Dissipative regime in low energy fission

Physics Procedia 31 ( 2012 ) 171 – 177

1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Institute for Reference Materials and Measurements.doi: 10.1016/j.phpro.2012.04.023

GAMMA-1 Emission of Prompt Gamma-Rays in Fission and Related Topics

Dissipative regime in low energy fission M. Mirea

Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, Bucharest, Romania

Abstract

In this presentation, a generalization of the time dependent pairing equations is presented by including the Landau-Zener effect in the superfluid model. These new equations allows a mixing of seniority one configurations that allows us to obtain a ground state at the end of the process. An application concerning the C-14 emission is offered and its fine structure is explained. These new equations are furthermore used to evidence a dynamical pair breaking effect that could explain the fine odd-even effect in cold fission. The partition of the dissipated energy between the two fission fragments will be also modeled. Finally, the time dependent pairing equations are used to deduce a model for non-adiabatic cranking inertia. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] Keywords: Fission; Time dependent pairing equations; Landau-Zener effect; Dissipation.

1. Introduction

The coupling of collective degrees of freedom with the microscopic ones causes dissipation and a modification of the adiabatic potential. The term dissipation usually refers to exchange of energy, linear momentum or angular momentum, by all kind of damping from collective motion to intrinsic heat. A measure of the dissipated energy can be obtained solving the time dependent pairing equations [1-4] (that are similar to the Hartree-Fock-Bogoliubov ones). A deep connection with the Landau-Zener transitions is included in the time dependent pairing equations: pairs undergo Landau-Zener transitions on virtual levels with coupling strengths given by the magnitude of the gap. A problem appears in solving these equations in the case of seniority one nuclear systems: the unpaired nucleon located on the blocked level will remain on the same level during the deformation of the system. Due to this phenomenon, the system cannot end in its ground state after disintegration.

In this presentation, a generalization of the time dependent pairing equations is presented by including the Landau-Zener effect in the superfluid model [5,6]. These new equations allows a mixing of seniority one configurations that allows us to obtain a ground state at the end of the process. An application

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172 M. Mirea / Physics Procedia 31 ( 2012 ) 171 – 177

concerning the C-14 emission is offered and its fine structure is explained. These new equations are furthermore used to evidence a dynamical pair breaking effect that could explain the fine odd-even effect in cold fission [7]. The partition of the dissipated energy between the two fission fragments will be also modeled [8]. Finally, the time dependent pairing equations are used to deduce a model for non-adiabatic cranking inertia [9].

2. The Landau-Zener effect

Single particle levels are functions of the deformation parameters that define the mean field. Levels characterized by the same quantum numbers associated with the same symmetries of the system cannot intersect and exhibit avoided level crossing regions. The transition probability of a nucleon from one adiabatic level to another is strongly enhanced in an avoided crossing region. This promotion mechanism is known as the Landau-Zener effect [10]. Using the time-dependent Schrodinger equation, the following system of coupled channel equations is obtained [11]:

( ) )1(,)()(1exp)()(101 ⎭

⎬⎫

⎩⎨⎧ −−= ∫∑ =

t

ijn

j ijji dthtci

c ττετε

where cj is the amplitude of the diabatic state j in the total wave function, εj is its single particle energy and hij is the interaction energy in an avoided levels crossing region. The probability to find the unpaired nucleon on the level i is pi=|c_i|2. These equations have already been used to explain the resonant fine structure in the fission cross section [12], the resonant-like structure of the inelastic cross section in heavy ion collisions [13], the fine structure in cluster emission [14,15] and in α-decay [16].

3. Generalization of the Landau-Zener effect in superfluid systems

An effect analogous to the Landau-Zener one can be obtained by generalizing the TDPE for the case of seniority one nuclear systems. The problem will be explored in the simplest possible way: a monopole pairing force, and a sufficiently weak pairing such that the nucleons are not redistributed to change significantly the mean field potential. Using quasiparticle creation and annihilation operators it is possible to construct some interactions that help us to promote the nucleon from one diabatic level to another. In the following, ak

+ and ak denote operators for creating and destroying a particle in the state k, respectively. The state characterized by a bar signifies the time-reversed partner of a pair. The parameters vk and uk are the occupation and vacancy amplitudes, respectively. Because only the relative phase between the parameters uk and vk matters, in the following, uk is considered to be a real quantity and vk a complex one.

