Transcript S. N. Ershov Joint Institute for Nuclear Research Halo Nuclei HALO: new structural dripline phenomenon with clusterization into an ordinary core nucleus and a veil.

S. N. Ershov
Joint Institute for Nuclear Research
Halo Nuclei
HALO:
new structural dripline phenomenon with clusterization
into an ordinary core nucleus and a veil of halo nucleons
– forming very dilute neutron matter
Chains of the lightest isotopes
( He, Li, Be, B, …) end up with two neutron
halo nuclei
Nuclear scale ~ 10-22 sec
Width 1 ev ~ 10-16 sec
Two neutron halo nuclei ( 6He , 11Li , 14Be , … )
break into three fragments and are all
Borromean nuclei
Life time for radioactive
nuclei > 10-12 sec
One neutron halo nuclei ( 11Be , 19C , … )
break into two fragments
b decays define the life time
of the most radioactive nuclei
( 6He, 11Li, 11Be, 14Be, 17B, … )
weakly bound systems
with large extension
and space granularity
“Residence in forbidden regions”
Appreciable probability for dilute nuclear matter extending
far out into classically forbidden region
11Li
n
~ 7 fm
n
~ 5 fm
2.3 fm
9Li
Separation energies
of last neutron (s) :
halo
<1
stable
6 - 8 MeV
Large size of halo nuclei
Two-neutron halo nuclei
( 11Li, 6He, 14Be, 17B, . . . )
Borromean system
is bound
e (11Li ) = 0. 3 MeV
e (11Be) = 0. 5 MeV
e ( 6He) = 0. 97 MeV
< r2 (11Li) >1/2 ~
( r.m.s. for A ~
3.5 fm
48 )
Borromean systems
none of the constituent two-body
subsystems are bound
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
4.0
This is very unusual for strongly interacting
systems held together by short-range interactions
Be B
RI ( fm )
3.5
Li
3.0
Interaction radii :
He
E / A = 790 MeV, light targets
2.5
1/3
1.18A
2.0
6
8
10 12 14 16 18
I. Tanihata et al.,
Phys. Rev. Lett., 55 (1985) 2676
A
(iii) very narrow momentum distributions, compared to stable nuclei, of both
E / A = 800 MeV
1200
11
9
C( Li, Li)
Counts
800
s = 21 MeV/c
neutrons and 9Li measured in high energy
fragmentation reactions of 11Li .
No narrow fragment distributions in breakup
on other fragments, say 8Li or 8He
( naive picture )
narrow momentum distributions
400
0
-400
-200
0
200
400
transverse momementum ( MeV / c )
large spatial extensions
(iv) Relations between interaction and neutron removal cross sections ( mb ) at 790 MeV/A
sI
A + 12C
s -2n
796 m 6
1060m 10
503m 5
722m 5
817m 6
9Li
11Li
4He
6He
s -4n
sI (A=C+xn) = sI (C) + s-xn
Strong evidence for the well defined
clusterization into
the core and two neutrons
220m 40
189m14
202m17 95m 5
Tanihata I. et al.
PRL, 55 (1987) 2670;
PL, B289 (1992) 263
(v) Electromagnetic dissociation cross sections per unit charge are orders of magnitude
halo
larger than for stable nuclei
11
100
6
Li
He
He
14
11
Be
Be
12
10
EMD
1
0.1
Evidence for a rather large difference between
charge and mass centers in a body fixed frame
8
2
S-2n
/ Zp
EMD
2
S-n
/ Zp
Be
9
Li
concentration of the dipole strength
at low excitation energies
12
C
1
10
Separation energy (MeV)
stable
Zp-normalized EMD cross section (mb)
8He
T. Kobayashi, Proc. 1st Int. Conf. On
Radiactive Nuclear Beams, 1990.
(peculiarities of low energy halo continuum)
100
0
0
Ng
10
20
Ex ( MeV )
soft DR
30
normal GDR
specific nuclear property of
extremely neutron-rich nuclei
s EMD   N ( EX ) s g ( EX ) dEX
s g ( EX ) 
dB(E1) / dEx ( e 2 fm 2 / MeV )
200
E1 strength ( arbitrary )
Virtual photon Intensity ( MeV )
-1
Large EMD cross sections
16 3
9 c
EX
dB ( E1)
dE X
11Li
0.6
M. Zinser et al.,
Nucl. Phys.
A619 (1997) 151
0.2
0
1
2
3
4
Ex ( MeV )
Ex ~ 1 MeV
~ 20 MeV
excitations of soft modes with
different multipolarity
collective excitations versus
direct transition from weakly
bound to continuum states
(vi) Ground state properties of 11Li and 9Li :
9Li
11Li
Spin J :
3/2 3/2 quadrupole moments : -27.4 m 1.0 mb
-31.2 m 4.5 m
magnetic moments :
3.4391 m 0.0006 n.m. 3.6678 m 0.0025 n.m.
Schmidt limit : 3.71 n.m.
Previous peculiarities cannot arise from large deformations
core is not significantly perturbed by the two valence neutrons
3.0
Nuclear charge radii
by laser spectroccopy
2.8
11
Li
2.4
9
Li
R. Sanches et al., PRL 96 (2006) 033002
L.B. Wang et al., PRL 93 (2004) 142501
Li
2.2
1
2
<r
2
>
ch
1/2
(fm)
6
2.6
2.0
1.8
6
4
He

