Numeric Code for the Symmetric Two-Center Shell Model:
Application for Fission.
by
P. STOICA
Institute of Physics and Nuclear Engineering,
P.O. Box MG-6, Bucharest, Romania
The theoretical analysis of fission processes is limited by the use of one-center shell
models. The simplest way to generalize the Nilsson model for such purposes is to use a
two-center shell model. A numerical code [1] for a superasymmetric two--center shell
model was developed recently in our Institute in order to study axially—symmetric
disintegration processes in a wide range of mass-asymmetries, including the alpha--decay
[2]. This work was based on the Frankfurt model [3] for the two-center oscillator
potential. However, the formalism used in this context is not appropriate for the study of
near-symmetric fission. The main difference lies in the fact that the system of
eigenvectors of the basic two-center oscillator for reflection symmetric systems is
characterized by two good quantum numbers, the parity and the projection of the intrinsic
spin: Ω. In the case of mass-asymmetries, the levels are characterized by only one good
quantum number, that is Ω. Accordingly, the mathematical formalism differs between the
two kinds of parameterization.
The code was developed for a nuclear shape parameterization given by two spheres of
equal radii smoothed joined by a neck given by the rotation of a circle around the axial
axis of symmetry. Accordingly, the potential is corrected by various terms that take into
account the influence of the neck, the spin terms and the depths of the potentials.
The numerical code is ready for applications and was successfully used for fission studies
[4]. The input parameters are the elongation, the A and Z parent values and the spin
constants. The single particle levels behave as Nilsson levels for small deformations in
order to reach finally the levels of the two separated fragments when the elongation tends
to infinity.
1. Introduction
Once discovered the fission process (Hans and Strassman, 1939), a range of
nuclear models have been applied and developed for explanation of this phenomenon; for
example: liquid drop model (LDM), Nilsson model, Strutinsky model etc.
This theory begin from the fact that Nilsson model does not allow the formation
of the separated fragments; it yields asymptotically to very prolate nuclei. We consider
that the symmetric two-center shell model better describes this process. More than that,
we have the possibility to obtain the wave functions analytically.
In this study we try to obtain the energy levels for the two-center potential model
for the fission process.
1
2. Theory
The Hamiltonian of this model in cylinder coordinate ξ, ρ, φ is:
H 
  2
1 
1 2
2 




  V (  )  V ( ) ,
2m   2    2  2  2 
(1)
The potential is described by the harmonic oscillator with two-centers:
V () 
1
m 2  2
2
(2)
V1 ( )   2 (  1 ) 2 ,   0

V ( )  
V ( )   2 (   ) 2 ,   0

1
 2
(3)
where ωρ,ξ are the oscillator frequencies.
In Fig. 1 one can see the form of the potential along the z-axis and the associated nuclear shape.
Fig. 1 - The form of the potential along the z-axis
and the associated nuclear shape on the fission process.
The Hamiltonian can be separated and the wave function can be written like this
 (  ,  ,  )  R(  )( ) ( )
(4)
So we obtain the equations:
2
 2

 2  A1 ( )  0
 

(5)
 2

1  1 2m
 2  2 V (  )  A2  R(  )  0
 2 
  

 

(6)
 2

2mE 2m
 2  2  2 V ( )  A2  ( )  0


 

(7)
where A1, A2 are the separation constants and the E representing the total energy, with
well-known solutions for (1)
 n ( ) 
1
2
e
in 
(8)
satisfying the condition  ( )  (  2 ) and for (2)
1
 2  (n   1)  2   
n
n
 e 2 (   )  Ln ( 2  2 )
Rn n (  )  
 (n   n  1) 
2
 m 
where    
 
2
(9)
1
2 m
 , Ln ( x)  Laguerre


satisfying the condition
R
2
polinom , ( x)  gamma
function and
(  ) d  finite .
0
Finally, the (φ,ρ) part of the wave function (6) can be written in this form
n  n
n N   ()
2
 n ( ) R N
  n
()
(10)
2
n  n
where N   2n   n and the phase ()
2
has been introduced for convenience.
And so, the partial energy
E  ,    N   1
(11)
is obtained solving the equation
3
H n N   E  , n N  .
(12)
To obtain the wave function and the energy for the two-center potential we must
brake the problem in two parts.

