Halo NucleiHalo Nuclei

S. N. ErshovS. N. Ershov

JointJoint Institute for Nuclear ResearchInstitute for Nuclear Research

new structuralnew structural driplinedripline phenomenon with clusterization phenomenon with clusterization into an ordinary core nucleus and a veil of halo nucleons into an ordinary core nucleus and a veil of halo nucleons

– – forming very dilute neutron matterforming very dilute neutron matterHALO:HALO:

Life time for radioactive

nuclei > 10-12 sec

Chains of the lightest isotopes ( He, Li, Be, B, …) end up with two neutron halo nuclei

Two neutron halo nuclei ( 6He , 11Li , 14Be , … ) break into three fragments and are all Borromean nuclei

One neutron halo nuclei ( 11Be , 19C , … ) break into two fragments

decays define the life timeof the most radioactive nuclei

Nuclear scale ~ 10-22 sec

Width 1 ev ~ 10-16 sec

weakly bound systemswith large extensionand space granularity

“Residence in forbidden regions”Appreciable probability for dilute nuclear matter extending far out into classically forbidden region

( 6He, 11Li, 11Be, 14Be, 17B, … )

11Li = 0. 3 MeV

11Be= 0. 5 MeV

6He= 0. 97 MeV

none of the constituent two-bodysubsystems are bound

Borromean systemsTwo-neutron halo nuclei( 11Li, 6He, 14Be, 17B, . . . )

Borromean system is bound

2.3 fm

~ 5 fm

~ 7 fmn n11Li

9Li

Separation energies of last neutron (s) :

halo stable < 1 6 - 8 MeV

Large size of halo nuclei< r2 (11Li) >1/2 ~ 3.5 fm

( r.m.s. for A ~ 48 )

-400 -200 0 200 4000

400

800

1200E / A = 800 MeV

C(11Li,9Li)

s = 21 MeV/c

Co

un

ts

transverse momementum ( MeV / c )

Peculiarities of halo nuclei: the example of 11Li (i) weakly bound: the two-neutron separation energy (~300 KeV)

is about 10 times less than the energy of the first excited state in 9Li .

(ii) large size: interaction cross section of 11Li is about 30% larger than for 9Li

6 8 10 12 14 16 182.0

2.5

3.0

3.5

4.0

He

LiBe B

1.18A1/3RI (

fm

)

A

E / A = 790 MeV, light targets

Interaction radii : 2

I I IR (proj) R (t arg)s

I. Tanihata et al., Phys. Rev. Lett., 55 (1985) 2676

This is very unusual for strongly interacting systems held together by short-range interactions

(iii) very narrow momentum distributions, compared to stable nuclei, of both neutrons and 9Li measured in high energy

fragmentation reactions of 11Li .

( naive picture )narrow momentum distributions

large spatial extensions

No narrow fragment distributions in breakup on other fragments, say 8Li or 8He

A + 12C sI s -2n s -4n

9Li 796 6 11Li 106010 22040 4He 503 5 6He 722 5 18914 8He 817 6 2021795 5

(iv) Relations between interaction and neutron removal cross sections ( mb ) at 790 MeV/A

sI (A=C+xn) = sI (C) + s-xn

Strong evidence for the well definedclusterization into

the core and two neutrons

Tanihata I. et al. PRL, 55 (1987) 2670; PL, B289 (1992) 263

0.1 1 10

1

10

100

12Be

14Be11Be

8He6He11Li

9Li

12C

S-2nEMD / Zp

2

S-nEMD / Zp

2

Zp-

no

rmalize

d E

MD

cro

ss s

ecti

on

(m

b)

Separation energy (MeV)

(v) Electromagnetic dissociation cross sections per unit charge are orders of magnitude larger than for stable nuclei

