Cranking mass parameters for fission
A formalism for semi-adiabatic cranking mass parameters is presented.
For the fission process of U, the time-dependent pairing equations of motion
were used to calculate
the excitation energy and to extract values of the
cranking inertia. A fission barrier is determined
by minimizing the action trajectory in a five dimensional configuration
space spanned by elongation, necking, deformations of fragments and
mass-asymmetry. The deformation energy is computed
in the the frame of the microscopic-macroscopic model.
The two center shell model with Woods-Saxon potentials is used in this context.
Values of the inertia for excited fissioning systems are reported.
A dependence between the cranking mass parameters and
the intrinsic excitation
energy is evidenced.
PACS: 24.10.-i Nuclear reaction models and methods; 25.85.-w Fission reactions
Keywords: Fission reactions, intrinsic excitation, inertia
In microscopic-macroscopic treatments of nuclear fission , the whole nuclear system is characterized by some collective coordinates associated with some degrees of freedom that determine approximately the behavior of many intrinsic variables. The basic ingredients in such an analysis is a shape parametrization that depends on several macroscopic degrees of freedom. The generalized coordinates of deformation vary in time leading to a split of the nuclear system in two separated fragments. Thus, these coordinates describe the change in time of the average field. Traditionally, in the large amplitude collective motion like fission or fusion, the calculation of the mass parameters are made within the cranking model [2, 3]. These parameters are calculated adiabatically. In reality, the nucleus can be excited with respect the ground state at any deformation. The excitation energy is shared between the collective kinetic energy of the fragments and the intrinsic energy released finally through gamma-ray and neutron emission. As mentioned in Ref. , no unique separation between these two kind of dumping can be realized. Several attempts were realized in order to investigate the change of the cranking mass parameters as function of the intrinsic excitation energy. In this respect, investigation based on the cranking model have been done for the rotational motion  and for the behavior of the temperature dependent mass parameter [6, 7, 8]. A theory for non-adiabatic cranking was proposed in Ref.  where the time-dependent many-body Schrodinger equation is solved and the time dependence of the collective motion is determined with the classical Lagrange equation of motion. It is worth to mention that in Ref.  an expression is derived for the collective kinetic energy containing corrections up to the fourth-order in the collective velocities. The fourth-corrections to the cranking approximation at hyperdeformations were found very large. The authors concluded that the perturbation treatment when applied to fission is invalidated.
In this paper, a new treatment for the cranking model is realized that takes into account the intrinsic excitation energy. In the next section, a semi-adiabatic cranking approximation is derived for even-even systems to investigate the general features of the cranking mass parameters when the nucleus is internally excited. To make the problem tractable, it is assumed that the system deforms slowly in time. Therefore, the matrix elements of the time-derivative of the wave functions are smalls. Because these matrix elements are responsible for quasi-particle excitations, our approximation allows to consider the system in a seniority-zero state during fission. In other words, the contributions to the inertia originating from seniority-two configurations are considered negligible. In section 3, the mass parameters are calculated. To obtain the inertia, a least action trajectory for the evolution of the nuclear system is determined. The single-particle energies for protons and neutrons were calculated along this path. Then, the intrinsic excitation energy of the fissioning system is evaluated within the time-dependent pairing equations of motion using different values of the inter-nuclear velocities. 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. Some behavior concerning the dependence of the cranking mass parameters versus the intrinsic excitation energy are extracted. In the last section, a discussion is made.
The single particle motion of a particle in an average field is governed by the Schrödinger equation that includes a Hamiltonian with pairing residual interactions. This Hamiltonian depends on some time-dependent collective parameters (), such as the internuclear distances between the nascent fragments, the mass-asymmetry, the fragment deformations or the necking parameter:
where are single-particle energies, is an monopole pairing interaction constant and denote creation operators. In order to obtain the time dependent pairing equations of motion, we shall start from the variational principle taking the following energy functional
and by assuming the many-body state formally expanded as a superposition of time dependent BCS seniority-zero and seniority-two adiabatic wave functions
is the seniority-zero Bogoliubov wave function. The excited seniority-two configurations are obtained by mean of quasiparticle creation and annihilation operators:
In definitions (2) and (3), and are amplitudes of the two kinds of configurations, is the chemical potential, and is the particle number operator. Because only the relative phase between the parameters (vacancy amplitudes) and (occupation amplitudes) matters, in the following is considered to be a real quantity and a complex one. To minimize the functional, the expression (2) is derived with respect the independent variables , , , (the amplitudes of the wave function), together with their complex conjugates, and the resulting equations are set to zero.
