Influence of the neutron-pair transfer on fusion

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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

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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. Quantum Diffusion Approach: Formalism - PowerPoint PPT Presentation

Transcript of Influence of the neutron-pair transfer on fusion

Page 1: Influence of the neutron-pair transfer on fusion

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

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Outline

I. Quantum Diffusion Approach: Formalism

II. Role of deformations of colliding nuclei

III. Role of neutron pair transfer

IV. Summary

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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.

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The formalism of quantum-diffusion approach

The full Hamiltonian of the system:

bbRVbbHPRHH couplingintercolltot ,,,

Page 5: Influence of the neutron-pair transfer on fusion

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

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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)

Page 7: Influence of the neutron-pair transfer on fusion

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)

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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

Page 9: Influence of the neutron-pair transfer on fusion

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

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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

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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).

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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.

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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

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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).

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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

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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.

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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.

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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.

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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

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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)

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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 ?

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Large enhancement of the excitation function for the 40Ca+96Zr reaction with respect to the 40Ca+90Zr reaction!

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Large enhancement of the excitation function for the 40Ca+96Zr reaction with respect to the 40Ca+90Zr reaction!

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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.

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Sargsyan et. al., PRC 84, 064614 (2011)Sargsyan et. al., PRC 85, 024616 (2012)

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Sargsyan et. al., PRC 84, 064614 (2011)Sargsyan et. al., PRC 85, 024616 (2012)

Page 27: Influence of the neutron-pair transfer on fusion

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

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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

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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

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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.

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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:

Page 34: Influence of the neutron-pair transfer on fusion

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:

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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)

Page 36: Influence of the neutron-pair transfer on fusion

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