Dissipation in a wide range of mass-asymmetries
MIHAIL MIREA
INSTITUTE OF PHYSICS AND NUCLEAR ENGINEERING
P.O. BOX MG6, BUCHAREST-MAGURELE, ROMANIA
Email: mirea@ifin.nipne.ro
In the following, all kind of binary partitions , including the alpha and cluster decays, are treated as fission
processes in a wide range of mass-asymmetries in a unitary manner. That means, the whole nuclear system
is characterized by some collective coordinates associated to some degrees of freedom which determine
approximately the behavior of many other intrinsic variables. The shape collective parameters vary and
manage the evolution of the system. The initial parent nucleus is split into two parts: the daughter (the
heavy fragment) and the emitted fragment (the light one). The decaying system provides a time dependent
single-particle potential in which the nucleons move independently.
We shall discuss the fine structure of very asymmetric processes and the dissipation for the fission in an
unitary manner.
The fine structure of alpha (and cluster decay) reveals essential variations of the light fragment emission
probability for different states of the parent, the daughter and the emitted nuclei. The kinetic energies of the
emitted particles has several determined values related to the nuclear structure of the parent nucleus and the
nascent fragments.
The fine structure of alpha decay was discovered in 1929 by Rosemblum by measuring the ranges of alpha
particles in air.
This phenomenon was explained many years latter by Mang (H.J. Mang, Z. Phys, 148, 528 (1957)) who
considered that the preformation probabilities of the emitted nucleus in a given channel is proportional with
the square overlap between the parent and the daughter wave functions in different states after the
disintegration.
In the case of cluster decay, a fine structure in the 14C radioactivity of 223Ra was first observed in 1989 by
the Orsay group who took advantage of the large solid angle SOLENO superconductor spectrometer
facility (L. Brillard et al., C.R. Acad. Sci. Paris 309 (1989) 1405). The fine structure is characterized by a
more intense branch to the first excited state of the daughter 209Pb. Surprisingly, they observed about 87%
transitions to the first excited state of the daughter and only 13 % transitions to the ground state.
This process being similar to the fine structure of the alpha decay observed by Rossemblum, this analogy
was emphasized in the theoretical investigations.
It is however of interest to note that theoretically, the fine structure in the case of cluster decay was
anticipated by Greiner and Scheid since 1986 (M. Greiner and W. Scheid, J. Phys. G 12, 229 (1986)). The
authors showed that ground state to ground state transitions in cluster radioactivity are only a channel
among many possibilities, stating implicitly that a fine structure must exist in this type of decay.
Other predictions were done by Silisteanu and Ivascu (I. Silisteanu and M. Ivascu, J. Phys. G 15, 1405
(1989)) who solved a system of coupled channel equations, the final channels being defined by a family of
potential barriers correlated with different excited modes.
The fine structure was also interpreted reasonably by taking into account simple nuclear structure
considerations (R.K. Sheline and I. Ragnarsson Phys. Rev. C 43, 1476 (1991), G. Ardisson and M.
Hussonnois, Radiochimica Acta 70/71, 123 (1995)). By considering that the fine structure of cluster decay
is similar to that observed in alpha decay, the probabilities to obtain transitions to different states in the
final channels are proportional with the square overlaps between the ground state of the parent and different
state of the daughter. Sheline and Ragnarsson obtained 4%, 64% and 7% for the square overlaps in the
ground state of the daughter, the first excited state and the second excited state.
Another attempt to investigate this phenomenon was performed by Gupta and col. taking use of the M3Y
(R.K. Gupta et al., J. Phys. G 19, 2063 (1993))potential and modifying the Q-value accordingly to each
channel. It was evidenced that the spectroscopic amplitudes for the first excited state must be 180 times
higher than that of the ground state. These results are not consistent with those obtained by Sheline and
Ragnarsson where this ratio is about one order of magnitude lower.
In the frame of the superasymmetric analytical fission model of Poenaru and Greiner (D.N. Poenaru and
W. Greiner, in Nuclear Decay Modes, Ed. D.N. Poenaru and M. Ivascu, IOP, 1996, p 317), the
specialization energy was invoked for the first time. Taking into consideration the conservation of the
angular momentum of the process, results in good agreement with the experiment were obtained.