This system is sometimes called time dependent Hartree-Fock-Bogoliubov equations . These equations were already used to determine the intrinsic excitation energy in fission [13, 14] or to investigate the pair-breaking mechanism . The following notations are used in Eqs. (2) and in the remaining part of the article:
where is the gap parameter for the seniority-zero state while are related to seniority-two states. are pairing moment components and are single particle densities.
An estimate of the intrinsic seniority-zero state excitation energy  can be obtained with the relation
As indicated in the Appendix, the next semi-adiabatic cranking formula can be obtained for the effective mass parameters :
where the values of and are solutions of the time dependent pairing equations (6)i and the index is associated to the collective coordinate .. are exactly the expected values of the Hamiltonian (1) for seniority-two configurations:
and are energy terms associated to single-particle states:
The mass parameters can be obtained from formula (2) only if the derivatives and are supplied. In the BCS stationary state, the derivatives and depend only on the collective variables and the cranking formula can be easily obtained. On the other hand, if the system follows a trajectory in the configuration space and the motion is characterized by some given collective velocities, then the derivatives depend also on . Therefore, the second term in Eq. (2) depends on the history of the system and can be calculated only if the variations of and are known. The inertia along the trajectory is
where one coordinate is taken as the independent variable. The other coordinates are taken to be function of . From definition, the collective kinetic energy is and the last term in Eq. (2) becomes:
This term depends only on the derivatives with respect to time and . Their expressions are given by the coupled channel system of equations (6). That means, the term (15) is practically independent on the collective velocity and depends only on the values of , and , that is on the structure of the system and its intrinsic excitation and represents the minimal collective kinetic energy. Therefore, the term (15) can be viewed as an ground collective kinetic energy. In other words, if the nucleus is internally excited and the derivatives of the probabilities , are different from zero, the system possesses a minimal collective kinetic energy. The term (15) can be considered as responsible for energy transfer between intrinsic and collective degrees of freedom.
The calculation addresses the fission of U. As already mentioned, the basic ingredient in our analysis is the nuclear shape parametrization. The nuclear shape parametrization used in the following is given by two ellipsoids of different sizes smoothly joined by a third surface obtained  by the rotation of a circle around the axis of symmetry. Five degrees of freedom characterize this parametrization: the elongation given by the inter-nuclear distance between the centers of the ellipsoids, the two deformations of the nascent fragments characterized by their eccentricities , the mass asymmetry given by the ratio between the major semi-axis of the fragments , and the necking parameter related to the median curvature ( being the radius of the intermediate circle and the sign associated to the curvature). The meaning of all geometric parameters can be understand by inspecting the Fig. 1.
As specified in Ref. , first of all, a calculation of the fission trajectory in our five-dimensional configuration space, beginning with the ground-state of the system up to the exit point of the barrier must be performed. This can be done by minimizing the action integral. For this purpose, two ingredients are required: the deformation energy and the tensor of the effective mass. The deformation energy was obtained  by summing the liquid drop energy with the shell and the pairing corrections. The macroscopic energy is obtained in the framework of the Yukawa plus exponential model  extended for binary systems with different charge densities . The Strutinsky microscopic corrections were computed on the basis of the Woods-Saxon superasymmetric two-center shell model . This model gives the single particle level diagrams by diagonalizing a Woods-Saxon potential, corrected within spin-orbit and Coulomb terms, in the analytic eigenvalue basis of the two center semi-symmetric harmonic model [19, 20]. The effective mass is computed within the cranking adiabatic approximation as given in . After minimization, the dependences between the generalized coordinates () in the region comprised between the parent ground state configuration and the exit point of the external fission barrier supply the least action trajectory. The ground-state corresponds to the lowest deformation energy in the first well. The least action trajectory is obtained within a numerical method. Details about the numerical procedure of minimization and about the model can be found in Refs. [14, 16, 21] and references therein. Plots of the minimal deformation energy surface as function of the necking coordinate and the elongation are displayed in Figs. 2 (a) and (b). In Fig. 2 (b) the minimal action trajectory is also plotted. The resulting U fission barrier is plotted on Fig. 2 (c) as function of the distance between the centers of the nascent fragments . Some nuclear shapes obtained along the minimal action trajectory are inserted in the plot.
The realistic level scheme along the least action trajectory were also obtained within the superasymmetric Woods-Saxon two-center shell model. Within the energy diagrams, the system (6) is solved having as initial conditions the stationary BCS state in the ground state configuration. Three values of the internuclear velocity were taken into account: , and fm/fs, corresponding to a time to penetrate the barrier comprised between s. In Fig. 3 (a), the intrinsic excitation energy given by Rel. (8) is plotted as function of the internuclear distance for the three values of . If the internuclear velocity increases, the excitation energy becomes larger.
