Valence Proton-Neutron Interaction

Development of configuration mixing, collectivity and deformation – competition

with pairing

Changes in single particle energies and magic numbers

Partial history: Goldhaber and de Shalit (1953); Talmi (1962); Federman and Pittel ( late 1970’s); Casten et al (1981); Heyde et al (1980’s); Nazarewicz, Dobacewski et al (1980’s); Otsuka et al( 2000’s); Cakirli et al (2000’s); and many others.

Measurements of p-n Interaction Strengths

Vpn

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?

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)]}

Proton-neutron Interaction Strengths

<p|V|n> = 1/ [ n + l + 1 ]

Orbit dependence of p-n interactions – simple ideas

82

50 82

126

High j, low n

Low j, high n

82

50 82

126

Z 82 , N < 126

11

Z 82 , N < 126

1 2

Z > 82 , N < 126

2

3

3

Z > 82 , N > 126

208Hg208Hg

Can we extend these ideas beyond magic regions?

Away from closed shells, these simple arguments are too crude. But some general predictions can be made

p-n interaction is short range similar orbits give largest p-n interaction

HIGH j, LOW n

LOW j, HIGH n

50

82

82

126

Largest p-n interactions if proton and neutron shells have similar fractional filling

Empirical p-n interaction strengths indeed strongest along diagonal.

82

50 82

126

High j, low n

Low j, high n

New mass data on Xe isotopes at ISOLTRAP – ISOLDE CERNNeidherr et al, preliminary

Empirical p-n interaction strengths stronger in like regions than unlike regions.

Direct correlation of observed growth rates of collectivity with empirical p-n interaction strengths

p-n interactions and the evolution

of structure

W. Nazarewicz, M. Stoitsov, W. Satula

Microscopic Density Functional Calculations with Skyrme forces and

different treatments of pairing

Realistic Calculations

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

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

sensitive aspects of the wave functions that reflect specific correlations

Sensitivity of the mass filter:DFT with different pairing interactions

The new Xe mass measurements at ISOLDE give a new test of the DFT

84

92

96

102

108

116

126

54

58

62

66

70

74

78

82

Z

N 84 96 114 126

54

58

62

66

70

74

78

82

Z

N

Experiment DFT

250 < Vpn < 300

r 350

b250

300 350 bVpn <

hp

hh

pp

SKPDMIX

Vpn (DFT – Two interactions)

84

92

96

102

108

116

126

54

58

62

66

70

74

78

82

Z

N 84 96 114 126

54

58

62

66

70

74

78

82

Z

N

Experiment DFT

250 < Vpn < 300

r 350

b250

300 350 bVpn <

hp

hh

pp

SLY4MIX

55 60 65 70 75 80

200

300

400

500

600

55 60 65 70 75 80

200

300

400

500

600

Vp

n /

keV

55 60 65 70 75 80

Pd

CdSn

TeXe

55 60 65 70 75 80

200

300

400

500

600

Neutron Number N

55 60 65 70 75 80

Vpn

Exp

Vpn

DFT

Vpn

Overlap

Martin’s model (Sn, Te, Cd, Xe, Pd)

Vpn =

A 1/ [ nik + lik + 1 ]} Vi2 Vk

2

Sum is over all pairs of the i-th proton and k-th neutron orbits near the Fermi surface, as seen in

the neighboring odd N and Z nuclei (in practice, levels up to

~ 300 keV). Parameters are A, B and the pairing gap parameter -

chosen once for all the nuclei. Interestingly, the p-n interaction

strength also depends on pairing! Might expect this simple, no-

collective-correlations, model to work for Sn but not much else

(because R4/2 >2 for Te, Cd, Xe, Pd).

Two regions of parabolic

anomalies.

Two regions of octupole

correlations.

Possible signature?

One last thing

Neidherr et al, preliminary

Summary of p-n interactions and Vpn

• p-n interactions and the evolution of collectivity

• Mass filters – Vpn – available far from stability

– Crossing pattern around doubly magic nuclei -- orbit overlaps– Maximum for similar shell filling – along diagonal in N-Z plot– Maximum in like quadrants (p-p or h-h vs p-h or h-p)– Correlation with growth rates of collectivity– Comparison with DFT calculations– Possible signature of octupole correlations

How to know what the nuclei are telling us

(using simple observables)

• R4/2, but you all know that already, right?

• Note: nuclear shapes have TWO variables so generally need TWO observables.

