Influence of the neutron-pair
transfer on fusion
V. V. Sargsyan*, G. G. Adamian, N. V. Antonenko
In collaboration with W. Scheid, H. Q. Zhang, D. Lacroix, G. Scamps
*Joint Institute For Nuclear Research
ECT*, Trento, Italy
Outline
I. Quantum Diffusion Approach: Formalism
II. Role of deformations of colliding nuclei
III. Role of neutron pair transfer
IV. Summary
The assumptions of the QD approach
The quantum diffusion approach based on the following assumptions:
1. The capture (fusion) can be treated in term of a single collective
variable: the relative distance R between the colliding nuclei.
2. The internal excitations (for example low-lying collective modes
such as dynamical quadropole and octupole excitations of the
target and projectile, one particle excitations etc. ) can be
presented as an environment.
3. Collective motion is effectively coupled with internal excitations
through the environment.
The formalism of quantum-diffusion approach
The full Hamiltonian of the system:



Htot  H coll R, P  Hinter b , b  Vcoupling R, b  b

The formalism of quantum-diffusion approach
The full Hamiltonian of the system:



Htot  H coll R, P  Hinter b , b  Vcoupling R, b  b
H coll
1 2  2 2

P 
R
2
2
 The collective subsystem
(inverted harmonic oscillator)

The formalism of quantum-diffusion approach
The full Hamiltonian of the system:



Htot  H coll R, P  Hinter b , b  Vcoupling R, b  b
H coll
1 2  2 2

P 
R
2
2
H inter    b b

 The collective subsystem
(inverted harmonic oscillator)
 The internal subsystem
(set of harmonic oscillators)

The formalism of quantum-diffusion approach
The full Hamiltonian of the system:



Htot  H coll R, P  Hinter b , b  Vcoupling R, b  b
H coll
1 2  2 2

P 
R
2
2
 The collective subsystem
(inverted harmonic oscillator)
H inter    b b
 The internal subsystem
Vcoupling  R  b  b 
 Coupling between the subsystems

ν
Adamian et al., PRE71,016121(2005)
Sargsyan et al., PRC77,024607(2008)
(set of harmonic oscillators)
(linearity is assumed)

The random force and dissipative kern
 The equation for the collective momentum contains dissipative kern and
random force:
t

dP
  2 R(t )   K (t   ) R( )d   F b (0), b (0), t
dt

0
22
K t ,   
cos t   
 
f t  f  t    ,  n ei (t t )
--- Dissipative Kern
--- Random force

The random force and dissipative kern
 The equation for the collective momentum contains dissipative kern and
random force:

t
dP
  2 R(t )   K (t   )R( )d   F b (0), b (0), t
dt

0
22
K t ,   
cos t   
 
f t  f  t    ,  n ei (t t )

--- Dissipative Kern
--- Random force
One can assumes some spectra for the environment and replace the
summation over the integral:


2
( w) 2
1
2
  dw  ( w)
...   dw 2
...

2
w
 0
 w
 
0
1
--- relaxation time for the internal subsystem
The analytical expressions for the first and
second moments in case of linear coupling
R(t )  At R0  Bt P0
 RR (t ) 
2 2 2

t
t

0
0
0
 d B   d B   d

  
coth
cos (    )
2
2


 
 2T 
3
At    i si ( si   )    e si t
Functions determine the dynamic
of the first and second moments
i 1
Bt 
1
3
 (s


i 1
i
i
  ) e si t
1  s1  s2 s1  s3 1
2  s2  s1 s2  s3 1
3  s3  s1 s3  s2 
1
si
--- are the roots of the following
equation
s   s 2   2   s /   0
Nucleus-nucleus interaction potential:
V R, Z i , Ai , J   VNucl
 J ( J  1)
 VCoul 
2
2R
2
Double-folding formalism used for nuclear part:
VNucl
 