The two situations plotted in Fig. 1 can be modeled. In the plot 1 (a), the single particle follows the diabatic level εj while in 1 (b) it remains on the adiabatic one ei. The interaction able to promote the unpaired nucleon from one adiabatic level to another must be given by some products of quasiparticel creation and annihilation operators.

In order to obtain the equations of motion, we shall start from the variational principle taking the following energy functional

)2(Ψ′+∂

∂−Ψ= H

tiHL δδ

M. Mirea / Physics Procedia 31 ( 2012 ) 171 – 177 173

and assuming the many-body state formally expanded as a superposition of n time dependent BCS seniority one diabatic wave functions. The functional contains several terms. The first one is the many body Hamiltonian with pairing residual interactions

( )∑ ∑> >

+++ −+=0 0,

* )3(.)]([)(k ik iikkkkkkk aaaaGaaaatqtH ε

The residual interactions between diabatic levels characterised by the same quantum numbers that are responsible for the Landau-Zener effect are assumed on the form:

)4(.)()(,

,)()()()(∑ ∏≠

≠

+++=′n

ijijik

jkkkjkijjiij aathtH αααα

The sum runs over diabatic levels i and j.

Figure 1: Ideal avoided level crossing region and possible transition states for an unpaired nucleon in the superfluid model. (a) The single particle follows the diabatic state. (b) The single particle remains on the same adiabatic level. Figure from Ref. [6].

After some calculations, as detailed in Ref. [6], a new set of pairing equations, which account for

configuration mixing results:

i ρl (m) =κ l (m)Δm* −κ l (m)

* Δm ,

i κ l (m) = 2ρl (m) −1( )Δm + 2κ l (m)εl − 2Gρl (m)κ l (m) ,

i pm = hmj Smj − Sjm( )j≠m

n

∑ ,


i Sjm = Sjm −Δm

2− Δ j

2

G+ εm −ε j( )+G ρk (m)

2 − ρk ( j )2

k≠ j∑

k≠m∑( )

⎧⎨⎪

⎩⎪(5)

−1

2Δmκk (m)

* +Δm*κk (m)( )

k≠m∑

ρk (m)2

κk (m)

2 −1⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+−

1

2Δ jκk ( j )

* +Δ j*κk ( j )( )

k≠ j∑

ρk ( j )2

κk ( j )

2 −1⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎫⎬⎪

⎭⎪

+ hmlSjl − hjlSlm⎡⎣ ⎤⎦+ hmj pj − pm( )l≠m

∑ .

where ρk are the single particle densities, κk are the pairing moment components, and pm denotes the probability to have an unpaired nucleon on the level m. ρk and pm are real quantities while κk and Sjm are complex ones. In analogy with the pairing moment components κk, Sjm can be called as unpairing moment components, having the property |Sjm|2=pjpm. The index k runs over a workspace that allows the pairing force to operate only within a finite number of active levels around the Fermi energy. If the pairing is neglected, the third equation and the fourth equations of the system reduce to the Landau-Zener equations for systems without residual interactions. So, the previous system of coupled channel equations represents a generalisation of the TDPE in the case of seniority one nuclear systems.

4. New dynamical pair breaking effect

The dynamical pair breaking effect emerges from a new set of coupled channel equations deduced [7] for the time-dependent probability to find the system in a seniority-one state or in a seniority-two one. In such an approximation, if the interactions produced in the avoided crossing regions or those due to the Coriolis coupling are not taken into consideration, other perturbations between two different states are not possible. In the following, the calculations are restricted only for perturbations produced in the avoided crossing regions. This approximation does not affect the essential features of the model but leads to a considerable simplification of the mathematical apparatus.

Figure 2: Ideal avoided crossing regions between two adiabatic levels εi and εj. Three possible transitions between configurations in an avoided crossing region in the superfluid model are displayed: (a) The configuration remains unchanged after the passage through the avoided crossing region.(b) A pair is broken. (c) A pair is created. Figure from Ref. [7].


Using quasiparticle annihilation and creation operators, it is possible to construct some interactions able to break a Cooper pair when the system traverses an avoided crossing region. The three situations plotted in Fig. 2 can be modeled within products of such creation and annihilation operators acting on Bogoliubov wave functions. In the plot 2 (a), the Cooper pairs remain on the adiabatic levels εi and εj after the passage through the avoided crossing region, in Fig. 2 (b) the pair destruction is illustrated, while in Fig. 2 (c) two fermions generate a pair after the passage through an avoided crossing region. Formally, to describe these three situations, an interaction in the avoided crossing can be postulated exactly as that described in the previous section. A new set of time dependent pairing equations emerges.