T. Kobayashi, Proc. 1st Int. Conf. OnRadiactive Nuclear Beams, 1990.

stab

le

halo

Evidence for a rather large difference between charge and mass centers in a body fixed frame

concentration of the dipole strength at low excitation energies

(peculiarities of low energy halo continuum)

soft DR normal GDR

Ex ~ 1 MeV ~ 20 MeV

0 1 2 3 4

0.6

0.2

dB

(E1)

/ d

Ex (

e 2

fm

2 /

MeV

)

Ex ( MeV )

11Li

M. Zinser et al., Nucl. Phys. A619 (1997) 151

excitations of soft modes withdifferent multipolaritycollective excitations versusdirect transition from weaklybound to continuum states

Large EMD cross sections

specific nuclear property ofextremely neutron-rich nuclei

0 10 20 300

100

200

E1 s

tren

gth

( a

rbit

rary

)

Ng

Vir

tual p

ho

ton

In

ten

sit

y (

MeV

)-1

Ex ( MeV )

EMD X X

XX

3

X

X

B E

N

16 1

E

E

E dE

d

dEE

g

g

s

s

s

9 c

(vi) Ground state properties of 11Li and 9Li :

Spin

J3/2

3/2 quadrupole moments : 27.4 1.0 mb 31.2 4.5 mmagnetic moments : 3.4391 0.0006 n.m. 3.6678 0.0025 n.m.

9Li 11Li

Schmidt limit : 3.71 n.m.

Previous peculiarities cannot arise from large deformationscore is not significantly perturbed by the two valence neutrons

4 6 8 10 12

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

11Li

9Li

6Li

6He

< r

2 ch >

1/2 (

fm)

A

4He

R. Sanches et al., PRL 96 (2006) 033002L.B. Wang et al., PRL 93 (2004) 142501

Nuclear charge radii by laser spectroccopy

1 2

C

(vii) The three-body system 11Li (9Li + n + n) is Borromean : neither the two neutron nor the core-neutron subsytems are bound

Three-body correlations are the most important:due to them the system becomes bound.

The Borromean rings, the heraldic symbol of the Princes of Borromeo, are carved in

the stone of their castle in Lake Maggiore in northern Italy.

p n p n

2 3 4 5 6 7 8 9

-12

-8

-4

0

1/2-

1/2+

15O14N13C12B11Be10Li9He

E* -

S n (

MeV

)

Atomic Number Z

low angular momentum motion for halo particles and few-body dynamics

Prerequisite of the halo formation :

1s - intruder level

10Li

11Be

1

2

3

2

1-

p

s02

parity inversion of g.s.

g.s. :

0~ 0.16 fm -3

proton and neutronshomogeneously mixed,no decoupling of protonand neutron distributions

N / Z ~ 1 - 1.5S ~ 6 - 8 MeV

decoupling of proton and neutron distributions

neutron halos and neutron skins

N / Z ~ 0.6 - 4S ~ 0 - 40 MeV

Peculiarities of haloPeculiarities of haloin low-energy continuumin ground state

weakly bound,with large extensionand space granularity

concentration of the transition strength near break up threshold - soft modes

BASICBASIC dynamicsdynamics of halo nucleiof halo nuclei

Decoupling of halo andnuclear core degrees of freedom

elastic scatteringsome inclusive observables(reaction cross sections, … )

nuclear reactions(transition properties)

