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
ECT*, Trento, Italy
*Joint Institute For Nuclear Research
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:
bbRVbbHPRHH couplingintercolltot ,,,
The formalism of quantum-diffusion approach
The full Hamiltonian of the system:
The collective subsystem (inverted harmonic oscillator)
bbRVbbHPRHH couplingintercolltot ,,,
22
2
221 RPH coll
The formalism of quantum-diffusion approach
The full Hamiltonian of the system:
The collective subsystem (inverted harmonic oscillator)
bbRVbbHPRHH couplingintercolltot ,,,
22
2
221 RPH coll
bbH inter The internal subsystem (set of harmonic oscillators)
The formalism of quantum-diffusion approach
The full Hamiltonian of the system:
The collective subsystem (inverted harmonic oscillator)
bbRVbbHPRHH couplingintercolltot ,,,
22
2
221 RPH coll
bbH inter The internal subsystem (set of harmonic oscillators)
νcoupling bbRV
Coupling between the subsystems (linearity is assumed)
Adamian et al., PRE71,016121(2005)Sargsyan et al., PRC77,024607(2008)
The random force and dissipative kern The equation for the collective momentum contains dissipative kern and random force:
tbbFdRtKtRdtdP t
),0(),0()()()(0
2
--- Random force
--- Dissipative Kern
ttK cos2,2
)(, e ttintftf
The random force and dissipative kern The equation for the collective momentum contains dissipative kern and random force:
One can assumes some spectra for the environment and replace the summation over the integral:
... 1...
)()( 0
22
2
0
22
wdw
wwwdw
tbbFdRtKtRdtdP t
),0(),0()()()(0
2
--- Random force
--- Dissipative Kern
ttK cos2,2
)(, e ttintftf
1 --- relaxation time for the internal subsystem
The analytical expressions for the first and second moments in case of linear coupling
00)( PBRAtR tt
)(cos2
coth2)( 22000
22
TdBdBdt
tt
RR
ts
iiiit
iessA
3
1
)(
ts
iiit
iesB
3
1
)(1
Functions determine the dynamic of the first and second moments
131211
ssss
132122
ssss
123133
ssss 0/22 sss
--- are the roots of the following equation
is
Nucleus-nucleus interaction potential:
Double-folding formalism used for nuclear part:
2
2
2)1(,,,
RJJVVJAZRV CoulNuclii
)()()( 22211121 rRrrFrrdrdVNucl
Nucleus-nucleus potential:1. density - dependent effective nucleon-nucleon interaction2. 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 frequency of oscillator is found from the condition of equality of classical action
. .c mE
bRbRexrinr inr exr
The real interaction between nuclei can be approximated by the inverted oscillator.
The capture cross sectionThe capture cross-section is a sum of partial capture cross-sections
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:
--- the reduced de Broglie wavelength
--- the partial capture probability at fixed energy and angular momenta
JEPJJEE capJ J
c ,12, c.m.2
c.m.c.m.
c.m.22 2 E
JEPcap ,c.m.
inr
tcap PRtPRGdPdRP 0,,,,lim 00
Propagator for the inverted oscillator
)()(2)()()]()()([2
1exp
)()()(210,,,,
222
200
tRRtPPttPPttRRtttt
ttttRPtRPG
PRRRPPPRPPRR
PRPPRR
For the inverted oscillator the propagator has the form:
)()(erfc
21lim
ttRP
RRtcap
The expression for the capture probability
--- the mean value of the collective coordinate and momentum
)( ,)( tPtR
)(21 tij --- the variances and PRji,tij , ,0)0(
Dadonov, Man’ko, Tr. Fiz. Inst. Akad. Nauk SSSR 167, 7 (1986).
Initial conditions for two regimes of interaction
10 12 14 16 18505560657075
U (M
eV)
R
0 , 00 PrR ex
)(2 , int..0int0 RUEPRR mc
1.
2.
rexrin
Rint
Ec.m. > U(Rint) -- relative motion is coupled with other degrees of freedom
Ec.m. < U(Rint) -- almost free motion
Nuclear forces start to act at Rint=Rb+1.1 fm, where the nucleon density of colliding nuclei reaches 10% of saturation density.
MeV 15MeV 2
16O+208Pb reaction
64 72 80 88 96 10410-5
10-3
10-1
101
103
cap
(mb)
Ec.m. (MeV)
16O+208Pb Reactions with spherical nuclear are good test for the verification of the approach.
Using these reaction we fixed the parameters used in calculation.
The change of the slope of the excitation function means, that at sub-barrier energies the diffusion becomes a dominant component.
Reactions with spherical nuclei
170 175 180 185 19010-310-210-1100101102103
cap
(mb)
Ec.m. (MeV)
48Ca+208Pb
95 100 105 110
100
101
102
103
cap
(mb)
Ec.m. (MeV)
40Ca+90Zr
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.
Sargsyan et. al., EPJ A45, 125 (2010)Sargsyan et. al., PRA 83, 062117 (2011)Sargsyan et. al., PRA 84, 032117 (2011)
Our approach takes into account the fluctuation and dissipation effects in the collisions of heavy ions which model the coupling with various
channels.
Role of deformation of nuclei in capture process
At fixed bombarding energy the capture occurs above or below the Coulomb barrier depending on mutual orientations of colliding nuclei !
