Transcript PPT - NUCLEAR REACTIONS VIDEO Project

Microscopic time-dependent analysis of
neutrons transfers at low-energy nuclear
reactions with spherical and deformed nuclei
V.V. Samarin
Joint Institute for Nuclear Research, Dubna
[email protected]
The aim of report is application of based quantum
mechanics equations for neutron transfers description.
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) .
2
We studied four nuclear reactions
(three from them more detail)
and used two methods:
6Не
1. Time-dependent
+ 197Au
6Не + 208Pb
Schrödinger equation
with spin-orbital
18O + 48Са
interaction and Pauli's
48Са + 238U
exclusion principle
2. Spherical and deformed
More detail and less detail
nuclei shell model for
external nucleons
1.
2.
3.
4.
3
Time-dependent Schrödinger equation with spin-orbital interaction
ˆ   b  (V ) pˆ 
V
Time-dependent Schrödinger equation with spin-orbital interaction LS

2 
  1 
  1   2
ˆ
i
  V (r , t )  VLS (r , t )   
   
t   2   2m
 2 
b
R02 
2
2m
2
R02c 2
 0,022 R02 
R0=1 fm
in Cartesian coordinates
2



b  V 1 V 1 
i
1   
  V ( r , t )  1  i 


t
2
m
2

x

y

y

x




b  V  2 V  2  b  V  2 V  2 
i 


 
,
2  y z
z y  2  x z
z x 
2



b  V  2 V  2 
i
2   
  V (r , t )   2  i 


t
2  x y
y x 
 2m

b  V 1 V 1  b  V 1 V 1 
i 


 
.
2  y z
z y  2  x z
z x 
is numerically solved by difference method [1-3] for external neutrons
of spherical nuclei at their grazing collisions with energies near to a Coulomb barrier.
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.
4
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.
Samarin V.V. Phys. Atom. Nucl., 2010, 73, p. 1416.
 2
b 1  
b i   
  1  
  V (, z )  i
V   1  i e i V  Vz   Vz    2  1

2
m
2

 
2
    

  z
 2
b 1  
b i   
  1  
  V (, z )  i
V    2  i e i V  Vz   Vz   1    2

2
m
2

 
2   z
    

1  f1 (, z)exp i(1 2)
N
f 1 (, z )   pk  ( z ) yk  ()
k 0
   1 2
2  f2 (, z)exp i(1 2)
N
f  2 (, z )   qk1 ( z ) yk1 ()
k 0
0
y
k
() yn ()d   kn
0
  0,1,
5
6Не
+ 208Pb, Еcm=18 MeV,
frontal collision
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 1p
3/2
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
+ 197Au, Еcm=18 MeV,
grazing collision
6
6Не
+ 208Pb, Еcm=20 MeV,
frontal collision
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
+ 197Au, Еcm=21 MeV,
grazing collision
7
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
p
Fig. 1.
Fig. 2.
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. 3.
10
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).
R2
R2
L2  pR2  pR1
 L1
 2 L1 ,
R1
R1
1 p : L1  1, L2  2 : 3d
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.
p 0,2
E=14 MeV
0,0
p
0,2
E=18 MeV
22
3d5/2,3/2
24
26
28
197
Level number in Au
3d5/2
2g7/2
0,1
0,0
3d3/2
=1/2 and =3/2
=1/2
=1/2 and =3/2
=1/2
3d3/2
Fermi
3d5/2
4s1/2
level
2g
3p3/21i13/2 3p1/22g9/2
1i11/2 7/2
20
R2
Fig. 1
p
197Au
R1
1p3/2
  2 MeV
0,1
6Не
4s1/2
Fermi
2g9/2
level
3p 1i13/23p1/2
1i11/2
3/2
20
Fig. 2
22
24
26
Level number in
11 28
197
Au
Pauli's exclusion principle limit transfers to occupied states
for reactions 18O+48Ca.
Preliminary: unlimited transfers
Begin of
collision
1d5/2
18O
48Ca
End of
collision
Neutron states in spherical shell model
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Pauli's exclusion principle limit transfers to occupied states
for reaction: 48Ca+248U
Preliminary: unlimited transfers
Begin of
transfer
48Ca
238U
2g9/2
48Ca
End of
collision
238U
Neutron states in spherical shell model
Neutron states in deformed shell model
are studied in next slides
Begin of
collision
Begin of
transfer
3d5/2
1f7/2
End of
collision
End of
collision
2g9/2
48Ca
238U
48Ca
238U
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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  r1 , t 
1  rN , t  
1


 M  r1 , rn , t  
det 

M!
 M  r1 , t 
 M  rN , t 

P   M  r1,
rM , t  M  r1,
rM , t  dV1...dVM
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
At reaction 48Ca+238U probabilities of neutrons
stripping and pick-up are commensurable.
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
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Visual simulation of
neutron transfers at
reaction 18O+48Ca.
Two external neutrons:
1d5/2 from 18O and
1f7/2 from 48Ca
with moment projection
=1/2, 3/2 are took into
account
Visual simulation of
neutron transfers at
reaction 48Ca+238U
Three external neutrons:
1f7/2 from 48Ca and
2g9/2, 1i11/2 from 238U
with 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
b
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 b2 and octupole deformation b4 = b2/2 (a) and at real nucleus 238U with
b2 = 0.215, b4 = 0.095 (b), dashed lines correspond unoccupied levels
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Neutron pick-up at reaction 48Ca+238U
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.
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Neutrons pick-up in low-energy nuclear reactions
40Ca+238U with deformed nucleus 238U
a
b
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
18
238U and initial velocity of 40Ca nuclei equal 45o (solid line) and 90o (dashed line).
Neutron stripping at reaction 48Ca+238U
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
19
nucleus equal 0o.
Neutron stripping at reaction 48Ca+238U
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.
20
Neutrons stripping in low-energy nuclear reactions
48Ca+238U with deformed nucleus 238U
a
b
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 21
0
(dotted line).
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
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