• How shapes change with N and Z: P-factor• Bubbles and shell structure (already discussed)• How shapes change with N and Z: AHV model• How shapes change with spin – E-GOS• IBA (Its really simple, don’t be scared)

How does R4/2 vary and why do we care

We care because it is the only observable whose value

immediately tells us something (not everything !!!)

about the structure.

We care because it is easy to measure, especially far

from stability

Other observables, like E(2+) and masses, are

measurable even further from stability. They can give

valuable information in the context of regional behavior.

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

Guide to R42 values in Collective Nuclei (R42 < 2 near closed shells)

Sph.

Def.

R42~2.2

R42~2.9

Axial Sym.

Axi

al A

sym

.

E(02 ) > E(2

)

E(02) < E(2)

Broad perspective on structural evolution

Quantum phase transitions as a function of proton and neutron number

86 88 90 92 94 96 98 100

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Nd Sm Gd Dy

R4/

2

N

Vibrator

RotorTransitional

Competition between pairing and the p-n interactions

A simple microscopic guide to the evolution of structure

(The next slides allow you to estimate the structure of any nucleus by multiplying and dividing two numbers

each less than 30)(or, if you prefer, you can get the same result from 10 hours of

supercomputer time)

NpNn p – n

PNp + Nn pairing

p-n / pairing

P ~ 5

Pairing int. ~ 1 MeV, p-n ~ 200 keV

P~5

p-n interactions perpairing interaction

Hence takes ~ 5 p-n int. to compete with one pairing int.

Comparing with the data

Comparison with the data

Simple ways to see underlying shell structure

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

A~100

52 54 56 58 60 62 64 66

1,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

3,2

Z=36 Z=38 Z=40 Z=42 Z=44 Z=46

R4/

2

Neutron Number

36 38 40 42 44 46

1,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

3,2

N=52 N=54 N=56 N=58 N=60 N=62 N=64 N=66

R4/

2

Proton Number

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

E(21+) [as 1/ E(21

+) ], is easier to measure, works as well !!!

Two very useful techniques:

Anharmonic Vibrator model (AHV)(works even for nuclei very far from vibrator)

E-GOS Plots(Paddy Regan, inventor)

Anharmonic Vibrator

Very general tool, BOTH for yrast states and structural evolution

R4/2 not exactly 2.0 in vibrational nuclei. Typ. ~2.2–ish. Write: E(4) = 2 E(2) + where is a phonon - phonon interaction.

How many interactions in the 3 – phonon 6+ state? Answer: 3How many in the n – phonon J = 2n state? Answer: n(n-1)/2

Hence we can write, in general, for the any “band”:

E(J) = n E(2) + [n(n – 1)/2] n = J/2

General. E.g., pure rotor, E(4) = 3.33E(2), hence = 1.33 E(2)

/E(2) varies from 0 for harmonic vibrator to 1.33 for rotor !!!

E-GOS PlotsEGamma Over Spin

Plot gamma ray energies going up a “band” divided by the spin of the initial state

E-Gos plots are, for spin, what AHV plots are for N or Z

Evolution of structure with J, N, or Z

E-GOS plots in limiting cases

Vibrator: E(I) = h

Hence E (I) = h

Rotor: E(I) = h2/2J [I (I + 1)]

Hence E(I) = h2/2J [4I - 2]

AHV (R4/2 = 3.0): E(4) =

2E(2) Hence E(I) = 2(I + 2)

R = 2 = constant

R4/2 = 3.0 (AHV)

Comparisons of AHV and E-GOS with data

J = 2 4 6 8 10 12 14

98-Ru 652 1397 2222 3126 4001 4912 5819

Egamma 652 745 825 904 875 911 907

AHV 652 1397 2235 3166 4190 5307 6517 ( /E(2) = 0.14

E-GOS 326 189 137 113 87.5 76 64.8

----------------------------------------------------------------------------------------------------------------166-Dy 77 253 527 892 1341 1868 2467

Egamma 77 176 274 365 449 527 599

AHV 77 253 528 902 1375 1947 2618( /E(2) = 1.29

E-GOS 38 44 46 46 45 44 42.8

102 Ru

R4/2 = 3.0 (AHV)

Vibr rotor

0+2+4+

6+

8+

Rotational states

Vibrational excitations

Rotational states built on (superposed on)

vibrational modes

Why do the energies of rotational states differ from the rotor model??

[You can see this by comparing with the rotor formula E ~ J ( J + 1). You can also see it deviations of E-GOS plots from the rotor

expression, or with the AHV.]

Look at the rotor formula:

E(J) = [h2 /2I] x [ J ( J = 1)] where I is the moment of inertia.