),,(sinsin)( 21..2
2
021
2
01..
mccapmccap EddE
12
The lowest Coulomb barrier
The highest Coulomb barrier
9 10 11 12 13 14 15 16 1755606570758085
side-side spherical pole-pole
U (M
eV)
R (fm)
16O + 238U
Reactions with deformed nuclei
50 55 60 65 70 75 80 85 9010-210-1100101102103
16O+154Sm 16O+144Sm
cap
(mb)
Ec.m. (MeV)112 120 128 136 14410-4
10-310-210-1100101102103
40Ar+154Sm 40Ar+144Sm
cap
(mb)
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)
Reactions with two neutron transfer
162 168 174 180 186 192100
101
102
103
with 2 neutrons transfer without transfer c
ap (m
b)
Ec.m. (MeV)
58Ni+130Te
150 160 170 180 190 200100
101
102
103
with 2 neutrons transfer without transfer c
ap (m
b)
Ec.m. (MeV)
58Ni+132Sn
160 165 170 175 180 18510-310-210-1100101102
with 2 neutrons transfer without transfer
cap
(mb)
Ec.m. (MeV)
40Ca+194Pt
150 156 162 168 174 180 18610-2
10-1
100
101
102
with 2 neutrons transfer without transfer
cap
(mb)
Ec.m. (MeV)
40Ca+192Os
Pair transfer ? Reactions with Q1n < 0 and Q2n > 0
Good agreement between calculations and experimental data is an argument of pair transfer
By describing sub-barrier capture, we demonstrate indirectly strong spatial 2-neutron correlation and nuclear surface enhancement of neutron pairing
Indication for Surface character of pairing interaction ?
108 114 120 126 13210-310-210-1100101102103
with 2 neutrons transfer without transfer
cap
(mb)
Ec.m. (MeV)
40Ca+124Sn
Enhancement or suppression ? 2n-transfer can also suppress capture
If deformation of the system decreases due to neutron transfer, capture cross section becomes smaller
)17.0Mo()25.0S()26.0Mo()31.0S(
)24.0Pd()25.0S()26.0Pd()31.0S(
298
234
2100
232
2108
234
2110
232
70 75 80 85 90 9510-410-310-210-1100101102
with 2 neutrons transfer without neutron transfer
cap
(mb)
Ec.m. (MeV)
32S+100Mo
80 85 90 9510-410-310-210-1100101102
with 2 neutrons transfer without neutron transfer
cap
(mb)
Ec.m. (MeV)
32S+110Pd
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
-4 0 4 8 1230
40
50
-6 -4 -2 0 2 4 6 8 100
1020304050607080
9Be + 124Sn
P BU
Ecm-Vb (MeV)
9Be + 144Sm 9Be + 209Bi 9Be + 208Pb 9Be + 89Y
P BU
Ecm-Vb (MeV)
30
40
50
60
70
-4 0 4 8 12 16 20 240
102030405060708090
6Li + 208Pb 7Li + 165Ho
P BU
6Li + 144Sm 6Li + 198Pt 6Li + 209Bi
7Li + 209Bi 9Li + 208Pb
P BU
Ecm-Vb (MeV)
There are no systematic trends of breakup in reactions studied! For some system with larger (smaller) ZT breakup is smaller (larger).?
expcap
thcapBUP 1The break-up probability:
Reactions with weakly bound projectiles
-4 0 4 8 1230
40
50
-6 -4 -2 0 2 4 6 8 100
1020304050607080
9Be + 124Sn
P BU
Ecm-Vb (MeV)
9Be + 144Sm 9Be + 209Bi 9Be + 208Pb 9Be + 89Y
P BU
Ecm-Vb (MeV)
30
40
50
60
70
-4 0 4 8 12 16 20 240
102030405060708090
6Li + 208Pb 7Li + 165Ho
P BU
6Li + 144Sm 6Li + 198Pt 6Li + 209Bi
7Li + 209Bi 9Li + 208Pb
P BU
Ecm-Vb (MeV)
There are no systematic trends of breakup in reactions studied! For some system with larger (smaller) ZT breakup is smaller (larger).?
expcap
thcapBUP 1The break-up probability:
Friction depending on the relative distance of colliding nuclei
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.
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.
215.012
if 2
BB
B
RRRR
RR
64 72 80 88 96 10410-5
10-3
10-1
101
103
constant friction R - dependent friction
cap
(mb)
Ec.m. (MeV)
16O+208Pb
-6-4-20
16O+16O
V-V B(M
eV)
16O+208Pb
0 1 2 3 4 5 6 7 8 9 10
1
2
(M
eV)
R-RB (fm)
Calculations with constant and R-dependent friction
45 50 55 60
10-3
10-1
101
103
R -dependent friction constant friction c
ap (m
b)
Ec.m. (MeV)
48Ca+48Ca
35 40 45 50 55
10-3
10-1
101
103
R -dependent friction constant friction c
ap (m
b)
Ec.m. (MeV)
36S+48Ca
35 40 45 50 55
10-3
10-1
101
103
R -dependent friction constant friction c
ap (m
b)
Ec.m. (MeV)
32S+48Ca
45 50 55 60 65
10-3
10-1
101
103
R -dependent friction constant friction c
ap (m
b)
Ec.m. (MeV)
40Ca+48Ca
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