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Microscopic time-dependent analysis of neutrons transfers at low-energy nuclear

reactions with spherical and deformed nuclei

samarin@jinr.ru

V.V. SamarinJoint Institute for Nuclear Research, Dubna

The aim of report is application of based quantum mechanics equations for neutron transfers description.

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Motivation

• Neutron transfers in the low energy nuclear reactions allow us to obtain new

isotopes of atomic nuclei with increased neutron content.

• The probability of neutron transfer is highest during so-called grazing nuclear

collisions. In this case the distances between the surfaces of the atomic nuclei

do not exceed the range of the action of nuclear forces (1–2 fm).

• The most probable transition is the one between the nuclei of the external, most

weakly bound neutrons.

• A new possibility for theoretical study of this reactions is provided by numerical

solution for the non-stationary Schrodinger equation for external neutrons [1].

• In this study the spin-orbital interaction and Pauli's exclusion principle were

taken into consideration for spherical and deformed nuclei.

1. V.Samarin, V. Zagrebaev, Walter Greiner. Phys. Rev., C 75, 035809 (2007) .

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We studied four nuclear reactions (three from them more detail)

and used two methods:

1. 6Не + 197Au2. 6Не + 208Pb3. 18O + 48Са4. 48Са + 238U

1. Time-dependent Schrödinger equation with spin-orbital interaction and Pauli's exclusion principle

2. Spherical and deformed nuclei shell model for external nucleons

More detail and less detail

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Time-dependent Schrödinger equation with spin-orbital interaction

21 1

1 1

2 2 2 2

( , )2 2

,2 2

b V Vi V r t it m x y y x

b V V b V Vi

y z z y x z z x

22 2

2 2

1 1 1 1

( , )2 2

.2 2

b V Vi V r t it m x y y x

b V V b V Vi

y z z y x z z x

1. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2010. V. 74. P. 567.2. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2011. V. 75. P. 964.3. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2012. V. 76. P. 450.

Time-dependent Schrödinger equation with spin-orbital interaction

is numerically solved by difference method [1-3] for external neutronsof spherical nuclei at their grazing collisions with energies near to a Coulomb barrier.

in Cartesian coordinates

22 20 02 2 2

0

0,0222

b R Rm R c

R0=1 fm 2

1 1

2 2

ˆ( , ) ( , )2 LSi V r t V r t

t m

ˆˆ ( )2LS

bV V p

5

2

1 2 1

1 1( , )

2 2 2i

z z

b bV z i V i e i V V V

m z

2

2 1 2

1 1( , )

2 2 2i

z z

b bV z i V i e i V V V

m z

1 1( , ) exp ( 1 2)f z i 2 2 ( , ) exp ( 1 2)f z i

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( , ) ( ) ( )N

k kk

f z p z y

1 2 0,1,

2 1 10

( , ) ( ) ( )N

k kk

f z q z y

0

0

( ) ( )k n kny y d

Samarin V.V. Phys. Atom. Nucl., 2010, 73, p. 1416.

Deformed nuclei shell model

Schrödinger equation for the nucleon energy levels and wave functions at arbitrary axial-symmetrical field with spin-orbit interactionare calculated by a numerical solution of a Schrödinger equation for an arbitrary axial-symmetrical field with spin-orbit interactions, basedon decomposing on Bessel functions and difference scheme along internuclear axis.

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Time-dependent Schrödinger equation solution is the way of visual simulation of neutron transfer for reactions with halo nuclei 6Не. At first, for frontal and grazing collisions with energy below Coulomb barrier you can see, that 6Не neutron wave function of 1p3/2 state is arranged between projectile and target nuclei. There is stable space structure, like to 3d state of Au or Pb, with zero angular momentums projection to internuclear axis.

6Не + 208Pb, Еcm=18 MeV, frontal collision

6Не + 197Au, Еcm=18 MeV, grazing collision

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At second, for frontal and grazing collisions with energy in vicinity of Coulomb barrier you can see, that 6Не neutron wave function of 1p3/2

state is arranged between projectile and target nuclei with some space structure too.

6Не + 208Pb, Еcm=20 MeV, frontal collision

6Не + 197Au, Еcm=21 MeV, grazing collision

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At next, for grazing collision with energy above of Coulomb barrier you can see, that 6Не neutron wave function of 1p3/2 state is arranged between projectile and target nuclei with stable space structure like to rotated 3d state of Au.