y

x

rC

j

rj

r2

r1

1, , , , CA C Α xr yr ,

1, ,

1 2

C 1C C C2 C1 2

A AA A A

A + A A1 2

1 2 1 2

r rx r r y r R r r r

The hyperspherical coordnates : , ,, , C C C CC x x y y

1

,

i i

2 23 3

i i2 2C C i i j

C

C

ji =1 i>j=1

x y A A A

arctan 02

C C

x y r R r r

x

y

A

Volume element in the 6-dimensional space

1

1

i i

i i

2 2 5i i i i i i

5

i5

2 2i i i

x y

x ysin cos

x y

x y

ddx d y x dx y d y d d

d

d

ddd

3/2

3/2

1 2

C

1x1y

1 2

C

Cx

Cy

Y - basis

T - basis

1 2

C

V - basis

r1C r2C

The T-set of Jacobin coordinates ( ) i iA /m m, 1 2 CA A + A A

is the rotation, translation and permutation invariant variable

i ii i iix ysin cos ,x y

The kinetic energy operator T has the separable form

5 11 1

2

2 5 62

2 2 2i

2

x yx y 2

T2 2

Km m m

is a square of the 6-dimensional hyperorbital momentum i5

2K

2

2i5 22

2 2 24

cot l x l y

sin cK

os

,

,4

x x y y

x x y y

i6 6 5

2i i5 5

0

0

K K

K

K l m l m

l m l m

P x, y = =

K =ΦK

Φ

K

, x x y y i5K

l m l mΦ are hyperspherical harmonics or K-harmonics.

,,

2

x yx y

x

x x y y

x x y

x y

yy

1 1

2 21

2

i5 sin cos cos K K

K -

l ll l

l l

l m l ml m l

l lmN Y x Y y PΦ

, z nP are the Jacobi polynomials, mlY x are the spherical harmonics

They give a complete set of orthogonal functions in

yxK l l 2n the 6-dimensional space on unit hypersphere

Eigenfunctions of are the homogeneous harmonic polynomials6

The functions with fixed total orbital moment x yL l l

,

,

, x y x x y y

x y

ix x

i5 y y 5K L M K

l l l m

m m

l ml m l m L ΦMΦ

a normalizing coefficient is defined by the relationx yK

l lN

, , x y x y

x x y y

' ' *i i i5 5 5K' L' M' K L M K K' L L' M M' ' '

l l l ll l l ld Φ Φ

The three equivalent sets of Jacobi coordinates areconnected by transformation (kinematic rotation)

, ,

j i i

j i

ji ji

ji ji

ji ji i

cos sin

sin cos

j i i

j i i

1 2 C

x x y

y x y

x x

yA

y

x yA A

Quantum numbers K, L, M don’t change under a kinematic rotation.

HH are transformed in a simple way and the parity is also conserved.

, ,

,

, , k k i i

k k

i i k kx x

i k5 5

x x

x y y K LK L M K L M

y y

yl

l

l

l l ll l l lΦ Φ

Reynal-Revai coefficients

The parity of HH depends only on yxK l l 2n

+ (positive), if K – even- (negative), if K - odd

The three-body bound-state and continuum wave functions (within cluster representation)

exp 2 3/2x , y J JC CM M i P R /

0x , yJ MT V E The Schrodinger 3-body equation :

1 1

2

xx y

yT2 m

where the kinetic energy operator :

1212V r VV r V r 1C 2C1C 2C and the interaction :

1 2 2 2 x yk k m E

is the hypermomentum conjugated to

The bound state wave function (E < 0 )

1/2 1/2S

SS MS M 1 2 - spin function of two nucleons

, x y i5x , y

X Y

X YL S K l

L SJ M K l l K L S

Ml J

l lχ Φ - /5 2

The continuum wave function (E > 0 )

,'

,

,

x y

x y

i5x y

i5

, x ,, yg g

S X Y X Y

L S ,L' S'S' M' K l l K' l' K L S

J M

K'K'

L S '

'

L

l

l l

l' l'

k k

i L' M' S' M' Φ M

Φ

J

χ- /5 2

Normalization condition for bound state wave function

x , y ,x y x yJ' M' J*

M J J' M M'd d

The HH expansion of the 6-dimensional plane wave

, , xx y y

3*

x y 52i5

2x yexp +

g

K K

KK LL

l l l li k k i ΦJ Φ +2

After projecting onto the hyperangular part of the wave function the Schrodinger equation is reduced to a set of coupled equations

Normalization condition for continuum wave function

'

*x y x y

x x y y 5 5

, , , ,

- -

x , y x , yx

- -

yS S

S S S S

S' M' S M

S S' M M' S M' 5S' M

k' k' k k

k' k k' k '

d d

' ''

