1
Pair correlation and pair collectivity
in neutron-rich nuclei
M. Matsuo (Niigata U.)
H. Shimoyama (Niigata U)
Y. Ootaki (Niigata U)
Y. Serizawa (Niigata U)
D.Y. Pang (Peking U.)
J. Lee (RIKEN)
Y. Aoki (RIKEN)
1. Surface enhancement & di-neutron correlation
2 T
2.
Two-neutron
t
transfer
t
f as a probe
b
p beyond
y
the N=82 shell closure
Sn isotopes
Bose condensation and strong coupling pairing:
BCS-BEC crossover
Leggett 1980, Nozieres & Schmitt-Rink 1985
BEC
Interaction:
In free space:
BCS
Strong
Interaction:
Bound pair (boson)
is already formed
Crossover
In medium: Condensate of bosons
In free space:
In medium:
Weak
No bound pair is
formed
Bound pairs (Cooper
pairs) are formed,
then condensate
Average distance
d
a  

Size of correlated pair
/d<<1
Pairing gap
/eF>1
Scattering length a & density
/kFa > 1
/d = 0.2 ~ 1.10
/d>1
/eF= 1.3 ~ 0.2
/eF<<1
/kFa = -1 ~ 1
/kFa < -1
k F3
 2
3
Strong coupling pairing in dilute matter &
BCS-BEC crossover
2
“Large”
Large pair gap vs
vs. Fermi energy /eF > 0.2
0 2 at low-densities
Monte-Carlo calculation
Mean-field calculation (BCS approx.)
/0 10
10-3~ 0.5x10-1
/eF
/0 10-4~ 2x10-1
1
crossover
0.1
0.01
10-4
Gezerlis & Carlson, PRC81,025803 (2010)
10-3
10-2
/0
10-1
10-0
MM, PRC73,044309(2006)
3
“Dilute matter” in n-rich nuclei
esp
Weakly bound
many
y neutrons
proton
s
Un
Up
neutrons
density
n
Neutron skin
Dilute(unsaturated) matter
n/0=1/2 – 10-2(?)
N
Nucleon
deensity
n/p >>N/Z
Tanihata et al.
PLB287 (1992)
p
r
11Li
密度
図
Ｒ
thikness < 1-2fm(?)
Neutron halo
L
Long
tail
t il
n/0 < 10-2(?)
r
Enhancement of pairing at surface
and
d tail
t il in
i neutron-rich
t
i h nuclei
l i
1. Spatially
p
y correlated halo neutrons in 11Li and 6He,, etc
G.F.Bertsch, H.Esbensen, Ann. Phys. 209(1991) 327
Coulomb break-up exp. on 11Li
n

n
Nakamura et al.
PRL96,252502 (2006)
nn=48+14-18 deg
Rc,2n=5.01+-0.32 fm
Charge radius Mueller et al.
PRL99,252501 (2007)
2. Medium & heavy
y mass n-rich nuclei with several weakly
y bound neutrons
Surface enhancement of pairing
Dobaczewski et al, PTPS146,70(2002), EPJA15,21(2002)
Spatial correlation
In n-rich nuclei
NB. In stable nuclei
Matsuo, Mizuyama, Serizawa PRC71,064326(2005)
Pillet, Sandulescu, Schuck, PRC76, 024310 (2007)
Ibarra et al. NPA288, 397 (1977)
Janouch & Liotta PRC27,896 (1983)
Pankratov, et al. PRC79, 024309 (2009)
etc
etc
4
5
132Sn
and beyond
y
Neutron orbits
One-neutron separation energy
es.p.HF
0
-5
82
h11/2
NNDC
(0.66)
(0
66)
(0.27)
-0.25
-1.99
f5/2
p1/2
p3/2
f7/2
-7.68
-10
Hartree-Fock for 132Sn
Skyrme SLy4
Pairing in the density functional theory
6
In the present work, we analyse neutron-rich Sn
S isotopes using
Skyrme energy density functional (HFB) + linear response theory (QRPA)
Given the Skyrme functional + pairing functional

  
E  ESkyrme [  ,  ,  , , j , s , J ]  E pair [  , ~, ~ * ]
Skyrme functional
Parameter set: SLy4
Ground state
DDDI pairing functional
Using the pair density
~* (r )    (r )  (r )
solving the Hartree-Fock Bogoliubov equation in the coordinate-space
Hartree Fock pot.
Hartree-Fock
pot pair pot
pot.
Collective excitations
Normal modes of time-dependent Hartree-Fock Bogoliubov equations
7
Models of pairing functional
DDDI-matter, reproducing matter pair gap matter(BCS)
DDDI a-18
0.59
v0= -458.4 ← ann= -18.5 fm
Vn [  n ]  v0 1  0.845 n / 0.08
0 71
0.71
on density
DDDI-mix Linear weak dependence
1
v0= -292 ← ann = -1.4 fm
Vn [  n ,  p ]  v0 1  0.5 n   p / 0.16 
DIDI volume No density dependence
DIDI-volume
v0= -195 ← ann = -0.63 fm
vol a-1
Vn [  n ,  p ]  v0



average pair
i gap iin Sn
S istopes
i t


Different low-density
low density limit
V( )
V(r)
vol a-1
DDDI a-18
Distance from nuclear center
S ti l correlation
Spatial
l ti off Cooper
C
pair
i
Spatial correlation in 142Sn
Probability distribution of the Cooper pair wave function
2
pair (r1 , r2 )   (r1 ) (r2 )