6Не + 197Au, Еcm=30 MeV,grazing collision

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At last, for grazing collision with energy more above of Coulomb barrier you can see, that 6Не neutron wave function of 1p3/2 state is arranged between projectile and target nuclei with some space structure like to rotated 3d state of Au too.

6Не + 197Au, Еcm=60 MeV, grazing collision

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If distances between nuclear centers and surfaces is decreasing, potential barrier between two potential walls is decreasing too ( Fig. 1), transfer probability is increasing ( Fig. 2). This reason lead to cross section of neutron transfer at 6He+197Au, which satisfactorily agrees with experimental data (Fig. 3).

Spacing interval between nuclear surfaces

Probabilities p of the transfer of the external neutron of the 6Не as functions of the minimum distance s between the surfaces of 6Не, 197Au nuclei for the energies in the center of mass system near barrier energies from 18 to 22 MeV ( ), 30 MeV ( ) and 60 MeV ( ).

Fig. 1.

Fig. 2. Fig. 3.

p

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1 21 : 1, 2 : 3p L L d R1R2

p

2 22 2 1 1 1

1 1

2 ,R R

L pR pR L LR R

1p3/2

197Au

3d5/2,3/2

6Не

20 22 24 26 280,0

0,1

0,2

3p3/2

1i13/2

3p1/2 1i

11/22g

9/2

3d3/2

4s1/2

2g7/2

3d5/2

Level number in 197Au

Fermilevel

E=14 MeV

p

=1/2 and =3/2 =1/2

20 22 24 26 280,0

0,1

0,2

Fermilevel

3d3/2

4s1/2

2g7/2

1i11/2

3d5/2

2g9/2

3p1/2

1i13/23p

3/2

Level number in 197Au

pE=18 MeV =1/2 and =3/2

=1/2

For reaction 6Не + 197Au, in frontal collision dominant transfer is: 1p3/2 (6Не) 3d5/2,3/2 (197Au).Calculated by time-dependent Schrödinger equation occupied states probabilities in Au for stripping neutron with full momentum projection are shown on Fig. 2.

Fig. 2

2 MeV

We may illustrate semiclassically dominant neutron transfer for neutron orbits with zero angular momentums projection to internuclear axis, which touching each other. Neutron energy and momentum for classical orbit in projectile equal approximately to neutron energy and momentum for classical orbit in target. In this case angular momentums relatively to centers of nuclei will be proportional to radii of nuclei (Fig. 1).

Fig. 1

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18O

Pauli's exclusion principle limit transfers to occupied statesfor reactions 18O+48Ca.

Preliminary: unlimited transfers

Begin ofcollision

End ofcollision

48Ca

1d5/2

Neutron states in spherical shell model

1d5/2

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Pauli's exclusion principle limit transfers to occupied statesfor reaction: 48Ca+248U Preliminary: unlimited transfers

End ofcollision

Begin oftransfer

Neutron states in spherical shell model

End ofcollision

Begin ofcollision

Neutron states in deformed shell model are studied in next slides

3d5/2

238U48Ca

48Ca 238U

Begin oftransfer

End ofcollision

1f7/2

48Ca

48Ca

238U

238U

2g9/2

2g9/2

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Fig. 1. Probabilities of neutrons pick-up (solid curves) and stripping (dashed curves) for 48Са at reactions 48Са + 18O (curves 1) и 48Са + 238U (curves 2) as function on minimum value of internuclear distance

Pauli's exclusion principle were taken into consideration by: 1. exception with transfer to occupied states in “frozen” nuclei shell structures (simple approximation). 2. time dependent many body wave function (M=2, 3)

(more correct approximation).

1 1 1

1

1

, ,1

, , det!

, ,

N

M n

M M N

r t r t

r r tM

r t r t

1 1 1, , , , ...M M M M MP r r t r r t dV dV

Fig. 2. Probabilities of neutrons pick-up (solid curves) and stripping (dashed curves) for 48Са at reactions 48Са + 238U as function on minimum value of internuclear distance

At reaction 48Ca+238U probabilities of neutrons stripping and pick-up are commensurable.