 

  dr1dr2 1 (r1 )F (r1  r2 )  2 ( R  r2 )
Nucleus-nucleus potential:
1. density - dependent effective nucleon-nucleon interaction
2. Woods-Saxon parameterization for nucleus density
Adamian et al., Int. J. Mod. Phys E 5, 191 (1996).
Approximation:
realistic nucleus-nucleus potential  inverted oscillator
 The real interaction between nuclei can be approximated by the
inverted oscillator.
The frequency of oscillator is found from
the condition of equality of classical action
Ec.m.
rin
Rb
rex
rin
Rb
rex
The capture cross section
The capture cross-section is a sum of partial capture cross-sections
 Ec.m.    c Ec.m. , J     2  2 J  1Pcap Ec.m. , J 
J
J
 2  2 2  Ec.m.  --- the reduced de Broglie wavelength
Pcap Ec.m. , J 
--- the partial capture probability at fixed energy and
angular momenta
The partial capture probability obtained by integrating the propagator
G from the initial state (R0,P0) at time t=0 to the finale state (R, P) at
time t:
Pcap  lim
t 
rin

 dR  dP GR, P, t R , P ,0
0


0
Propagator for the inverted oscillator
 For the inverted oscillator the propagator has the form:
G P, R, t P0 , R0 , t  0 
1
2  RR (t ) PP (t )   2PR (t )





1
2
2













exp 

t
R

R
(
t
)


t
P

P
(
t
)

2

t
P

P
(
t
)
R

R
(
t
)
PP
RR
PR

2
 2[ RR (t ) PP (t )   PR (t )]

R(t ),
P(t )
1
 ij (t )
2
--- the mean value of the collective coordinate and
momentum
--- the variances and
 ij (t  0)  0,
i, j  R, P 
Dadonov, Man’ko, Tr. Fiz. Inst. Akad. Nauk SSSR 167, 7 (1986).
 The expression for the capture probability
Pcap
 R(t ) 
1
 lim erfc 