2
One neutron is fixed at r1=7 fm (slightly outside)
Plotted on the plane of r2
z2
Probability for Cooper pair to be
correlated at short distances r < a few fm
is significantly enhanced at R > Rsurf
x2
DDDI a-18
8
Large probability at short relative distances
Probability distribution in R-r

  2
2 2
Pc ( R , r )  R r  d |  pair ( R  , R  r2 ) |

r
2
10
142Sn
R
(fm))
8
6
4
2
0
r (fm)
Probability for Cooper pair to be
correlated at short distances r < a few fm
is significantly enhanced at R > Rsurf
R
n
r
n
9
di-neutron correlation vs. single-j pairing
Single-j J=0 pair (3p3/2)2
R
(fm)
full HFB
r (fm)
10
Coherence length, not the best way to see
Rms radius of ‘Cooper
Cooper pair
pair’ as a function of R
142Sn
  r2 
2
r
 Pc ( R, r )dr

R
Pc ( R , r ) dr
n
r
n
V0 varied
R
It is influenced by geometrical effect of finite volume
[1] Pillet, Sandulescu, Schuck, Berger, PRC81 (2010)
But also by the surface spatial correlation.
[2] Hagino et al J Phys.G37 (2010)
r
12
11
Pair contact probability r < 2.6 fm
Probability of pair at short relative
distances within the interaction range
reffff
p( R) 
n
R
 Pc ( R, r )dr
Effective
range 2.6fm
r < reff
n
R
0
142Sn
DDDI a-18
r (fm)
Volume pairing a-1
Single-j Cooper pair
13
Pair transfer as a p
probe to the
surface pairing
Pair transfers in neutron-rich Sn isotopes: 132Sn~
Pair removal
(p t)
(p,t)
E. Khan et. al, PRC69, 014314 (2004); ibid PRC80,044328 (2009)
B. Avez, et. al, PRC78,044318 (2008)
MM, Y.Serizawa PRC82, 024318 (2010)
H. Shimoyama, MM, arXiv:1106.1715
Pair addition
(t p)
(t,p)
14
Pair transfer exp.
p on n-rich nuclei
11Li
I. Tanihata et al. PRL 100, 192502 (2008)
11Li
6,8He
+p→
9Li
Sensitivity to nn-correlation
+t
A. Chatterjee et al. PRL 101, 032701 (2008)
A. Lemasson et al. PLB 697, 454 (2011)
6He
+ 65Cu → 67Cu* + 4He → 66Cu + 4He + n
→ 66Cu* + 5He → 66Cu + 4He + n
Dominance of 2n-transfer
2n transfer
32Mg
K. Wimmer et al. PRL 105, 252501 (2010)
30Mg
+t→
32Mg
+p
Shape coexisting 0+2 state
Collective twotwo-neutron transfers in surperfulid
nuclei
15
This standard picture of the two‐neutron transfer modes may be modified ? Pairing vibration
Pairing vibration
+
+
02
weak
+
02
02
weak
+
+
0gs
A-2
0gs
+
0gs
Strong ~2
Pairing rotation
A
strong
A+2
Pairing rotation
superfluid phase
D. R. Bes , et. al, NP(1966)
,
, (
)
R. A. Broglia , at. Al, NP(1973)
open-shell
p
nuclei
E
Pairing
vibration Pairing rotation
17
Im⊿
R ⊿
Re⊿
16
Ground state pair transfer
02
2n-add/removal : transition density and strength


0 gs , N  2  (r )  (r ) 0 gs , N

B (P0)  0 gs
21
21



 0 HFB  (r )  (r ) 0 HFB  ~ (r )

gs-gs
gs
gs transition
   
~ (r )r 2 dr
Y

(
r
)

(
r
)
d
r
0

4


gs
 00


0gs
0gs
134Sn
Dobaczewski et al, PRC53, 2809 (1996)
Pair transfer transition density (form factor)
02
add/remove 136
Sn
Hartree-Fock single-particle
energy in 142Sn
es.p.
sp
142-150Sn
0
120-130Sn
134-140Sn
2n add
2n‐add
(0.09)
(0.01)
‐0.23
‐0.84
p1/2
p3/2
‐2.62
f7/2
[MeV]
N=８２
‐5
-10
R1/2
‐7.68
h11/2
Ground state pair transfer strength in >132Sn
2n-add/removal transfer amplitude and strength


0 gs , N  2   (r )  (r ) 0 gs , N
02



 0 HFB   (r )  (r ) 0 HFB  ~ (r )
21

 