2 2

2 2

1

2 g gg g gg gg

K' KKK KK' 'K K

dE

mχ χ

dV V

where 3 2/K + and partial-wave coupling interactions

' ' 121i i5 52g g g gK K' K K'V Φ ΦV r V r V r 1C 2C1C 2C

the boundary conditions: 1 5/2g Κ

Kχ

At the system of differential equations is decoupled since effective potentials can be neglected

The asymptotic hyperradial behaviour of 'K K'V g g The simplest case : ' ' x y, , 0 , 0, 0K K K l lg g

two-body potentials : a square well, radius R ij jk kiV V V

1

i i i5 5 5

/2

jk i

jk

0

2

0

2 2

00 00000 000

/

3

3 sin cos sin

V V x

V

R

d Φ Φ

d d

a general behaviour of three-body effective potential if the two-body potentials are short-range potentials

1

2 2

2 2

1

20K

dE

m dχ g

1

5 2exp - expx -, yK J M /

χ g

if E < 0

' ' '

1

(-) (+)

5x 2y sin cos, x, , y A BS

K K' K +2 K +2K K K

S M /

K

' 'χ H

k

HS

k

gg g gg g

if E > 0

1( ) expK +2H i

Correlation density for the ground state of Correlation density for the ground state of 66HeHe

cigar-like configuration dineutron configuration

n n

C

rnn

R(nn)-C

, ,22 2

nn nn-C nn nn JM nn nn-nn-M

-C Cnn Cd1

P r R r R2J + 1

r Rd

31

2

A*3

1 2

Aa

1 + 2 + 3 + Agr , elastic ( 4-body )1 + 2 + 3 + A* , inelastic ( 4-bodies)

a + A

complex constituents

Cross section

events with undestroyed core peripheral reactions

Study of halo structure

2

i f i f1 2 C

ifi

2

vA*dk dk dk d P Tk P

s

2

x

* y

y

x

2

a

k

2

k

2E

Reaction amplitude Tfi (prior representation)

1

2C

A

kxky

kf

a

, , , grf A ii ix ayf , UT k k k k p,t

p,t

V

halo ground state wave function

target ground state wave function

distorted wave for relative projectile-target motion

exact scattering wave function

NN - interaction between projectile and target nucleons

optical potential in initial channel

gr

ii k

,f x y,k k k

p,tV

aAU

, , , grx yf if iAi aU, k kkT k p,t

p,t

V

DW: low-energy halo excitations small kx & ky ; large ki & kf

(no spectators, three-body continuum, full scale FSI )

Kinematically complete experiments• sensitivity to 3-body correlations (halo)• selection of halo excitation energy• variety of observables• elastic & inelastic breakup

a FS

I

A

123

A*

,, , , ,gr grx y if fi if k kT k k pt

p,t

V

Reaction amplitude Tfi (prior representation)

Method ofMethod of hyperspherical harmonics:hyperspherical harmonics: 3-body3-body boundbound andand continuumcontinuum statesstates

effective interactionseffective interactions

( ( NNNN && N-coreN-core ) )

binding energybinding energyelectromagnetic momentselectromagnetic moments

electromagnetic formfactorselectromagnetic formfactorsgeometrical propertiesgeometrical propertiesdensity distributionsdensity distributions

. . . . . . .. . . . . . .

Transition densitiesTransition densities

Nuclear structure

,, , , ,gr grx y if fi if k kT k k pt

p,t

V

y

x

rC

j

rj

r2

r1

Three-body models

1, , , , CA C Α xr yr ,

, xS

yp

L p pJ

p p

r -k k

rY r

r r

s

Reaction mechanismReaction mechanism

,, , , ,gr grx y if fi if k kT k k pt

p,t

V

no

co

nsi

sten

cy w

ith

nu

clea

r st

ruct

ure

in

tera

ctio

ns

free NN scattering

complex, energy, density dependent

effective NN interactions Vpt

Distorted wave approachDistorted wave approach

nucleus-nucleus elastic scattering and reaction cross sections

optical potential

distorted waves (k) self-consistency

One-step processOne-step process

2 4 60

2

4

6

2+

0+

1-

E* ( MeV )