He
1.6
C
4
6
8
A
10
12
(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.
N / Z ~ 0.6 - 4
eS ~ 0 - 40 MeV
N / Z ~ 1 - 1.5
eS ~ 6 - 8 MeV
r0 ~ 0.16 fm -3
proton and neutrons
homogeneously mixed,
no decoupling of proton
and neutron distributions
p
E* - Sn ( MeV )
9
10
11
decoupling of proton and
neutron distributions
neutron halos and
neutron skins
p
n
12
He Li Be B
13
C
14
N
15
n
Prerequisite of the halo formation :
O
low angular momentum motion for halo
particles and few-body dynamics
0
-4
+
-8
1/2
-12
1/2
2
3
4
5
6
7
8
Atomic Number Z
-
9
1s - intruder level
parity inversion of g.s.
g.s. :
Peculiarities of halo
in ground state
in low-energy continuum
weakly bound,
with large extension
and space granularity
concentration of the transition
strength near break up threshold
- soft modes
elastic scattering
some inclusive observables
(reaction cross sections, … )
x
r1
y
rC
r2
rj
j
nuclear reactions
(transition properties)
Decoupling of halo and
nuclear core degrees of
freedom
BASIC dynamics
of halo nuclei
(
 r1 ,
)
(
,rA  C  ,
) ( )
, ΑC  x ,y
1 xC
2

yC
xC  r1 - r2 ,
C
yC  rC -
3
r   xi xC2  yi yC2   A i ( ri - R )
i =1
2
y1
x1

C
r1C

xi xi  r sini ,
2
yi yi  r cosi
Volume element in the 6-dimensional space
d xi d yi  xi2 d xi yi2d yi d  xi d  y i 
1
(  )
x
C
V - basis
1 3
  A i A j ( ri - rj )
A i> j=1
 xC 

,
0




C
y
2
C


r is the rotation, translation and permutation invariant variable
2
r2C
2
 C, xC , xC , yC , yC
 C  arctan 
Y - basis
1
A  A1 + A2  AC )
A1r1  A 2 r2
1
, R  ( A1r1  A 2 r2  A C rC )
A1 + A 2
A
The hyperspherical coordnates : r,
T - basis
1
The T-set of Jacobin coordinates ( A i  m i / m,

1
(  )
x
y
3/2
3/2
r 5 d r d  i5
y
r 5 d r sin 2 i cos 2 i d i d  xi d  yi
The kinetic energy operator T has the separable form
2
2
 1