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Visual simulation of neutron transfers at reaction 18O+48Ca.Two external neutrons:1d5/2 from 18O and1f7/2 from 48Cawith moment projection=1/2, 3/2 are took into account

Visual simulation of neutron transfers at reaction 48Ca+238UThree external neutrons:1f7/2 from 48Ca and2g9/2, 1i11/2 from 238Uwith moment projection=1/2, 3/2 are took into account

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Nucleons transfers in low-energy nuclear reactions 40,48Ca+238U with deformed nucleus 238U

a) Some upper energy levels for neutron states with module of total angular

momentum projection to symmetry axis at hypothetic nuclei 238U with quadrupole

deformation 2 and octupole deformation 4 = 2/2 (a) and at real nucleus 238U with

2 = 0.215, 4 = 0.095 (b), dashed lines correspond unoccupied levels

a b

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Visual simulation of neutron pick-up at reaction 40Ca+238U during frontal collision at energy in the center of mass system E=192 MeV.In the beginning external neutron of 238U is in initial

state 1j15/2 with angular

momentum projections on symmetry axis =5/2. Angle between symmetry axes of deformed nucleus 238U and initial velocity of 40Ca nucleus equal 45o.

Neutron pick-up at reaction 48Ca+238U

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Neutrons pick-up in low-energy nuclear reactions 40Ca+238U with deformed nucleus 238U

a) Probability density of the external neutron of 238U for initial state 1j15/2 with angular momentum projections on symmetry axis =5/2 during frontal collision with the 40Ca at energy in the center of mass system E=192 MeV. Angle between symmetry axes of deformed nucleus 238U and initial velocity of 40Ca nucleus equal 45o.b) The probabilities of neutron pick-up at reaction 40Са+238U as a function of minimum distance s between nuclear surfaces. Angles between symmetry axes of deformed nucleus 238U and initial velocity of 40Ca nuclei equal 45o (solid line) and 90o (dashed line).

a b

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Visual simulation of neutron stripping at reaction 48Ca+238U during frontal collision at energy in the center of mass system E=175 MeV.

In the beginning external neutrons of 48Ca is in initial state 1f7/2. Angle

between symmetry axes of deformed nucleus 238U and initial velocity of 48Ca nucleus equal 0o.

Neutron stripping at reaction 48Ca+238U

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Visual simulation of neutron stripping at reaction 48Ca+238U during frontal collision at energy in the center of mass system E=192 MeV.In the beginning external neutrons of 48Ca is in

initial state 1f7/2. Angle

between symmetry axes of deformed nucleus 238U and initial velocity of 48Ca nucleus equal 45o.

Neutron stripping at reaction 48Ca+238U

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Neutrons stripping in low-energy nuclear reactions 48Ca+238U with deformed nucleus 238U

a) Probability density of the external neutrons of 1f7/2 shell of 48Ca during a frontal collision with the 238U at energy in the center of mass system E=192 MeV. Angle between symmetry axes of deformed nucleus 238U and initial velocity of 40Ca nucleus equal 45o.b) The probabilities of neutron stripping at reaction 48Са+238U (a) as a function of minimum distance s between nuclear surfaces. Angles between symmetry axes of deformed nucleus 238U and initial velocity of Ca nuclei equal 45o (solid line), 90o (dashed line) and 0 (dotted line).

ba

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Conclusion

• A new possibility for theoretical study of this reactions is provided by numerical solution for

the non-stationary Schrödinger equation for external neutrons.

• In this study the spin-orbital interaction and Pauli's exclusion principle were taken into

consideration. Time-dependent Schrödinger equation is numerically solved by difference

method for external neutrons of spherical nuclei 6He, 18O, 48Са and deformed nucleus 238U

at their grazing collisions with energies near to a Coulomb barrier.

• The probabilities of transfer of neutrons at reactions 6He+197Аu, 18O+48Ca, 40,48Ca+238U are

determined as function on minimum internuclear distances.

• The calculation results of cross section for formation of the 198Au isotope in the 6Не+197Au

reaction agree satisfactorily with the experimental data in vicinity of the Coulomb barrier.

• At reactions 6He+197Аu, 18O+48Ca, neutrons are predominantly transferred from a smaller

nucleus to the greater nucleus. At reaction 48Ca+238U probabilities of neutrons stripping and

pick-up are commensurable.

• Nonstationary quantum approach applied in this work may be used for internal nucleons

too. It may be useful for nucleons transfer experimental data analysis.

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Thank you for

attantion!

Dubna