5. Excitation of fission fragments and effective mass

A simple recipe to fix the average number of particles of both fission fragments at scission was elaborated in Ref. [8]. A condition that the sum of BCS occupation probabilities of the levels of both fragments is equal to the number of nucleons was introduced by means of Lagrange multiplier in the energy functional. Within this dynamical projection method it is possible to obtain microscopically the intrinsic excitation energy of each nucleus issued in a fission process.

Using the same Lagrange function, a semi-adiabatic cranking approximation is derived [9] for even-even systems to investigate the general features of the cranking mass parameters when the nucleus is internally excited. It is assumed that the system deforms slowly in time. Therefore, the matrix elements of the time-derivative of the wave functions are small. Because these matrix elements are responsible for quasi-particle excitations, our approximation allows considering the system in a seniority-zero state during fission. At each deformation, instantaneous values of the single-particle densities and of the pairing moment components are deduced. The semi-adiabatic effective mass is evaluated using these values. If the BCS approximation is used, the cranking formula obtained from the time dependent pairing equations become similar to the classical one [17].

6. Results

A fine structure in the 14C radioactivity of the 223Ra was observed in 1989 [18]. In the first experiment, the results indicate that 15% of 14C decays are transitions on the ground state of the daughter, while 8% are transitions on the first excited state. In Ref. [19], using the M3Y potential, it was evidenced that the preformation probability must be more favorable for the excited state than for the ground state with a factor of 180. Such a value cannot be accounted from classical models. On another hand, the fine structure in the 14C radioactivity was explained in Refs. [6,11] by an enhanced transition probability of the unpaired neutron from the adiabatic level Ω=3/2 emerging from 1i11/2 to the adiabatic level with the same spin projection Ω that emerges from 1j15/2, in terms of the Landau-Zener effect. The time dependent pairing equations were solved in Ref. [6]. The Woods-Saxon two-center shell model is used to compute the single particle levels. The two-center shell model, based on the semi-symmetric two-center oscillator [20,21], represent a valuable tool for the investigation of the alpha-decay, of the cluster emission [22,23], of the fission process [24-25] and of the superheavy element synthesis. In the case of fission process, the results concerning the heights of barriers are in good agreement with experimental. The branching ratio r between the partial half-life for transitions to the ground state of the daughter and the partial half-live to the first excited state was calculated for cluster decay. The experimental values of r range between 5.4 and 5.9. Our theoretical value is r=5.48, which is in an excellent agreement with experimental data. It is important to note that some simple arguments in analogy with the results obtained in [26,27] can be given


in the favor of the approach presented in this paper. In the previous references a parent metastable state is generated and a tunneling time of the order of (10-21-10-22) s is computed.

In thermal neutron induced fission, the charge and mass distributions of fragments exhibit a fine

structure showing a preference for an even number of protons and neutrons. We will expect that this phenomenon will be better evidenced in cold fission, where the excitation energy of the fragments is so small that no neutrons can be emitted. It is not true. The experimental evidences are strange. Up to an excitation energy of 6-8 MeV, the odd partitions dominate [28,29]. It is interesting to note that at zero excitation energy, the probability to find the system in a seniority-two state is practically one. Such a behavior is not expected because in cold processes the system doesn't possess enough energy to break a pair and because the penetrability is hindered for odd-systems due to the specialization energies associated to the two unpaired nucleons. Up to now, the odd-even structure in fission was explained within statistical arguments, as for example in Ref. [30]. The odd-even structure in fission fragment yields can be explained within the dynamical pair breaking effect. The probability to break a pair can be determined dynamically by taking into account only the interaction available in the avoided crossing regions. If one assumes that the odd-even effect in the fission fragments distribution is strongly correlated to the seniority-two state probability, this phenomenon can be alternatively explained by solving the coupled-channel system of time-dependent pair breaking equations described above.

The time dependent equations for pair breaking were solved for different values of the inter-nuclear velocities. For each inter-nuclear velocity it was possible to evaluate the dissipated energy and the probability to have a seniority two configuration. The probability to obtain a seniority two configuration Podd and the dissipated energy E* were obtained at the scission point, for a distance between the centers of the two fragments of R = 18 fm. The calculated dependence of Podd versus E* exhibited a clear decrease in the cold fission regime in accordance to experimental findings. These calculations give an explanation for the dominance of the odd configurations for excitation energies lower than 6-8 MeV.