 2 5 
1
1 2 i 
Ty   - 2 K ( 5 )   6
  x 
 2

2m   x
y 
2m  r
r r r
2m

2
K
2
(  ) is a square of the 6-dimensional hyperorbital momentum




K ( )  - 4cot (  )

l ( x)
l ( y)

 sin 
cos 
i
5
2
2
i
5
2
2
Eigenfunctions of
2
2
 6 are the homogeneous harmonic polynomials
( ) = 0
(  ) - K ( K  4) Φ l m , l m (  ) = 0
 6 PK ( x, y ) =  6 r K ΦKx
l mx , ly my
K
l mx , ly my
ΦKx
2
2
i
5
x
x
y
i
5
y
K
i
5
(  ) are hyperspherical harmonics or K-harmonics.
i
5
They give a complete set of orthogonal functions in
the 6-dimensional space on unit hypersphere
Pn(
 ,b
()
ly  2n )
1
1

 lx  , ly  
2
2

1
K - lx - l y
2
( ) (sin ) (cos ) P ( ) (cos 2 )
)
( z ) are the Jacobi polynomials, Y ( x ) are the spherical harmonics
lx mx , ly my
ΦK
( K  lx 
( )  N
i
5
lx ly
K
lx
Ylx mx x Yly my y
lm
ly
The functions with fixed total orbital moment L  lx  ly
ΦKxL My ( i5 ) 
l ,l
(l

m ,m
x
x
x mx l y my L M ) ΦK
l mx , ly my
y
l ly
N Kx
a normalizing coefficient
( )
i
5
is defined by the relation
i
i
i
x y
x y
d

Φ

Φ

(
)
(
 5 K' L' M' 5 K L M 5 )   K K'  L L'  M M'  lx lx'  ly ly'
l ', l ' *
l ,l
+ (positive), if K – even
- (negative), if K - odd
The parity of HH depends
only on K  lx  ly  2n
The three equivalent sets of Jacobi coordinates are
connected by transformation (kinematic rotation)
  x x j  - cos ji  x xi - sin ji  y yi
j
i
i


  y j y j  sin ji  xi xi - cos ji  yi yi
 ji   ji ( A 1 , A 2 , A C )
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.
lx , l y
i
ΦK Li M
( i5 ) 

l ,l
xk
yk
lx , l y
k
 lx , l y lx , l y  K L ΦK Lk M
k
k
i
i

Reynal-Revai coefficients
( )
k
5
The three-body bound-state and continuum wave functions
(within cluster representation)
J M  C ( C )J M ( x , y ) exp i ( P R ) / ( 2 )
The Schrodinger 3-body equation :
3/2
(T  V - E )J M ( x , y )  0
 1

1
y 
  x 

2m   x
y

and the interaction : V  V r 12  V r 1C  V r 2C
12
1C
2C
where the kinetic energy operator : T  -
2
( )
( )
( )
The bound state wave function (E < 0 )

J M ( x , y )  r -5 / 2
L S K l X lY
 S M   1/2
S
1
 1/2
l ,l
χ KL Sl X lY ( r ) ΦKxL y ( i5 )   S 

JM
 - spin function of two nucleons
2  S MS
The continuum wave function (E > 0 )
S' M' ( kx , ky , x , y )  (r )
S
-5 / 2

g ,g'
l ,l
χ KL Sl X,L'lY ,S'K' l' X l'Y (, r ) ΦKxL y ( i5 )   S 

JM
 i K' ( L' M' L S' M' S J M ) ΦK'xL'
l' , l'y
  k x2  k y2 
1
( )
i
5
2m E is the hypermomentum conjugated to r
The HH expansion of the 6-dimensional plane wave


exp i ( kx x + ky y ) 
( 2 ) 3
(r )
2
 i K J K +2 (r ) ΦKxL
l , ly
g
( )Φ l , l ( )
i
5
x
y*
KL

5
Normalization condition for bound state wave function
*
d
x
d
y

J' M' ( x , y )J M ( x , y )   J J'  M M'

Normalization condition for continuum wave function
*
d
x
d
y

S'
M' S ( k'x , k'y , x , y )S M S ( kx , ky , x , y ) 

 S S'  M
S
M'S
 ( k'x - kx )  ( k'y - ky )   S S'  M
S
M'S





'



