.Rev. Lett. 97, 132501 (2006)


3. Measuring the evaporated particles

With this method we do not measure directly the final residue but the particles emitted from the compound nucleus

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

We need high efficiency detectors for:

charged particlesneutrons


Observation of excited states in 50Fe

50Fe sum of gates (*)

Cr50

24 26E (keV)

cou

nts

σ(Fe)/ σ(Cr) ≈ 10-4

Fe50

26 24

50Cr sum of gates (*)

S. M. Lenzi et al., Phys. Rev. Lett. 87, 122501 (2001)

Coulomb energy differences: Theoretical methods


Cranked shell model and alignment

CSM: good qualitative description of the data

Approximations:• one shell only• fixed deformation• no p-n pairing

alignment

Cranked shell-model

J.A. Sheikh, P. Van Isacker, D.D. Warner and J.A. Cameron, Phys. Lett. B 252 (1990) 314


Ingredients for the Shell Model calculations

1) an inert core2) a valence space3) an effective interaction that mocks up the general hamiltonian in the restricted basis

s1/2

p1/2p3/2

d3/2

d5/2

f7/2

s1/2

f5/2

p3/2

p1/2

8

20

28

2

N or Z

the valence space

inert core

The choice is determined by the limits in computing time and memory: large dimension of the matrices to be diagonalised.

Current programs diagonalise matricesof dimension ~109


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 2007-2

00

-150

-100-5

00

50

100

57

911

1315

1719

2123

2527

Exp

erim

ent

fp-s

hell

Mod

el

-200

-150

-100

-50

0

50

100

5 7 9 11 13 15 17 19 21 23 25 27

Experiment

fp-shell Model

Experiment

Shell Model

100 0 -100

25

21

17

13

9

5

MED

MED (keV)

2J

51Fe 51MnEnergy (MeV)6

3

0

Alignment in A=51D.D. Warner et al., Nature Physics 2 (2006) 311

M.A. Bentley et al, PRC 62 (2000) 051303

J.Ekman et al, EPJ A9 (2000) 13

Alignment

100 keV


Improving the description of Coulomb effects

Can we do better? What is missing?

Monopole term of the Coulomb energy Vcm

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

A.P. Zuker

CmCMC VVV

Multipole term of the Coulomb energy VCM:

Between valence protons only

interaction with the core


The radial term

CC R

e)Z(ZE

5

13 2Coulomb energy of a charged sphere:

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

C

CCC R

enZnZEZEJE

5

230

2

If RC changes as a function of the angular momentum…

2

2

22

2

25

3

02

5

3

0

112

5

30

C

C

C

CC

CCCCCr

R

JRenZn

R

JRRenZn

RJRenZnEJEJE

Radial contribution to the MED

2

nTz

J


254823V23

4825 Mn

Evidencing the monopole radial effect

Multipole (alignment) effects are cancelled out

radial term

Most important contribution

The nucleus changes shape towards band termination

327

327 // ff 3

273

27 // ff

M.A. Bentley et al., PRL 97, 132501 (2006)

VCr


Electromagnetic single-particle effects: the ℓ·s term

slR

Ze

cmggE

NlsCls

2

222

1

50 times smaller than the nuclear spin-orbit term!!!

d3/2

j=l-½

d3/2

j=l-½

f7/2

j=l+½

f7/2

j=l+½ΔEp ~ 220 keV

ℓ

ℓ

s

s

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

Acts differently on protons and neutrons


Negative parity

The two 11/2 states have very different configurations

The T=1/2 mirror nuclei A=35

35Ar 35Cl

11/2-2

15/2-1 15/2-

2

19/2-

23/2-

11/2-2

15/2-1

15/2-2

19/2-

Measured very large MED values for all high spin states!

13/2-13/2-

F. Della Vedova et al., PRC 75, 034317 (2007)

Large MED for the 13/2- state is found in A=35

>300 keV!!!

J. Ekman et al., PRL 92 (2004) 132502

13/2-


MED in the sd shell: the EMSO

The mirror pair35Ar – 35Cl

The mirror pair39Ca – 39K

Exp. data: Th. Andersson et al., EPJA 6, 5 (1999)

Calculations in the sdfp space

Large and similar contributionsfrom the multipole Coulomb and the electromagneticspin-orbit terms

Small effects due to the orbital term

Puzzling results…deformation effects?