B (P0)  0 gs  Y00  ( r )  (r )dr 0 gs  4  ~ (r )r 2 dr
134Sn
Ground state 2n transfer is
significantly increased in
very n-rich
i
isotopes(
( >140Sn)
S )
A
21
gs-gs transition
0gs
Pair transfer strength
02
0gs
add/remove 136
Sn
Pair gap squared
2
132Sn
17
Pair transfer modes in Skyrme HFB +QRPA
Given the Skyrme functional
 + pairing functional

 
E  ESkyrme [  ,  ,  , , j , s , J ]  E pair [  , ~, ~ * ]
Hartree-Fock-Bogoliubov
Hartree
Fock Bogoliubov ground state
Linear response eq. Landau-Migdal approx.
 (r ) 
(r ' )  Vext (r ' ) 
 ~ 









(
r
)
dr
'
R
(
r
,
r
'
,
)
(
r
'
)
0


 

*
*
~ (r ) 


 (r' )






Response function (ph,
(ph pp,
pp hh)
 
R0 ( r , r ' ,  )   dEG ( r , r ' , E )G ( r ' , r , E   )
C
particle-hole density
 (r , t ) 

(r ) (r )
pair -addition density
~* (r, t )    (r )  (r )
pair
i -removall density
d
it
~(r, t )   (r ) (r )
pair-addition transition density


L , N  2 YLM (r )  (r ) 0 gs , N





pair-removal transition density


L , N  2 YLM (r ) ( r ) 0 gs , N
N
02
02
21+
21+
0gs
0gs
Add/remove
N+2
18
19
Pair vibration: Excited 0+ pair transfer
Strength function for the pair transfer
02
02
0gs
Pair addition/removal operator
A‐2
25
10
addition
5
0
1
2
3
4
5
6
134Sn
15
7
removal
8
9
strength
h
strength
h
10
15
A+2
20
120Sn
15
10
A
25
20
5
0gs
0gs
5
0
5
10
15
20
E ( MeV)
E ( MeV)
addition
1
2
3
4
5
6
7
8
9
removal
20
E ( M V)
E ( MeV)
The pair addition strength to the pair vibration 0+ state in 134Sn is large. The pair vibrational state of 134Sn is a narrow resonance though it is located
above the one‐neutron separation energy.
21
As we explain below, this pairing vibrational mode in 134Sn is quite anomalous.
20
Anomalous Pairing Vibration
02
02
0gs
0gs
A
gs‐gs
A+2
+
Pair addition strength of pairing vibrational mode
Ratio ( 02 vs gs ) less than 10 % in stable nuclei
R. A. Broglia, O. Hansen, and C. Riedel (1973)
5
Sn isotopes
1
138Sn
s
strength
4
22
3
Large strength ~several
several times times
0.8
136Sn
134Sn
140Sn
BPad 0; gs  PV 
BPad 0; g
gs  g
gs
0.6
132Sn
2
0.4
Anomalous
1
0.2
0
0
100
110
120
A
6090％
130
140
150
100
110
120
130
A
140
150
Anomalous Pairing Vibration
Transition density of pairing vibrational mode
21
Hartree-Fock single-particle energy
in 134Sn
es.p.
2n‐add
(0.66)
Very long tail r～ 15 fm
r2P(ad)00(r) [ffm‐1]
0.6
0.5
120‐130Sn
0.4
132‐140Sn
p1/2
p3/2
‐2.14
f7/2
[MeV]
‐5
N=８２
0.3
‐7.68
0.2
0.1
h11/2
The transition densities to the pair vibrational mode of 132‐140Sn have a long tail
have a long tail.
0
-0.1
0
23
0
(0.27)
‐0.36
2
4
6
8 10
r [fm]
12
14
16
By adding two‐neutrons in the weakly bound p orbits, we can h
have a long tail.
l
il
The weakly bound p orbits play an important role !!
22
Relation to the enhanced ground
state transfer
02
02
0gs
0gs
A
gs‐gs
A+2
25
es.p.
Pair ad
ddition sstrength
Sn isotopes
(0.66)
20
0
15
Ground state transfer
p orbits occupied
in ground states
p orbits occupied
in excited states
10
(0.27)
‐0.36
p1/2
p3/2
‐2.14
f7/2
[MeV]
‐5
5
N=８２
pairing vibration
0
‐7.68
100
24
2n‐add
110
120
130
140
150
h11/2
A
The anomalous pairing vibration in 132‐140Sn appears as a precursor of large enhancement of the ground state transfer beyond A=140. 23
DWBA cross section
Ags ( p , t ) A-2gs
Code TWOFNR
One-step DWBA
Z
Zero-range
approx.
24
Conclusions
Neutron rich-nuclei exhibit enhanced neutron p
pairing
g in the
surface region.
Spatial
Spat a correlation
co e at o of
o Cooper
Coope pa
pair at short
s o td
distances
sta ces
A promising probe is two-neutron transfers in neutron-rich
>132Sn isotopes.
132--140—
140 Sn
 Enhanced ground state transfer in 132
 Anomalous pairing vibration in 134--140Sn as a precursor
Weakly bound p-orbits play a central role
Ref. H.Shimoyama, MM, arXiv:1106.1715
MM, Y.Serizawa PRC82, 024318 (2010)