'

(
)
(
5
5 )
5
After projecting onto the hyperangular part of the wave function
the Schrodinger equation is reduced to a set of coupled equations
2


 d 2  (   1) 
 2
  VK g , K g ( r ) - E  χ K g ( r )  -  VK g , K'g ' ( r ) χ K'g ' ( r )
2
r
K'g '  K g

 2m  d r

where   K + 3 / 2 and partial-wave coupling interactions
( )
( )
( )
VKg , K'g ' ( r )  ΦK g ( i5 ) V12 r 12  V1C r 1C  V2C r 2C
the boundary conditions:
χ Kg ( r  )
ΦK' g ' ( 5i )
r 1  r Κ  5/2
The asymptotic hyperradial behaviour of VK g , K'g '
(r)
The simplest case : K  K ', g  g ', K  0 , l x  0, l y  0
two-body potentials : Vij  Vjk  Vki  a square well, radius R
00
00
V00 ( r )  3  d i5 Φ 000
( i5 ) Vjk ( xi ) Φ 000
( i5 )
 /2
 3  d  sin  cos  Vjk ( r sin ) 

r  
2
2

R /r

1
d  2
r

1
a general behaviour of three-body effective potential

if the two-body potentials are short-range potentials
r
At r   the system of
differential equations is
decoupled since effective
potentials can be neglected
2



 2m

 (   1) 
 d2


E
 χ Kg ( r )  0


2
2
r

dr


if E < 0
χ K g ( r   ) exp ( - r )  J M ( x , y )
if E > 0
χ K g , K'g ' ( r   )
S M ( kx , ky , x , y )
S
r  H
(-)
K+2
(r )  K g , K'g '
1
r
5/ 2
exp ( - r )
- S K g , K'g ' H K(+)+ 2 (r ) 
( A sin (r )  B cos (r ) )
r 5/ 2
1
H K( +2) (r )
exp (  ir )
r
1
Correlation density for the ground state of
P ( rnn , Rnn-C )  r R
2
nn
2
nn - C
6He
2
1
d nn dnn-C JM ( rnn , Rnn-C )


2J + 1 M
n
rnn
n
R(nn)-C
C
cigar-like configuration
dineutron configuration
1
1
2
3
a
A
3
2
A*
Study of halo structure
events with undestroyed core
peripheral reactions
complex constituents
a+A
1 + 2 + 3 + Agr , elastic ( 4-body )
1 + 2 + 3 + A* , inelastic (  4-bodies)
Cross section
2 )
(
2
s
d k 1 d k 2d k C d k A*  (  i -  f )  ( Pi - Pf ) Tfi


v 
i
Reaction amplitude Tfi (prior representation)
Tfi  (
a
ky
C
kf
A
-)
(
kf , kx , ky
)
(
V
U

,

,

 p,t aA  gr i
p,t
 
1
kx
gr 
2
( )
 ( ) (k , k , k )
)
(k )
i
halo ground state wave function
target ground state wave
function
 i(  ) k i  projectile-target motion
-

f
x
y
distorted wave for relative
exact scattering wave function
Vp,t 
NN - interaction between
projectile and target nucleons
UaA 
optical potential in initial channel
Reaction amplitude Tfi (prior representation)
(-)
Tfi  
(k , k , k )  V
f
x
y
p,t
()
- UaA  , gr ,  i
p,t
(k )
i
DW: low-energy halo excitations small kx & ky ; large ki & kf
(no spectators, three-body continuum, full scale FSI )
-)
( )
k f , gr , (
FSI
Tfi   f(
a
A
1
2
3
A*
-)
(
kx,ky
)
(
V

,

,

 pt  gr i
p,t
)
(k )
i
Kinematically complete experiments
• sensitivity to 3-body correlations (halo)
• selection of halo excitation energy
• variety of observables
• elastic & inelastic breakup
Tfi   f(
-)
( k ) ,
f
(
,

gr
-)
(k , k )  V
x
y
pt
gr
p,t
Nuclear structure
Transition densities

(-)
(k , k ) 
x
y
p
Three-body models
(
 r1 ,
)
(
) ( )
,rA  C  ,
, ΑC  x ,y
y
rC
 ( r - rp )
r rp
YL ( rp ) s Sp  