Alternatively, in a competitive way, it was proposed that the fine structure can be explained appealing to
the Landau-Zener promotion mechanism of the unpaired nucleon from one level to another in the avoiding
crossing regions. (M. Mirea and F. Clapier , Europhys. Lett. 40, 509 (1997), M. Mirea Phys. Rev. C 57,
2484 (1998))
It is evidenced that the fine structure can be explained by investigating the modality in which the levels
bunched initially in shells are reorganized during the decay to reach the final configuration.
Why this alternative mechanism must be introduced in the theory? To postulate the possible existence of
the Landau-Zener effect in cluster decay, we must analyze the behavior of the barrier which is penetrated
by the system during the process. In this plot, the deformation energy is computed in the frame of the
Yukawa plus exponential model extended for binary systems with different charge densities as function of
the normalized elongation R (distance between the center of the nascent fragments). Here the
parameterization is given by two intersected spheres and we have the reaction 223Ra->14C+209Pb. In the case
of ground state to ground state transitions, the barrier is represented with thick line. If the system is excited
during the scission process, that means the final channel will be characterized by an excited mode, the
barrier must be increased by the excitation energy which in our case is exactly the difference in
specialization energies. This specialization energies must be determined using a level scheme, in the
following with the help of the superasymmetric two center shell model (M. Mirea Phys. Rev.C 54, 302
(1996)).
Any excitation can be treated by increasing the potential barrier.
What is the specialization energy? Wheeler introduced this concept in order to explain the slower
spontaneous fission rates of odd parents in comparison with even ones (J.A. Wheeler, in Niels Bohr and the
Development of Physics, Ed. W. Pauli et al, Pergamon, 1955, p163). He postulates the possibility that the
height of the barrier in such processes is increased by a quantity due to the conservation of some motion
integrals related to some symmetries of the decaying system. So the specialization energy measure a greater
energy as then is when the unpaired nucleon occupies all the time the lower empty level.
To present this competitive way in investigating the fine structure, and furthermore to describe the
dissipation, we follows several steps.
1. Describe the nuclear shape parameterization
2. To present some features of the superasymmetric two center shell model
3. To investigate the single particle level scheme
4. To explain the Landau-Zener effect
5. Results concerning the fine structure of 14C from 223Ra, 14C from 225Ac, alpha from 211Po, fission of
236
U.
6. Conclusions
The nuclear shape parameterization
The nuclear shape parameterization is given by smoothly joining two intersected spheres of different radii
R1 and R2 with a third surface obtained by rotating a circle of radius R3 around the symmetry axis. This
nuclear shape has the most important degrees of freedom encountered in fission: the elongation given by
the distance between the centers of the fragments (R), the mass asymmetry which is characterized by the
radius R2 and the necking as the generalized coordinate C (the inverse of its radius R3).
This parameterization is suitable for fission because a single spheres and two fragments are allowed
configurations and the flatness of the neck (Brosa condition) is an independent variable. Using this
parameterization, we can describe not only the disintegration processes, but also necked ground states or
diamond ones. The equation of the parameterization look like this. We have 3 equations, one for each
spherical surface and another one for the median surface. From now on, the elongation will be also
characterized by a normalized elongation Rn=(R-Ri)/(Rf-Ri). The values Ri and Rf characterizes two special
and very important configurations: the starting point of the fission process (the initial fragment starts to
emerge) and the scission point for two intersected spheres with a constant radius of the light fragment. In
the case of cluster and alpha decay, we will use a simple parameterization given only by two intersected
spheres because some insights showed that the neck is not very important in this kind of decays.
The theoretical study of fission processes is limited by difficulties encountered in the simulation of the
level scheme: central potentials can not describe adequately the split of one nucleus and the sum of single
particle levels for very large deformations reaches infinity. One center potential is not adequate to
characterize the split of one nucleus in two fragments. To describe fission, the simplest generalization of
the Nilsson model is to use a two centre shell model, in this case the nucleons belong to two nuclear
fragments.
