Masses and Structure in Exotic Nuclei

R. F. CastenWNSL, Yale

Eurorib’10, June, 2010

1 12 4 2( ; )B E

1000 4+

2+

0

400

0+

E (keV) Jπ

Structural Evolution: Simple Observables - Even-Even Nuclei

1 12 2 0( ; )B E

)2(

)4(

1

12/4

E

ER

Masses

1300 2+

0+

2+

6+. . .

8+. . .

Vibrator (H.O.)

E(I) = n ( 0 )

R4/2= 2.0

n = 0

n = 1

n = 2

Rotor

E(I) ( ħ2/2I )I(I+1)

R4/2= 3.33

Doubly magic plus 2 nucleons

R4/2< 2.0

10 20 30 40 50 60 70 80 90 100110120130140150

10

20

30

40

50

60

70

80

90

100

Pro

ton

Num

ber

Neutron Number

80.00

474.7

869.5

1264

1600

E(21+)

10 20 30 40 50 60 70 80 90 100110120130140150

10

20

30

40

50

60

70

80

90

100

Pro

ton

Num

ber

Neutron Number

1.400

1.776

2.152

2.529

2.905

3.200

R4/2

Broad perspective on structural evolution

The remarkable regularity of these patterns is one of the beauties of nuclear structural evolution and one of the challenges to nuclear theory.

Whether they persist far off stability is one of the fascinating questions for the future

R. B. Cakirli

Structural evolution – rapid structural change

Spherical-deformed trans.

Near N ~ 90

Cakirli

Often, esp. in exotic nuclei, R4/2 is not available.

E(21+) is easier to measure, works as well !!!

R4/2 across this region

84 86 88 90 92 94 961.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Ba Ce Nd Sm Gd Dy Er Yb

R4/

2

N

Better to use in

the form

1/ E(21+)

Vibrator

Rotor

!

56 58 60 62 64 66 68 701.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

N=84 N=86 N=88 N=90 N=92 N=94 N=96R

4/2

Z84 86 88 90 92 94 96

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Ba Ce Nd Sm Gd Dy Er Yb

R4/

2

N

Physics from different perspectives

Onset of deformation Onset of deformation as a phase transition

mediated by a change in shell structure

Mid-sh.

magic

“Crossing” and “Bubble” plots as indicators of phase transitional regions mediated by sub-shell changes

Cakirli and Casten, PRC 78, 041301(R) (2008)

Sn – Magic: no valence p-n interactions

Both valence protons and

neutrons

The importance of the p-n interaction

Microscopic origins of

phase transitional

behavior

Potentials involved In

Phase transitions

Valence pn interactions

Can we Measure p-n Interaction Strengths?

dVpn

Average p-n interaction between last protons and last neutrons

Double Difference of Binding Energies

Vpn (Z,N)  =  ¼ [ {B(Z,N) - B(Z, N-2)}  -  {B(Z-2, N) - B(Z-2, N-2)} ]

Ref: J.-y. Zhang and J. D. Garrett

Vpn (Z,N)  = 

 ¼ [ {B(Z,N) - B(Z, N-2)} -  {B(Z-2, N) - B(Z-2, N-2)} ]

p n p n p n p n

Int. of last two n with Z protons, N-2 neutrons and with each other

Int. of last two n with Z-2 protons, N-2 neutrons and with each other

Empirical average interaction of last two neutrons with last two protons

-- -

-

Valence p-n interaction: Can we measure it?

Cakirli based on Zhang and Garrett

Empirical interactions of the last proton with the last neutron

Vpn (Z, N) = ¼{[B(Z, N ) – B(Z, N - 2)]

- [B(Z - 2, N) – B(Z - 2, N -2)]}

dVpn has singularities for N = Z in light nuclei• Wigner energy, related to SU(4),

supermultiplet theory, spin-isospin symmetry.

• Physics is high overlaps of the last proton and neutron wave functions when they fill identical orbits.

• Expected to vanish in heavy nuclei due to: Coulomb force for protons; spin-orbit force which brings UPOs into different positions in each shell; protons and neutrons occupy different major shells.

8 10 12 14 16 18 20 22 24 26 28

1000

2000

3000

4000

Vpn (

keV

)

Neutron Number

Ne Mg Si S Ar Ca

This effect should not persist in heavy nuclei.

Does it? In a way, yes!