J
effective interactions
( NN & N-core )
Method of hyperspherical harmonics:
3-body bound and continuum states
x
r1
( )
 ,  ,  i(  ) k i
r2
rj
j
binding energy
electromagnetic moments
electromagnetic formfactors
geometrical properties
density distributions
.. . . . . .
no consistency with nuclear structure interactions
Tfi   f(
-)
( )
k f , gr , (
-)
(
kx,ky
)
(
V

,

,

 pt  gr i
p,t
)
(k )
i
Reaction mechanism
One-step process
Distorted wave approach
effective NN interactions Vpt
distorted waves  (k)
complex, energy,
density dependent
optical potential
free NN scattering
nucleus-nucleus elastic scattering
and reaction cross sections
Halo scattering on nuclei
6
208
He + Pb
E / A = 240 MeV/A
E / A = 240 MeV
8
Pb
-
1
50
+
2
0
+
0
2
4
6
8
6
4
2
no FSI
+
2
-
1
+
0
0
0
E* ( MeV )
1
2
En (MeV)
6
4
-
+
2
1
2
+
0
0
2
4
6
E* ( MeV )

4
-
1
no FSI
2
+
2
+
0
0
0
3
1
2
Enn (MeV)
6
3
12
He + C
T. Aumann et al., Phys. Rev., C59 (1999) 1252.
d s /dE  (arb.units)
d s /dE* (mb/MeV)
8
6
d s /dE nn (arb.units)
100
208
6
d s /dE n (arb.units)
He +
d s /dE  n (arb.units)
d s /dE* (mb/MeV)
6
6
no FSI
4
+
2
2
-
1
+
0
0
0
1
2
E (MeV)
3
no FSI
4
+
2
2
-
1
+
0
0
0
1
2
En (MeV)
3
n
x
θx
C
n
n
x
y
n
n
y
x
C
C
Y-system
T-system
Continuum Spectroscopy
Two-body breakup
Three-body breakup
Excitation Energy
E 
2
k X2 /2  X
E 
2
k X2 /2X 
2
kY2 /2Y
Orbital angular momenta
Yl X mX ( X )
Yl ( X )  Yl ( Y )
 X
Y
 LM L
Spin of the fragments
 S1  SC SM
N = 2n + l X
S
 S1  S2  SC SM
Hypermoment
K = 2n + l X  lY
S
ABC of the three-body correlations
Permutation of identical particles must not produce an effect on observables.
Manifestation depends on the coordinate system where correlations are defined.
Y-system
n
n
y
x
T-system
n
n
x
y
Neutron permutation changes
k X  -k X ;  XY   -  XY
1. Angular correlation is
symmetric relative 
→
XY
 
no effect
e  e X /E ,
C

k X  kY ; e  1 - e
 XY  XY
no recoil: mC
1. Angular correlation → no effect
2. Energy correlation is approximately
symmetric relative
where
y
x
C
C
2. Energy correlation
V-system
n
n
ε = 1/2
E  e X  e Y , 0  e 
ABC of the three-body correlations
n
Energy correlations of elementary modes :
2+ excitations with the hypermoment K = 2
(1 -e )
Y system
Tsystem

l XT lYT L
n

(2, 0, 2, 0 ) 
( 1, 1, 1, 1 ) 
(2, 0, 0, 2 )
(0, 2 ) 
(2, 0 ) 
( 1, 1 ) 
 l l l l
Y Y
X Y
l XY lYY

l XY lYY L
n
T T
X Y
KL
2
Y system
1
2
1
2
Y Y
(
l
 x ,ly )
( L, S , l Tx , l Ty )
1:
2 :
3 :
ly
n
y
C
T system
Wph (e)
W ph ( e ) e
y
3
e (1 - e ) d e dE
lx
n
x
C
3
1
(2, 0)
(1, 1)
1
(0, 2)
0
0
Y system
Y system
3
2
Wph (e)
W ph ( e )   dE
d 2s
n
x
(1, 1)
1 (0, 2)
(1, 1)
1
(2, 0)
0
0
0.5
e
1
0
0
0.5
e
1
2