t  2
  RR (t ) 
Initial conditions for two regimes of interaction
75
Ec.m. > U(Rint) -- relative motion is coupled
with other degrees of freedom
1.
R0  Rint , P0  2 Ec.m.  U ( Rint )
U (MeV)
70
65
2.
60
55
50
Ec.m. < U(Rint) -- almost free motion
Rint
rex
rin
10
12
14
16
R0  rex , P0  0
18
R
Nuclear forces start to act at Rint=Rb+1.1 fm, where the nucleon density of
colliding nuclei reaches 10% of saturation density.
  2 MeV
  15 MeV
cap (mb)
16O+208Pb
10
3 16
10
1
10
-1
10
-3
10
-5
reaction
 Reactions with
spherical nuclear are
good test for the
verification of the
approach.
 Using these reaction
we fixed the
parameters used in
calculation.
208
O+ Pb
64
72
80
88
96 104
Ec.m. (MeV)
The change of the slope of the excitation function means, that at sub-barrier
energies the diffusion becomes a dominant component.
10
3
10
2
10
1
10
0
48
10
-1
10
-2
10
-3
208
Ca+ Pb
cap (mb)
cap (mb)
Reactions with spherical nuclei
170 175 180 185 190
Ec.m. (MeV)
10
3
10
2
10
1
10
0
40
90
Ca+ Zr
95
100
105
110
Ec.m. (MeV)
 Reactions with the spherical nuclei more clearly shows the behavior of the excitation
function.
The features of quantum diffusion approach
1. The coupling with respect to the relative coordinate results in a
random force and a dissipation kernel.
2. The integral term in the equations of motion means that the system is
non-Markovian and has a “memory” of the motion over the trajectory
preceding the instant t.
3. Predictive power.
Our approach takes into account the fluctuation and dissipation effects
in the collisions of heavy ions which model the coupling with various
channels.
Sargsyan et. al., EPJ A45, 125 (2010)
Sargsyan et. al., PRA 83, 062117 (2011)
Sargsyan et. al., PRA 84, 032117 (2011)
Role of deformation of nuclei in capture process
85
The lowest Coulomb
barrier
U (MeV)
The highest Coulomb
barrier
80
 2
 2
0
0
238
U
70
65
55
1
O+
75
60
2
16
side-side
spherical
pole-pole
9 10 11 12 13 14 15 16 17
R (fm)
 At fixed bombarding energy the
capture occurs above or below the
Coulomb barrier depending on
mutual orientations of colliding
nuclei !
 cap ( Ec.m. )   d1 sin 1  d 2 sin  2 cap ( Ec.m. ,1 , 2 )
Reactions with deformed nuclei
40
154
Ar+ Sm
40
144
Ar+ Sm
112 120 128 136 144
Ec.m. (MeV)
cap (mb)
cap (mb)
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
10
3
10
2
10
1
10
0
16
10
-1
10
-2
154
O+ Sm
16
144
O+ Sm
50 55 60 65 70 75 80 85 90
Ec.m. (MeV)
 The effect depends on the charges and deformations of the colliding
nuclei.
 The used averaging procedure seems to work correct.
Sargsyan et. al., PRC 85, 037602 (2012)
Role of neutron transfer
 Neutrons are insensitive to the Coulomb barrier and, therefore, their
transfer starts at larger separations before the projectile is captured by
the target nucleus.
 It is generally thought that the sub-barrier capture (fusion) cross
section increases because of the neutron transfer.
 The present experimental data (for example 60Ni + 100Mo system,
Scarlassara et. al, EPJ Web Of Conf. 17, 05002 (2011)) specify in complexity of
the role of neutron transfer in the capture (fusion) process and
provide a useful benchmark for theoretical models.
Why the influence of the neutron transfer is strong in some
reactions, but is weak in others ?
Large enhancement of the excitation function for the
40Ca+96Zr reaction with respect to the 40Ca+90Zr reaction!
Large enhancement of the excitation function for the
40Ca+96Zr reaction with respect to the 40Ca+90Zr reaction!
The model assumptions
 Sub-barrier capture depends on two-neutron transfer with positive
Q-value.
 Before the crossing of Coulomb barrier, 2-neutron transfer occurs and
lead to population of first 2+ state in recipient nucleus (donor nucleus
remains in ground state).
 Because after two-neutron transfer, the mass numbers, the
deformation parameters of interacting nuclei, and, respectively, the
height and shape of the Coulomb barrier are changed.
Sargsyan et. al., PRC 84, 064614 (2011)
Sargsyan et. al., PRC 85, 024616 (2012)
Sargsyan et. al., PRC 84, 064614 (2011)
Sargsyan et. al., PRC 85, 024616 (2012)
2
10
1
10
0
10
-1
10
-2
10
-3
10
3
10
2
10
1
10
0
40
Ca+
194
Pt
with 2 neutrons transfer
without transfer
cap (mb)
10
10
2
10
1
10
0
10
-1
10
-2
40
Ca+
192
Os
with 2 neutrons transfer
without transfer
160 165 170 175 180 185
150 156 162 168 174 180 186
Ec.m. (MeV)
Ec.m. (MeV)