I is linear in BETA, the quadrupole deformation. There is a centrifugal force. If the potential in beta is SOFT, the nucleus can stretch, and beta will increase, so I will increase, so the energies will Decrease relative to the rotor expression with fixed I.

Phase transitional critical point

nucleus

What to do with very sparse data???

Example: 190 W. How do we figure out its structure?

W-Isotopes

N = 116

IBA Model, flexible 2- parameter model for a wide variety of structures. Three benchmark symmetries:

Symmetry triangle (see lectures last year).

= 2.9R4/2

R42 not enough. Need another observable: the second 2+

Backups

Neutron rich Kr isotopes (Z = 36)

E N 2 4 R42 P factor Structure 86 50 1565 2250 1.44 0 Magic88 52 775 1644 2.12 1.33 Vib ???90 54 707 ---- ----92 56 769 1804 2.34 2.40 Vib ???94 58 665 1519?? 2.28 2.67 Vib ???

Sr (Z = 38)

88 50 1836 --- Doubly Magic?90 52 831 1655 1.9992 54 814 1673 2.0694 56 836 2146 2.57 !!! Why 4+ higher?96 58 814 1792 2.2098 60 144 434 3.01 !!! Deformed

BEWARE OF FALSE BEWARE OF FALSE CORRELATIONS!CORRELATIONS!

BEWARE OF FALSE BEWARE OF FALSE CORRELATIONS!CORRELATIONS!

Proton-neutron Interaction Strengths

Simple estimate of the p-n matrix element

<p|V|n> = 1/ [ n + l + 1 ]

The IBA – a flexible, parameter-efficient, collective model

• Enormous truncation of the Shell Model: valence nucleons in pairs coupled to L = 0 (s bosons) and L = 2 (d bosons), simple interactions • Three dynamical symmetries, intermediate structures, symmetry triangle

• Two parameters (except scale)

)2(

)2()0(

1

22

E

EE)2(

)4(

1

1

E

EVibrator Rotor

γ - soft

Mapping Structure with Simple Observables – Technique of Orthogonal Crossing Contours

Burcu Cakirli et al.Beta decay exp. + IBA calcs.

SU(3)U(5)

O(6)

3.3

3.1

2.92.7

2.5

2.2

-3.0

-1.0-2.0

-0.1

+0.1

+1.0

+2.0

+2.9

U(5) SU(3)

O(6)

R4/2

)2(

)2()0(

1

2

E

EE

= 2.3 = 0.0

156Er

119 123 127 131 135 139100

200

300

400

500

84Po

90U

86Rn

88Ra

Vpn

(Zev

en,N

odd)

(keV

)

75 77 79 81 83 85 87 89 91

200

300

400

500

54Xe

52Te

56Ba

58Ce

Vpn

(Zev

en,N

odd)(

keV

)

74 76 78 80 82 84 86 88 90 92200

300

400

500

Neutron Number

54Xe

52Te

56Ba

58Ce

Vpn

(Zev

en,N

even

)(ke

V)

What happens as the numbers of valence neutrons and protons increase?

Case of few valence nucleons:Lowering of energies, development of multiplets.

R4/2 ~2-2.4

This evolution is the emergence

of collective behavior

Lots of valence nucleons of both types:emergence of deformation and therefore rotation (nuclei

live in the world, not in their own solipsistic enclaves)

R4/2 ~3.33

0+2+4+

6+

8+

Rotor

E(I) ( ħ2/2I )I(I+1)

R4/2= 3.33

Deformed nuclei – rotational spectra

Think about the striking regularities in these data.

Take a nucleus with A ~100-200. The summed volume of all the nucleons is ~ 60 % the volume of the nucleus, and they

orbit the nucleus ~ 1021 times per second!

Instead of utter chaos, the result is very regular behavior, reflecting ordered, coherent, motions of these nucleons.

This should astonish you.

How can this happen??!!!!

Much of understanding nuclei is understanding the relation between nucleonic motions and collective behavior

6+ 690

4+ 330

0+ 0

2+ 100

J E (keV)

?Without

rotor

paradigm

Paradigm

Benchmark

700

333

100

0

Rotor J(J + 1)

Amplifies structural

differences

Centrifugal stretching

Deviations

Identify additional

degrees of freedom

Value of paradigms

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: ~ linear segments with drops after closed shellsDiscontinuities at first order phase transitionsS2n = A + BN + S2n (Coll.) Curvature in isotopic chains – collective effects: deviations from linearity are a few hundred keV

Use any collective model to calculate the collective contributions to S2n.

Binding Energies