F. Della Vedova et al., PRC 75, 034317 (2007)


254823V23

4825 Mn

EMSO effect in the fp and f7/2 shell

The mirror pair 61Ga – 61Zn

Data: L-L. Andersson et al., PRC (2005)

25 /f

23 /p

306131Ga

VCM

VCM+lsVCM+ls+ll

VCM+ls+ll+VCr

Exp

27 /f

234825 Mn

23 /d

The mirror pair 48Mn – 48V

M.A. Bentley et al., PRL 97, 132501 (2006)


Are Coulomb corrections enough?

Another term of nuclear nature is needed, but it has to be big!

sum of Coulombterms25

492424

4925 CrMn


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

If the energy differences are due only to VC one expects very small numbers for all J couplings for VB

A. P. Zuker et al., Phys. Rev. Lett. 89, 142502 (2002)

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

ππ πν νν

42Ti 42Sc 42Ca


Learning from MED and TED in A=42

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=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 VBf7/2 (1)

estimate VBf7/2 (2)

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

)( 2/7.. fVV oh

CC Calculated

Simple ansatzfor the application tonuclei in the pf shell:

)2( keV100)1( JVBpf

A. P. Zuker et al., Phys. Rev. Lett. 89, 142502 (2002)

)0( keV100)2( JVBpf

J=2 anomaly


Evidence of ISB of the nuclear interaction

The multipole Coulomb contribution gives information on the nucleon alignment

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!!!!!

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

A=49MED (keV)

Very good quantitative agreement between theory and data

Now, without changing the parametrization, see how the rest of the MED for nuclei along the f7/2 shell are described by the calculations…

Coulomb energy differences (CED):

Results


MED and TED in the shell model framework

A = 47 24472323

4724 VCr

A = 4925

492424

4925 CrMn

A = 53 27532626

5327 FeCo

A = 48 25482323

4825 VMn

-40

-20

0

20

40

60

80

100

120

140

0 2 4 6

-40

-20

0

20

40

60

80

100

120

140

0 2 4 6

A = 5428

542626

5428 FeNi

A = 51 26512525

5126 MnFe

Good quantitative description of datawithout free parameters

M.A. Bentley and S.M. Lenzi,Prog. Part. Nucl. Phys. (2006) in press


Rotational T=1/2 analogue states A=47/49

7 particles/holes in the f7/2 shell

Monopole effects: Cr

Multipole effects: CM and VB

All terms contribute significantly to the MED

821

23

25

27

215

23

25

27

2

31maxJ

Band termination state:

Deformed nuclei Rotational bands


T=1 A=54/42 triplet: MED and TED

A.Gadea et al., PRL 97, 152501 (2006)

no collectivity: only multipole effects: smooth recoupling and J=2 anomaly

2 particles / holes

A=54 A=42

A=54 A=42J=2 anomaly


More results of MED measurements

P.E. Garrett et al., PRC 75, 014307 (2007)

Rising and GANIL: T=2 mirror in A=36P. 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


Summary and outlookThese studies allow to learn about:

Mechanism of nucleon alignment at the backbending

Evolution of the radii along a rotational band

Evidence of isospin-non-conserving terms in the nuclear interaction

Importance of the single-particle effects: - test interactions and basis - information on the configurations

25 /f

23 /p

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


Thanks to…

M. A. Bentley (York)N. Marginean, A. Gadea, F. Della Vedova, (LNL and Padova)J. Ekman, D. Rudolph (Lund)P.E. Garrett (Guelph)

Experiments:

Theory:

A.P. Zuker, F. Nowacki (Strasbourg)


Monopole Coulomb single-particle effects:

2) the ℓ2 (orbital) termJ. Duflo and A.P. Zuker, Phys. Rev. C 66 (2002) 051304(R)

keV

31254

2331

1213

NA

NNZ.E

/

/cs

Cll

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:

3

20

N

Zcs

f7/2

p3/2150 keV

Eg. in the fp shell:

proton s.p. relative energy is increasedby 150 keV


Alignment and shell model

066

JJaaaa A

Define the operator

Calculate the difference of the expectation value in both mirror as a function of the angular momentum

JJJJJ, 'Z'Z AAAΔ

51Fe-51Mn

M.A.Bentley et al. Phys Rev. C62 (2000) 051303

“Counts” the number of protons coupled to J=6

In 51Fe (51Mn) a pair of protons (neutrons)align first and at higher frequency align the neutrons (protons)

1/23/25/2

7/2

p n p n

51Fe 51Mn

Documents

Carpathian Summer School of Physics 2007 Sinaia, Romania, August 20th-31st, 2007 Mirror Nuclei: symmetry breaking and nuclear dynamics Silvia M. Lenzi