58
Ni+
130
Te
with 2 neutrons transfer
without transfer
cap (mb)
cap (mb)
cap (mb)
Reactions with two neutron transfer
10
3
10
2
10
1
10
0
58
132
Ni+ Sn
with 2 neutrons transfer
without transfer
162 168 174 180 186 192
150 160 170 180 190 200
Ec.m. (MeV)
Ec.m. (MeV)
Pair transfer ?
 Reactions with Q1n < 0 and Q2n > 0
 Good agreement between calculations and experimental data is
an argument of pair transfer
cap (mb)
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
10
40
Ca+
124
Sn
with 2 neutrons transfer
without transfer
108
114
120
Ec.m. (MeV)
126
132
 By
describing
subbarrier
capture, we
demonstrate indirectly
strong spatial 2-neutron
correlation and nuclear
surface enhancement of
neutron pairing
 Indication for Surface
character of pairing
interaction ?
Enhancement or suppression ?
 2n-transfer can also suppress capture
 If deformation of the system decreases due to neutron transfer,
capture cross section becomes smaller
32
S(  2  0.31)110Pd(  2  0.26)34 S(  2  0.25)108Pd(  2  0.24)
32
S(  2  0.31)100Mo(  2  0.26)34 S(  2  0.25) 98 Mo(  2  0.17)
32
100
S+
2
Mo
with 2 neutrons transfer
without neutron transfer
70
75
80
85
Ec.m. (MeV)
90
95
cap (mb)
cap (mb)
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
32
110
S+ Pd
with 2 neutrons transfer
without neutron transfer
80
85
90
Ec.m. (MeV)
95
Summary
 The quantum diffusion approach is applied to study the
capture process in the reactions with spherical and
deformed nuclei at sub-barrier energies. The available
experimental data at energies above and below the
Coulomb barrier are well described.
 Change of capture cross section after neutron transfer
occurs due to change of deformations of nuclei. The
neutron transfer is indirect effect of quadrupole
deformation.
 Neutron transfer can enhance or suppress or weakly
influence the capture cross section.
Reactions with weakly bound projectiles
The break-up probability:
9
Be +
th
exp
PBU 1   cap
 cap
124
Sn
70
50
PBU
PBU
60
40
50
40
4
Ecm-Vb (MeV)
9
Be
9
Be
9
Be
9
Be
-4
-2
+
+
+
+
8
144
Sm
209
Bi
208
Pb
89
Y
0
2
4
Ecm-Vb (MeV)
6
165
Pb
Ho
12
PBU
PBU
80
70
60
50
40
30
20
10
0
-6
0
208
7
Li +
Li +
30
30
-4
6
8
10
90
80
70
60
50
40
30
20
10
0
-4
6
Li +
Li +
6
Li +
144
6
198
7
209
Li +
9
Li +
0
4
8
12
16
Ecm-Vb (MeV)
Sm
Pt
209
Bi
Bi
Pb
208
20
24
 There are no systematic trends of breakup in reactions studied!
 For some system with larger (smaller) ZT breakup is smaller (larger).?
Reactions with weakly bound projectiles
The break-up probability:
9
Be +
th
exp
PBU 1   cap
 cap
124
Sn
70
50
PBU
PBU
60
40
50
40
4
Ecm-Vb (MeV)
9
Be
9
Be
9
Be
9
Be
-4
-2
+
+
+
+
8
144
Sm
209
Bi
208
Pb
89
Y
0
2
4
Ecm-Vb (MeV)
6
165
Pb
Ho
12
PBU
PBU
80
70
60
50
40
30
20
10
0
-6
0
208
7
Li +
Li +
30
30
-4
6
8
10
90
80
70
60
50
40
30
20
10
0
-4
6
Li +
Li +
6
Li +
144
6
198
7
209
Li +
9
Li +
0
4
8
12
16
Ecm-Vb (MeV)
Sm
Pt
209
Bi
Bi
Pb
208
20
24
 There are no systematic trends of breakup in reactions studied!
 For some system with larger (smaller) ZT breakup is smaller (larger).?
  2 if R  RB
V-VB(MeV)
 Frictions is a result of the overlapping of the
nuclear densities.
 For the light systems, the coupling parameter
should depend on the relative distance
between the colliding nuclei and, as a result
the friction becomes coordinate-dependent.
0
 (MeV)
Friction depending on the relative
distance of colliding nuclei
2
-4
16
O+
2
3
-6
208
16
O+ O
Pb
1
0
1
4
5
6
7
8
9
10
R-RB (fm)
cap (mb)
2
  R  RB  
2
1  0.15 R  RB 
 Comparing the results, obtained with the
analytic expressions (constant friction) for the
equations of motions with the numerical one
(coordinate- dependent), one can assume that
the linear coupling limit is suitable for the
heavy systems and not very deep sub-barrier
energies.
16
-2
10
3
10
1
10
-1
10
-3
10
-5
16
208
O+
Pb
constant friction
R - dependent friction
64
72
80
88
Ec.m. (MeV)
96 104
10
3
10
1
10
-1
10
-3
48
48
Ca+ Ca
cap (mb)
cap (mb)
Calculations with constant and R-dependent
friction
R -dependent friction
constant friction
45
50
55
10
3
10
1
10
-1
10
-3
36
R -dependent friction
constant friction
35
60
48
S+ Ca
40
10
3
10
1
10
-1
10
-3
40
48
Ca+ Ca
cap (mb)
cap (mb)
Ec.m. (MeV)
R -dependent friction
constant friction
45
50
55
60
Ec.m. (MeV)
65
10
3
10
1
10
-1
10
-3
32
45
50
Ec.m. (MeV)
55
48
S+ Ca
R -dependent friction
constant friction
35
40
45
50
Ec.m. (MeV)
55