Using the time dependent pairing equations completed with dynamical projections on particle numbers

[8], it was found that if the heavy fragment is close to magic numbers, its excitation energy could be half than that of the partner. The calculations were realized for the channel 102Zr+132Te of the 234U fission. This result is consistent with experimental finding addressing the neutron multiplicities. A similar conclusion could be also drawn from statistical [31] or phenomenological models [32].

In Ref. [9] it was shown that the effective mass decreases when the excitation energy increases. It was also shown that the diagonal components of mean field Hamiltonian that enter in the effective mass depend on structure of the system and its intrinsic excitation and represents the minimal collective kinetic energy shared among all degrees of freedom. In other words, if the nucleus is internally excited the system possesses a minimal collective kinetic energy. The mass parameters obtained from the TDPE have a similar shell structure as those computed with classical cranking model but exhibit in general lower values. A similar behavior is obtained from the Gaussian overlap approximation [33].

Acknowledgements

This work was supported by CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0068.


References

[1] Blocki J, Flocard H, Nucl. Phys A. 1976; 273: 45. [2] Koonin SE, Nix JR, Phys. Rev. C 1976; 13: 209. [3] Schutte G, Wilets L, Nucl. Phys. A 1975; 252: 21 [4] Mirea M, Tassan-Got L, Stephan C, Bacri CO, Nucl. Phys. A 2004; 735: 21 [5] Mirea M, Mod. Phys. Lett. A 2003; 18: 1809 [6] Mirea M, Phys. Rev. C 2008; 78: 044618 [7] Mirea M, Phys. Lett. B 2009; 680: 316 [8] Mirea M, Phys. Rev. C 2011; 83: 054608 [9] Mirea M, Bobulescu RC, J. Phys. G 2010; 37: 055108 [10] Hill DL, Wheeler JA, Phys. Rev. 1953; 89: 1102 [11] Mirea M, Phys. Rev. C 1998; 57: 2484 [12] Mirea M, Tassan-Got L, Stephan C, Bacri CO, Bobulescu RC, Phys. Rev. C 2007; 76: 064608 [13] Cha MH, Park JY, Scheid W, Phys. Rev. C 1987; 36: 2341 [14] Mirea M, Clapier F, Europhys. Lett. 1997; 40: 509 [15] Mirea M, Eur. Phys. J. A 1999; 4: 335 [16] Mirea M, Phys. Rev. C 2001; 63: 034603 [17] Brack M, Damgaard J, Jensen AS, Pauli HC, Strutinsky VM, Wong CY, Rev. Mod. Phys. 1972; 44: 320 [18] Brillard L, El Ayi AG, Hourani E, Hussonnois M, Le Du JF, Rosier LH, Stab L, C.R. Acad. Sci. Paris 1989; 39: 1105 [19] Gupta RK, Horoi M, Sandulescu A, Greiner M, Scheid W, J. Phys. G 1993; 19: 2063 [20] Mirea M, Phys. Rev. C 1996; 54: 302 [21] Mirea M, Nucl. Phys. A 2006; 780: 13 [22] Mirea M, Sandulescu A, Delion DS, Nucl. Phys. A 2011; 870-871: 23 [23] Mirea M, Sandulescu A, Delion DS, Proc. Rom. Acad. Serie A 2011; 12: 203 [24] Mirea M, Delion DS, Sandulescu A, Phys. Rev. C 2010; 81: 044317 [25] Mirea M, Tassan-Got L, Centr. Europ. J. Phys. 2011; 9: 116 [26] Serot O, Carjan N, Strottman D, Nucl. Phys. A 1994; 569: 562 [27] Carjan N, Serot O, Strottman D, Z. Phys. A 1994; 349: 353 [28] Schwab W, Clerc H-G, Mutterer M, Theobald JP, Faust H, Nucl. Phys. A 1994; 577: 674 [29] Hambsch F-J, Knitter H-H, Budtz-Jorgensen C, Nucl. Phys. A 1993; 554: 209 [30] Avrigeanu V, Florescu A, Sandulescu A, Greiner W, Phys. Rev. C 1995; 52: R1755 [31] Schmidt K-H, Jurado B Phys. Rev. Lett. 2010; 104: 212501 [32] Manailescu C, Tudora A, Hambsch F-J, Morariu C, Oberstedt S, Nucl. Phys. A 2011; 867: 12 [33] Gozdz A, Pomorski K, Brack M, Werner E, Nucl. Phys. A 1985; 442: 26

Dissipative regime in low energy fission

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