For the nuclear shape parameterization presented above the microscopic potential is split into several parts
(M. Mirea, Phys. Rev. C 54, 302 (1996)):
V=V0+Vas+Vn+VLs+VL2-Vc
V0 is the two center harmonic potential where eigenvectors can be analytically obtained by solving the
Schrodinger equation. Unfortunately, analytical solutions can be obtained only for semi-symmetric shapes:
sphere + spheroid with the same semi-axis perpendicular on elongation. Solving the Schrodinger equation,
an orthogonal system of eigenfunctions is obtained which depends on cylindrical coordinates φ, ρ and z.
Four conditions must be fulfilled (for the continuity of the wave function after penetrating the separation
plane, for normalization and continuity of eigenvalues). The eigenvalues depend on the quantum numbers
n,m and ν. ν is a real quantum number and belongs to the Hermite function Hν (along the z-axis). This
quantum number has some properties: when the fragments are separated at infinity they becomes integers.
If ν1 becomes integer, that means the wave function will be located in the first potential well, otherwise ν2
is integer and the wave function will be located in the second potential well. We have only one of this two
possibilities, that help us to determine where a wave function of the initial fragment will be located after the
separation. As we can see, when Rn=0, so that the configuration is given by a spherical parent,
ν1=ν2=integer.
The second term is that due to the mass asymmetry, which help us to transform the semi-symmetric shape
(sphere +spheroid) to that given by two intersected spheres. Different parameterization were used for this
term for which the potential along z is presented in this figure, beginning with the spherical fragment and
finishing with the scission point for Rn=0., 0.25, 0.5, 0.7 and 1. In the following, we will use the sequence
of potential barrier presented in figs f to j.
The third term is due to the necking, and have an analytical relation from geometrical conditions. This
relations gives a correction term to simulate the neck. Here the equipotential lines of the well for Rn=1,
R3=1 as function of ρ and z.
The energy diagram for neutrons is plotted in this figures as function of the normalized elongation. In the
left we have the levels of the parent 223Ra considered spherical. In the right, we have the levels for the
daughter 209Pb and 14C in the first and second column, respectively. Here the two fragments are separated.
We used the spectroscopic notations to characterize the levels. At the beginning, the Ra unpaired neutron is
on the 3/2 level emerging from i13/2, somewhere at a given deformation. Adiabatically, after the separation,
this level reaches the g 9/2 state of the daughter, this state being the ground state. It is possible that during
the process the unpaired neutron skip from one level to another in the avoiding level region due to the
Landau-Zener effect. We have four avoiding crossing regions.
What is the Landau-Zener effect?
The realistic two center diagram presented before provides an instrument to study the role of individual
orbitals during the disintegration process. Levels with same symmetry (same quantum numbers) cannot
cross during an adiabatic process and exhibit avoided level crossings. The point of nearest approach
between two such levels is known as avoided level crossing. If the internuclear distance varies (or another
generalized coordinate) the probability that a neutron jump between two such levels is strongly enhanced in
the region of avoided level crossings. This promotion mechanism represents the Landau-Zener effect.
An ideal avoided crossing region is plotted in this figure. The distance between the levels at the nearest
approach define the interaction energy between these levels, this interaction energy being obtained directly
from the superasymmetric two centre shell model. If the transition is made very slowly, the neutron on the
adiabatic level Ei will remains on the same level after passing trough the avoided crossing region. If the
transition is made very fast, the nucleon will jump on the next adiabatic level, it will follow the adiabatic
state epsilon. The probability to find the nucleon on one final single particle state can be computed by
assuming that diabatic wave functions φi exist for these two levels. The total wave function is in the two
level approximation a superposition of the two wave function φi . This total wave function is introduced in
the time dependent Schrodinger equation and two coupled equations are obtained.
In our case three levels must be taken into consideration and the equations have the next form. All the
necessary ingredients are plotted on the plots: the diabatic levels obtained from the superasymmetric two
center shell model, where the values in the avoiding crossing regions are interpolated by spline functions,
the interaction matrix elements, the relative velocity of crossing the avoided level regions which intervene
in the time derivative.