+
Realistic calculations:
mixing of configurations
Wph ( Enn / E )
J =2
T system
n
1
n
x
(2,0,0,2)
y
(2,0,2,0)
(1,1,1,1)
C
0
0.0
Wph ( ECn / E )
2
0.5
For low-lying states :
only a few elementary modes
1.0
Enn / E
Y system
n
1
y
(2,0,1,1)
x
(1,1,1,1)
(2,0,2,0)
0
0.0
0.5
n
6
C
1.0
ECn / E
Quantum numbers : ( L, S , lx , ly )
Energy correlations for
2+ resonance in 6He
He + 208 Pb
E/A = 208 MeV
n
n
x
n
6
n
y
y
x
C
C
2
2
0 < E < 1 MeV
+
2
-
1
0
0.0
1
+
2
+
0
0.5
0
0.0
1.0
0.5
Enn / E
W ( cosCn)
W ( cosnn )
1.0
0.5
0.5
+
2
-
0.0
-1
0
cos (nn)
+
0
6
1.0
ECn / E
1.0
1
150
-
1
d s /dE* (mb/MeV)
Wph ( ECn / E )
Wph ( Enn / E )
0 < E < 1 MeV
1
Energy and angular
fragment correlations
+
+
2
-
1
0
1
0.0
-1
0
0
cos (Cn)
1
208
Pb
100
50
0
1
+
He +
2
2
+
0
1
+
3
-
4
E* (MeV)
5
6
7
n
n
x
n

n
y
y
x
C
C
2
2
1 < E < 3 MeV
Wph ( ECn / E )
Wph ( Enn / E )
1 < E < 3 MeV
1
-
1
+
0
0
0.0
+
2
0.5
1.0
-
1
1
+
0
0
0.0
0.5
Enn / E
0.5
-
1
+
0.0
-1
■
W ( cosCn)
W ( cosnn )
1.0
0.5
-
1
+
0
2
0
0
cos (nn)
1.0
ECn / E
1.0
+
+
2
1
0.0
-1
+
2
0
1
cos (Cn)
L.V. Chulkov et al., Nucl. Phys. A759, 23 (2005)
n
n
x
n

n
n
n
x
n
n
y
y
x
y
C
C
2
0
0.0
0.5
-
1
0
0.0
1.0
2
W ( cosCn)
1.0
-
1
0.5
+
2
0
cos (nn)
-
+
1
0.0
-1
-
+
2
1
2
0
cos (Cn)
1
-
1
0
0.0
+
+
0
0.5
0
0.0
1.0
0.5
1.0
0.5
0.5
-
+
1
2
0.0
-1
1.0
ECn / E
0
cos (nn)
-
+
2
+
0
1
+
2
Enn / E
+
0
1
1
1.0
0.5
0
+
6 < E < 9 MeV
0
1.0
ECn / E
1.0
W ( cosnn )
0
+
0.5
Enn / E
0.0
-1
+
2
Wph ( ECn / E )
2
1
Wph ( Enn / E )
0
+
C
6 < E < 9 MeV
W ( cosnn )
Wph ( ECn / E )
Wph ( Enn / E )
3 < E < 6 MeV
-
1
C
2
3 < E < 6 MeV
1
x
W ( cosCn)
2
+
y
1
0.0
-1
1
+
0
0
cos (Cn)
1
Theoretical calculations
?
Experimental data
detector
assume 4 -measurements of fragments
m1V12 m2V22 mCVC2
E 


2
2
2
E
Inverse kinematics:
V2
V1
VCM
VC
ECM
forward focusing
T. Aumann, Eur. Phys. J. A 26 (2005) 441
"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)".
CONCLUSIONS
 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 INCOGNITA “
Special thanks to B.V. Danilin & J.S. Vaagen