6He + 12C

ds/

dE

* (m

b/M

eV)

2 4 6 80

50

100

0+2+

T. Aumann et al., Phys. Rev., C59 (1999) 1252.

E / A = 240 MeV

E* ( MeV )

6He + 208Pb

ds/

dE

* (m

b/M

eV)

1-

0 1 2 30

2

4

6

8

no FSI

0+

2+

1-

En (MeV)

ds/

dE

n (

arb

.un

its)

0 1 2 30

2

4

6

no FSI

0+ 2+

1-

Enn

(MeV)

ds/

dE

nn (

arb

.un

its)

0 1 2 30

2

4

6

8

no FSI

0+

2+

1-

E (MeV)

ds/

dE

(a

rb.u

nit

s)

0 1 2 30

2

4

6

6He + 208PbE / A = 240 MeV/A

no FSI

0+

2+

1-

En (MeV)

ds/

dE

n (

arb

.un

its)

Halo scattering on nuclei

Continuum SpectroscopyContinuum Spectroscopy

Two-body breakup Three-body breakup

n

C

x

θx

n nx

C

yxy

n n

C

xyxy

T-s

yste

m

Y-system

2X/22

XE k 2 2X Y/2 /22 2

X YE k k

X X Xl mY X Y

LX Yl l LM

Y Y

SSM1 CS S

SSM1 2 CS S S

Excitation EnergyExcitation Energy

Orbital angular momentaOrbital angular momenta

Spin of the fragmentsSpin of the fragments

HypermomentHypermoment

XN = 2n + l X YlnK +2= l

ABC of the ABC of the three-bodythree-body correlations correlations

Permutation of identical particles Permutation of identical particles must not producemust not produce an an effecteffect on observables on observables..ManifestationManifestation depends on the coordinate system where correlations are defined depends on the coordinate system where correlations are defined ..

n nx

C

yxy

T-system

Neutron permutation changes

;X X XY XYk k

1. Angular correlation is symmetric relative

XY

2. Energy correlation → no effect

n n

C

xyxy

Y-system

n n

C

x yxy

V-system

no recoil: Cm

;X Y

XY XY

1k k

1. Angular correlation → no effect

2. Energy correlation is approximately

symmetric relative ε = 1/2

, , /X X YE E 0 where

0

1

2

3

3

21

Wp

h (

)

T system

0

1

2

(0, 2)

(2, 0)

Y system1

(1, 1)

0 0.5 10

1 (0, 2)(1, 1)

(2, 0)

Y system

2

Wp

h (

)

0 0.5 10

1(1, 1)

Y system3

Energy correlations of elementary modes :2+ excitations with the hypermoment K = 2

1

2

ph

dW dE

d dE

s

n n

C

xy

n nx

C

y

- yxph

llW 1

ABC of the ABC of the tthree-bodyhree-body correlations correlations

1: , ,

3 : , ,

, ,

: , ,

,

,

,

,

2

Tsystem

(2

(

0 0

1

2 )

(2

1

0

2

1

0 )

1

( )

)

T Tx yL S l l

,

,

,

,

(0 2

( 1

(2 0 )

Y sy

1

m

)

)

s e

)

t

( Y Yx yl l

Y Y

Y Y

T T Y T TYX Y X Y

X Y

l l l lX Y X Y

l l

L Ln n

KL

l l l l

0.0 0.5 1.00

1

2

(1,1,1,1)

(2,0,1,1)

Wp

h (

EC

n /

E )

ECn

/ E

(2,0,2,0)

0.0 0.5 1.00

1

2

(2,0,2,0)(1,1,1,1)

Wp

h (

En

n /

E )

Enn

/ E

(2,0,0,2)