The velocity was obtained from very crude assumptions. It is supposed that all the energy available to the
process is that given by the zero point energy. It is also supposed that this zero point kinetic energy is split
into two parts: one is lost to surpass the barrier and the remainder one is kept as kinetic energy of the whole
system. By increasing the kinetic energy, the part lost to surpass the barrier decreases the Gamow factors
for transitions. To determine the internuclear velocity (the velocity of passage through the avoided crossing
regions) is it supposed (considered) that a fraction (a part) of the total available energy (the zero point
vibration energy) is maintained (remains ) mass kinetic energy. The best results are obtained by considering
the kinetic energy to be 70 keV. Solving the system of differential equations we obtain the probabilities to
obtain the nucleon in the adiabatic levels ε1, ε2 and ε3. Calculating the Gamow factors we obtain the next
Geiger-Nuttal plots where squares are the theoretical values, circles the experimental values and the empty
squares the theory without taking into account the Landau-Zener effect. The agreement between
experimental and theoretical values is very good. The partial half lives are calculated in the frame of the
superasymmetric fission model using preformation probabilities obtained with the Landau-Zener method.
As a first conclusion: the fine structure depends not only on the initial and final structure but also on the
dynamical characteristics of the system.
14
C from 225Ac was also treated because some experiments were planned to measure the kinetic energy of
the emitted nucleus accurately enough to infer the levels of the daughter (M. Mirea, Europ.Phys.J.A4, 335
(1999)). It is known that the g.s. of 225Ac has the spin 3/2 but it is controversial if the nucleon emerges from
the level h9/2 or f7/2. If the g.s. proton is on the h9/2, it is clear that the nucleon will be in the final state 211Po
in a hole configuration 1h11/2 which is very unfavorable from energy point of view. The process is
hindered. So, the ground state must be in f. Solving the system of equations, with the same velocity of the
internuclear distance, normalizing all the values of the half lives to the reference 224Ra, we obtain the next
Geiger-Nuttal plot. The theory is with squares. We obtained HF=7 (to excite all the 8 low lying levels in the
interval 0.7-1.3 MeV) and HF=4000 for the first excited single particle level and HF=03 for the ground
state. The ground state of Po is the nearly spherical orbital 2h9/2.
So, as conclusion, the Landau-Zener effect can explain the fine structure in cluster decay. So the
characteristics of the fine structure depend not only on the structure of the parent, the daughter and the
emitted fragment but also on the dynamics of the process.
The alpha decay is also treated (M. Mirea, Phys. Rev. C, 63, 034406 (2001)) within molecular models (a
nuclear molecule is a system of two or more nuclei which are bound together in quasibound states of
quasimolecular potential). The 211Po alpha decay will be treated because this is the first nucleus
investigated theoretically by Mang. Using the superasymmetric two center shell model (modified in order
to obtain the stiffness of the alpha particle at the scission point) the next energy level diagram is obtained as
function of the normalized elongation.
Up to now, for cluster decay, only the radial coupling was treated (the Landau Zener effect). In the case of
alpha decay, another effect can compete: the Coriolis or rotational coupling which causes transitions
between two levels for which the value of Omega differs by one unity and it is proportional to the angular
momentum operator of the single particle J± matrix element. The final system of coupled equations is
presented here with a term due to radial coupling and terms due to rotational couplings (sum over levels
having the same Ω and two sums over levels which have Ω which differ by one unit). I is the total spin of
the system.
Few words about the initial conditions of the system. In this figure, detailed parts of the level diagram are
presented. With thick lines are presented the levels with different values of Ω, ½, 3/2, … In the parent
ground state the nucleon is located on the orbital g9/2. If we take the effect Landau-Zener alone (only the
radial coupling), the g.s. of the daughter after the scission is obtained if the nucleon follows the level Ω=1/2
(adiabatically), and the excited states are obtained due to the avoided crossing levels marked with an arrow.
The levels emerging from g1/2 arrives on very excited states and are not possible. Due to the Coriolis
coupling, a neutron located can skip with some probability on levels with Ω ½ contributing to the final
channels, An neutron located on Ω=5/2 can skip on 3/2 and afterwards on ½, but the probability to
contribute to the final channel is very small. So that we considered sufficient to take into consideration that
the neutron emerges from g9/2 with equal probability to have Ω=1/2 and 3.2 and the levels g9/2, p1/2, f5/2,
2p3/2 and i15/2 were taken into consideration to solve the system.