2 2
H


 V1 ( ),   0
 1
2m  2
H 
2
2
 H 2      V2 ( ),   0
2m  2

(13)
The Schrödinger equation will be written
H 1 1E ( )  E 1E ( ),   0,
(14)
H 2 2 E ( )  E 2 E ( ),   0,
having the conditions:

0
1)
  2 E d    1E d  1
2

2
0
2)  1E and

 1E continuous on   [0, )

3)  2 E and

 2 E continuous on   (,0]

4)  1E ( )  0   2 E ( )  0
5)


 1E ( )
  2 E ( )


 0
 0
Solving the Schrödinger equation we obtain the wave function that has a good
asymptotically behavior:
4
(15)

 2

(  1 ) 2 H  (  1 ),   0
 c1 exp 

 2

 E ( )  
2
c exp   (   ) 2 H   (   ),   0

 
1
1
 2
 2


 m
where c1 and c2 are the normalization constants,   
 
functions.
(16)



1
2
Matching the wave function   (z ) and the derivatives
and H (x) = the Hermite
d ( )
at  =0 and using
d
the relation
dH ( )
 2H -1 ( )
d
(17)
we obtain easily (z1=αξ1)
(c1  c2 )H ( z1 )  0



 (c1  c2 )z1H ( z1 )  2H2 1 ( z1 )  0
(18)
Denoting the solutions of the above equations by ν2k+1 and ν2k, corresponding to
the old and even parity levels, with
 H ( z1 )  0
 2 k 1


 z1H 2 k ( z1 )  2 2 k H2 2 k 1 ( z1 )  0
(19)
We obtain the following solutions

 2

c
exp
(  1 ) 2 H n  (  1 ),   0


n

 2

 E n ( )  
2
() n c exp   (   ) 2 H   (   ),   0

 n
n
1
1

2




The energy En is given by
5
(20)
1

E n    n  
2

(21)
and the normalization coefficient by
1

2
cn   j n , n , z1 
2

(22)
where
2 
H ( z )
H ( z1 )

j n , n . z1  e  z1 H 1 ( z1 )H ( z1 )    1 1 H ( z1 )   
H 1 ( z1 )




(23)
Finally, the total energy can be written in the following form, where we included
the part of the energy which proceed from the momentum-dependent part of the potential
that consists of a spin orbit coupling term
VLS (l , s)  c1 ( j 2  l 2  s 2 )
(24)
and an l2-like term
VL2  c2 l 2 :
(25)
3

E nlj  E n  El  Elj    n    c 2 l 2  c1 ( j  l 2  s 2 )
2

1
where ω=ωρ=ωξ, n=0,1,2,…, l= 0, n  1 , s  , j  l  s, l  s .
2
(26)
3. Code description
The most important subroutine is ENERG1 that calculates the shell levels’
energies, spins and parities of the two fragments of fission with the two-center potential
for the symmetric case. This subroutine is found in the subroutine REZTLATS from the
main program that is executed for many times to simulate the fission. Thus, the total
energies are calculated for every nΔ distance that separates the centers of the two fission
fragment (where n=0,1,2…). For every cycle we must obtain the form’s geometrical
parameters, coupling constants, normalization constants and critical point PCT4 for
scission. In Annex 1 one can see the logical branch.
6
4. Some results and future approach
In conclusion, from theory we observe that the two-center oscillator potential has
a good asymptotic behavior, consequently, applied at fission, it naturally permits the
formation of two fragments that may have strong deformations.
In Fig. 2 one can see the neutrons’ distribution of the energy levels between the
two fragments along of the fission process obtained for 236U with this numeric code.
Fig. 2 Behavior of the energy levels along of ξ-axis for neutrons
In future we will try to develop a numeric code which will describe the fission
process for the symmetric and asymmetric two-center shell model for deformed fission
fragments obtained from spherical and deformed parent nucleus.
7
REFERENCES
[1] M. Mirea, Phys. Rev. C 54 (1996) 302
[2] M. Mirea, Phys. Rev. C 63 (2001) 034603
[3] J. Maruhn and W. Greiner, Z. Phys. 251 (1972) 251
[4] M. Mirea and al, Europ. Phys. J. A 11, (2001) 59
[5] M.Brack, et al., Funny Hills: The Shell-Correction Approach to Nuclear Shell Effects and Its
Applications to the Fission Process, Reviews of Modern Physics, vol. 44, nr. 2, Apr. 1972.
[6] E.Badralexe si A.Sandulescu, Potentialul de oscilator cu doua centre,
ST. CERC. FIZ. ,TOM 25, nr. 9, P. 1087, Bucuresti, 1973.
[7] E.Badralexe, M.Rizea and A.Sandulescu, Symmetric two-center model wave functions, REV. ROUM.
PHYS., TOME 19, No. 1, P. 63-80, BUCAREST, 1974.
REV. ROUM. PHYS., TOME 19, NO. 1, P. 63-80, BUCHAREST, 1974.
[8] Joachim Maruhn and Walter Greiner, The Asymmetric Two Center Shell Model, Z.Physik 251, 431457 (1972).
8
ANNEX 1
S. ENERG1
MODSIM
PPUNCT
COOS
DRTMI
DRTMI
PVPE2
PFIS1
VOCOSI
PFIS2
NENGLS1
REPARM1
REPARE1
P2C1
PR2R2F
PR1R2F
DRTMI
KNORM1
FNIU11
FNIU22
JCPTN
NORMA
I2N1N2
VECT
JCPTNN
I1N1N2
IGAUSS
DHERM
DHEP
FIN5
FIN3
F
FIN11
DHIPER
FIN1
GAMAI
I3N1N2
GAMMA
DONARE
PROCEDURE
9
FUNCTION