J = 2+

Energy correlations for 2+ resonance in 6He

x y( , , , )Quantum numbers L S l l:

n nx

C

y

n n

C

xy

T system

Y system

6 208He + Pb

E/A = 208 MeV

Realistic calculations:mixing of configurations

For low-lying states :only a few elementary modes

1 2 3 4 5 6 70

50

100

150

E* (MeV)

0+

6He + 208Pb

2+ 1-

ds/

dE

* (m

b/M

eV)

0.0 0.5 1.00

1

2

0+1-

2+

Wp

h (

En

n /

E )

Enn

/ E

0 < E < 1 MeV

0.0 0.5 1.00

1

2

0 < E < 1 MeV

0+1-

2+

Wp

h (

EC

n /

E )

ECn

/ E

-1 0 10.0

0.5

1.0

0+1-

2+

W (

co

s n

n )

cos (nn

)

-1 0 10.0

0.5

1.0

0+1-

2+

W (

co

s C

n)

cos (Cn

)

n nx

C

yxy

n n

C

xyxy

6 208 / 240+ ,He Pb E A MeV

Energy and angular fragment correlations

0.0 0.5 1.00

1

2

0+

1-

2+

Wp

h (

En

n /

E )

Enn

/ E

1 < E < 3 MeV

0.0 0.5 1.00

1

2

1 < E < 3 MeV

0+

1-

2+

Wp

h (

EC

n /

E )

ECn

/ E

-1 0 10.0

0.5

1.0

0+

1-

2+

W (

co

s n

n )

cos (nn

)

-1 0 10.0

0.5

1.0

0+1-

2+

W (

co

s C

n)

cos (Cn

)

L.V. Chulkov et al., Nucl. Phys. A759, 23 (2005)■

n n

C

xyxyn nx

C

yxy

0.0 0.5 1.00

1

2

0+

1-

2+

Wp

h (

En

n /

E )

Enn

/ E

6 < E < 9 MeV

0.0 0.5 1.00

1

2

6 < E < 9 MeV

0+

1-

2+

Wp

h (

EC

n /

E )

ECn

/ E

-1 0 10.0

0.5

1.0

0+

1-

2+

W (

co

s n

n )

cos (nn

)

-1 0 10.0

0.5

1.0

0+

1-

2+

W (

co

s C

n)

cos (Cn

)

0.0 0.5 1.00

1

2

0+ 1-

2+

Wp

h (

En

n /

E )

Enn

/ E

3 < E < 6 MeV

0.0 0.5 1.00

1

2

3 < E < 6 MeV

0+1-

2+

Wp

h (

EC

n /

E )

ECn

/ E

-1 0 10.0

0.5

1.0

0+

1-

2+

W (

co

s n

n )

cos (nn

)

-1 0 10.0

0.5

1.0

0+1-

2+

W (

co

s C

n)

cos (Cn

)

n n

C

xyxyn nx

C

yxy

n nx

C

yxy

n n

C

xyxy

assume 4 -measurements of fragments

C1 222 2

C1 2 mm m

2 2

V

2

VVE

VCM

V1V2

VC

detector

Inverse kinematics: forward focusing

CME E

"The angular range for fragments and neutrons covered by the detectors corresponds to a 4 measurement of the breakup in the rest frame of the projectile for fragment neutron relative energies up to 5.5 MeV (at 500 MeV/nucleon beam energy)".

T. Aumann, Eur. Phys. J. A 26 (2005) 441

Theoretical calculations Experimental data?

The remarkable discovery of new type of nuclear structure at driplines, HALO, have been made with radioactive nuclear beams.

The theoretical description of dripline nuclei is an exciting challenge. The coupling between bound states and the continuum asks for a strong interplay between various aspects of nuclear structure and reaction theory.

Development of new experimental techniques for production and /or detection of radioactive beams is the way to unexplored

“ “ TERRA INCOGNITATERRA INCOGNITA “ “

CONCLUSIONSCONCLUSIONS

Special thanks to B.V. Danilin & J.S. Vaagen