The inertia along the trajectory was calculated within three different approaches: the adiabatic cranking model of Ref. [5, 22, 23], the formula (2) within stationary BCS parameters and and the same formula within and values given by Eq. (6). The total effective mass is the sum of partial values obtained for neutron and proton subsystems.
In Fig. 3 (c), the inertia along the minimal action path calculated within the adiabatic cranking model is displayed with a thick line. In the ground state of the first well, the inertia is very small. Large values of of the cranking inertia are obtained around the exit point of the outer fission barrier, close to the scission configuration. A value approaching the reduced mass is obtained after the scission point. A similar behavior, exhibiting very large values of the inertia around the touching configuration, was obtained for light systems in Ref. , where a version of the Woods-Saxon two center shell model based on the molecular orbital approach is used. In the same plot, a thin line gives the inertia obtained within formula (2) by using the stationary values and . This semi-adiabatic inertia exhibits a similar shell structure as the adiabatic cranking model. The semi-adiabatic model gives in general lower values than the adiabatic cranking one, excepting the regions of the ground state and of the scission point where the values are larger.
In Fig. 3 (b) a comparison is made between inertia computed within relation (2) for different values of the internuclear velocity. The same values of the internuclear velocities were taken as those used to determine the excitation energies. The inertia along the minimal action trajectory has the largest values for the lowest taken into consideration. In general, the values obtained for fm/fs represent an upper limit for the magnitude of the inertia. It must be mentioned that for this collective velocity, the calculated intrinsic excitation energy is practically negligible. For higher values of , when the excitation energy becomes important, the magnitude of the inertia is considerably lower. The values obtained for fm/fs and fm/fs are situated in the vicinity of those obtained within the stationary values. For the three collective velocities, the shell structure resemble.
In Ref. , a temperature dependent cranking model that includes a parameter for the dissipation was used. It was shown that the increase of the temperature smoothen and lower the mass parameter. In the high temperature limit, the cranking results are quite close to the irrotational flow results. Our calculations also show that the inertia decreases when the excitation energy increases.
Several approximations were derived in the literature to compute the collective inertia. In the frame of the response theory  the Hamiltonian is expanded around a particular value of the macroscopic coordinate and the first derivative is treated with the time dependent perturbation approach. Another method is related to the generator coordinate method proposed in Ref.  within the Gaussian Overlap Approximation (GOA). This method allows to obtain a representation of any operator in the collective space. In general, the GOA mass is about 2/3 times smaller than the adiabatic cranking mass  but both quantities exhibits a similar shell structure. Adiabatic mass parameters for fission were derived also from the Time-Depedent Hartree-Fock-Bogololiubov (TDHFB) theory . A comparison between values of the collective mass tensor obtained with three diferent models, i.e., cranking, GOA and adiabatic TDHFB, showed that the adiabatic TDHFB mass exhibits more pronounced variations than the cranking and the GOA masses. As expected, the GOA gives smaller values than the cranking model. Our results show that for stationary values of and , the inertia is lower than the adiabatic cranking values.
In Ref.  the second collective energy level was calculated for the rare-earth and actinide nuclei using the Bohr-Sommerfeld quantization rule. The results overestimates the experimental values in average within a factor 2. This discrepancy ca be caused either by the shape of the potential in the ground state or either by a systematic too low value of the cranking inertia. In our calculations, in the first well has larger values than that obtained within the adiabatic cranking model.
In conclusion, a semi-adiabatic formalism based on the time-dependent pairing equations was described. The values of the inertia obtained within this model exhibit a strong dependence on the intrinsic excitation energy. In general, the semi-adiabatic inertia have a similar shell structure as the adiabatic cranking model.
Work supported by the Grant CNCSIS IDEI 512 of the Romanian Ministry of Education and Research.