Rare-earth region

128 130 132 134 136 138 140

200

300

400

UThRaRnPo

Vpn

(ke

V)

Neutron Number

92 94 96 98 100 102 104 106 108 110

200

300

400

Hf

WYbErDyGd

Vpn

(ke

V)

Neutron Number

Sm

62Sm: dVpn(max) at N=94: 12 valence protons, 12 valence neutronsGd 14-14 (?) , Dy 16-16 (?), Er 18-18, Yb 20-20,

Hf, W 22,24, and 24,24

dVpn has peaks for Nval ~ Zval !!!!considering only the number of valence particles,

a possible mini-valence Wigner energy !!R. B. Cakirli, R. F. Casten and K. Blaum, to be published

Agreement is remarkable. Especially so since these DFT calculations reproduce known masses

only to ~ 1 MeV – yet the double difference embodied in dVpn allows one to focus on

sensitive aspects of the wave functions that reflect specific correlations

M. Stoitsov, R. B. Cakirli, R. F. Casten, W. Nazarewicz, and W. Satula PRL 98, 132502 (2007)

Exp.

Models

Masses, Separation energies

A “de-linearization” approach

Two-neutron separation energies

Sn

Ba

Sm Hf

Pb

5

7

9

11

13

15

17

19

21

23

25

52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132

S(2

n)

MeV

Neutron Number

Normal behavior: ~ drops after closed shells with linear segments in betweenDiscontinuities at first order phase transitions

Note that the range of values in S2n is ~ 18 MeV

Binding Energies

Many Methods to estimate

• Mass models – semi-empirical

• Microscopic calculations – many approaches including RMFT, DFT, etc

• Collective models – e.g., IBA, for the collective contribution to binding

• A new approach – pattern recognition techniques, aided by a linear subtraction method

Collective contributions to masses can vary significantly for small parameter changes in collective models, especially for well-deformed

nuclei where collective binding can be quite large.

S2n(Coll.) for alternate fits to Er with N = 100

Gd – Garcia Ramos et al, 2001

Masses: a new opportunity – complementary observable to spectroscopic data in pinning down structure. Strategies for best doing that are still being worked out. Particularly important far off stability where data will be sparse.

Sn

Ba

Sm Hf

Pb

5

7

9

11

13

15

17

19

21

23

25

52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132

S(2n

) MeV

Neutron Number

Cakirli, Casten, Winkler, Blaum, and Kowalska, PRL 102, 082501(2009)

Pattern Recognition “Physics-free” (therefore not biased) – can it tell us any physics???

Analyse the 2-D “surface” of an observable by Fourier transforms

Extrapolate to predict observables in unknown regions

How good is it? To date and longer term prospects

84 90 96 102 108 114 120 1268

10

12

14

16

18

20

22

Nd Hf

S2n

(MeV

)

Neutron Number

Pb

Try on separation energies. Test case: mask portion of data

Pattern recognition for separation energies – Not so good. Rapidly accumulate errors of a few MeV . Can we do better?

More sensitive tests of nuclear models

Remove the linear dependence

isolate and amplify collective effects

S2n-TOTAL = S2n-Linear + S2n-coll.

84 90 96 102 108 114 120 1268

10

12

14

16

18

20

22

Nd Hf

S 2n(M

eV)

Neutron Number

Pb

S2n-coll. (Z=50-82, N=82-126)

Subtract linear function, A + BN, from S2n plot

84 90 96 102 108 114 120 1260

2500

5000

S

2n-c

olle

ctiv

e (ke

V)

Neutron Number

Te Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt Hg Pb

The range of S2n is now ~ 2-3 MeV

84 90 96 102 108 114 120 1268

10

12

14

16

18

20

22

Nd Hf

S2n

(MeV

)

Neutron Number

Pb

Cakirli, Casten and Blaum, in progress

84 90 96 102 108 114 120 126

54

58

62

66

70

74

78

82

Z

N

0

1.2

2.4

3.6

4.2

S2n-collective

84 90 96 102 108 114 120 12652

56

60

64

68

72

76

80

Z

N

8.20

13.2

18.2

22.0

S2n

(MeV)

S2n-total

Greatly enhanced sensitivity of S2n-coll.

Now use pattern recognition to fit the S2n-coll values. Then add back the linear function

Work is in progress. Where will it lead? We don‘t have a clue. We will see. Work just beginning.

7000

12000

17000

22000

84 88 92 96 100 104 108 112 116 120 124

N

S2n

WEr

Pb

Nd

Very good agreement. Extrapolating ~ 12 mass units

Frank, Morales, Cakirli, Casten and Blaum, in progress

Principal Collaborators

• R. Burcu Cakirli• Klaus Blaum• Magda Kowalska• Alejandro Frank• Irving Morales

Backups

Correlations of Collective Observables

4+

2+

0+