In order to simplify the calculations, the variations of the parameter and as function of the seniority number, called blocking effect, is neglected. A treatment involving the blocking effect was realized in Ref.  in order to generalize the Landau-Zener effect and in Ref.  to evidence a new dynamical pair breaking effect. Within the wave function (3), using the notation (4) and the following identities
the energy functional (2) becomes
where the dot represents the time derivative. In order to obtain the equations of motion, the previous expression must be derived with respect the independent variables ant their complex conjugates and the resulting expressions must be set to zero. The assumption of a slowly varying deformation with respect the time will be used. In these circumstances, the probability to find the system in a real or virtual seniority-two state is much lower than the probability to find the system in a seniority-zero one. These statements can be translated in the following conditions: and . In this context, as prescribed in Ref.  and using the derivatives with respect and , two equations are obtained:
and another for its complex conjugate. The notations (7) are used. From Eq. (23) and its complex conjugate, the system (6) follows and the expressions involving the product between and apearing in equation (22) are determined:
The discussion found Refs.  reveal that the approximation used to obtain Rel. (23) helps us to describe approximately the effect of the residual interaction on dissipation and we eliminate only the terms related to the collective kinetic energy. The collective kinetic energy will be treated separately by solving equations involving different seniority states.
In order to determine the excitations between configurations the equations (22) must be derived with respect the amplitudes , , , , and . The next three equations follows:
If the system deforms slowly, the contributions of the type in Eqs. (25), (26) and (27) can be neglected and the solutions for seniority-two states given by the Lagrange method of variation of constants are:
Finally the probabilities to find the system in a seniority-two state are:
The total energy of the system is;
If we consider that the collective kinetic energy is
and substituting the time derivative with derivatives with respect the generalized coordinates
and considering that it follows
where are the effective mass parameter.
Using the identities
the dependences with respect the derivative of the Hamiltonian are evidenced:
It must be noticed that by replacing the energies with the approximate values  , by neglecting the differences between the sums of -terms and by using the stationary values and in the previous expression, it is straightforward to obtain the usual cranking mass parameter for an adiabatic BCS state . In this context, the following identities are also needed:
-  Nix J R 1972 Ann. Rev. Nucl. Sci. 22 65.
-  Inglis D R 1954 Phys. Rev. 96 1059.
-  Inglis D R 1955 Phys.Rev. 97 701.
-  Schutte G and Wilets L 1975 Nucl. Phys. A 252 21.
-  Brack M, Damgaard J, Jensen A, Pauli H, Strutinsky V and Wong W 1972 Rev. Mod. Phys. 44 320.
-  Iwamoto A and Greiner W 1979 Z. Phys. A 292 301.
-  Iwamoto A and Maruhn J A 1979 Z. Phys. A 293 315.
-  Schneider V, Maruhn J A and Greiner W 1986 Z. Phys. A 323 111.
-  Liran S, Scheefer H J, Scheid W and Greiner W 1975 Nucl. Phys. A 248 191.
-  Chaffin E F and Dickmann F 1976 Phys. Rev. Lett 37 1738.
-  Koonin S E and Nix J R 1976 Phys. Rev. C 13 209.
-  Blocki J and Flocard H 1976 Nucl. Phys. A 273 45.
-  Mirea M, Tassan-Got L, Stephan C and Bacri C O 2004 Nucl Phys. A 735 21.
-  Mirea M, Tassan-Got L, Stephan C, Bacri C O and Bobulescu R C 2007 Phys. Rev. C 76 064608.
-  Mirea M arXiv:0907.1347v1; Phys. Lett. B submitted.
-  Mirea M 2008 Phys. Rev. C 78 044618.
-  Davies K T R and Nix J R 1976 Phys. Rev. C 14 1977.
-  Mirea M, Bajeat O, Clapier F, Ibrahim F, Mueller A C, Pauwels N and Proust J 2001 Eur. Phys. J. A 11 59.
-  Mirea M 2006 Nucl. Phys. A 780 13.
-  Maruhn J and Greiner W 1972 Z. Phys. 251 431.
-  Mirea M, Tassan-Got L, Stephan C, Bacri C O, Stoica P and Bobulescu R C 2005 J. Phys. G 31 1165.
-  Ledergerber T and Pauli H-C 1973 Nucl Phys A 207 1.
-  Mirea M, Bobulescu R C and Marian P Rom. Rep. Phys. in print.
-  Diaz-Torres A, Gasques L R and Wiescher M 2007 Phys. Lett B 652 255.
-  Hofmann H 1976 Phys. Lett. B 61 423.
-  Brink D M and Weiguny A 1968 Nucl. Phys. A 120 59.
-  Gozdz A, Pomorski K, Brack M and Werner E 1985 Nucl Phys. A 442 26.
-  Baran A, Sheikh J A and Nazarewicz W 2009 Int. J. Mod. Phys. E 18, 1054.
-  Dudek J, Dudek W, Ruchowska E and Skalski J 1980 Z. Phys. A 294 341.
-  Ogle W, Wahlborn S, Piepenbring R and Fredriksson S 1971 Rev. Mod. Phys 42 424.