The STCSM provides the ingredient to calculate the single particle transitions: probabilities due to radial
and Coriolis couplings. In this picture we plotted the difference between the adiabatic states Ω=1/2 the
arrows locating the avoided crossing regions. The diabatic E states and the adiabatic ε ones. The interaction
matrix elements between diabatic states. An example for 3 levels with Ω=½ and one level with Ω= 3/2, the
interaction due to rotational coupling. Asymptotically we have the values associated to spherical shapes.
We obtained the ratio between intensities to first and second excited states on intensity of the g.s. These
results agrees with the theoretical one.
This calculation also suggest that the fine structure in alpha decay can be explained quantitatively using
molecular models and the dynamic of the process direct the phenomenon.
Using the previous method, the residual interactions are neglected. Initially, we assumed that the pairs
reached the lowest levels during the disintegration. Without this assumption, it follows that the system can
not end in its ground state due to the fact that nucleons can not jump between levels with different quantum
numbers, then these levels cross. Using the Time Dependent Hartree-Fock-Bogoliubov method, the pairing
residual interaction allows such transitions. However, even in this picture if single particle potentials are
used, levels with the same quantum numbers are avoided to mix, i.e., unpaired nucleons are blocked on
levels with same quantum numbers. If the nucleon is on a adiabatic level εi, this nucleon is blocked on this
level during the deformation of the nucleus if the deformation vary very rapidly. The HFB methods allows
only to describe mean values of several operators and it is not designed to give matrix elements of many
body operators like scattering amplitude correctly.
A way to bypass the problem is given in the following. We use a solution of the Hartree-Fock-Bogoliugov
time dependent equations (J. Blocki and H. Flocard, Nucl.Phys.A273, 45 (1976), S.E. Koonin and J.R. Nix,
Phys.Rev.C13, 209 (1978), J.W. Negele et al., Phys.Rev.C 17 1089 (1978)) for a time dependent single
particle potential and introducing the Landau-Zener effect to replace the effect given by a part of the
residual interactions. Solving the equations the response of the system is estimated when the shape nuclear
parameterization is changed. To deduce the equations we start with the variational principle taking the
Lagrangean as:
L=<φ|H-iħ∂/∂t+H’+λN|φ>
And assuming the many body state as a superposition of BCS seniority one diabatic wave functions. H is
the many-body Hamiltonian with pairing and H’ is the residual interaction between diabatic levels
characterized by the same quantum numbers, which lead to avoided crossing regions and allows slippage as
in the classical Landau-Zener effect. After variation we obtain the system of equations where pm are the
single quasiparticle occupation probability of orbital m and ρk is the pairwise occupation probability of
orbital k (M. Mirea, Mod.Phys.Lett.A, 18 1809 (2003)).
The initial conditions were obtained for the minimal value of the potential given by the microscopicmacroscopic method and using the ground state BCS formalism. We obtained the probabilities for the
different tunneling velocities to find spectroscopic amplitude on different states of the daughter for the 14C
emission from 223Ra. The ratio between the probabilities for transitions to first excited state and ground
state agrees with the theoretical value for a velocity of 10 5 fm/fs.
The dissipation (or damping) is the energy flow between collective and intrinsic excitations.
The probability to obtain a binary partition (A1Z1,A2Z2) is ruled by the barrier penetrability (T.
Ledergerbier and H.C. Pauli, Nucl. Phys. A 207 1 (1973)). Using the last action trajectory principle, it is
possible to obtain the path in the configuration space followed by the fissioning system, and the associated
probability of penetrating the barrier. The quantum penetrability
P=1/ħ∫(2EdB)1/2dr
is calculated by using the semi-classical Wentzel-Kramers-Brillouin (WKB) approximation. The region of
interest is classical forbidden The two turning points are fixed by the same values of the potential barrier.
Several quantities intervene in this integral:
Ed=deformation energy, computed within the microscopic macroscopic approach (liquid drop plus
Strutinsky correction based on the superasymmetric two center shell model).
B=tensor of inertia
The inertia tensor is computed in the frame of the Werner-Wheeler approximation, that means, the flow of
the fluid is idealized as nonrotational, nonviscous and hydrodinamical. Along the trajectory the inertia can
be reduced to
B=BRR+2BRC(dC/dR)+2BRR2(dR2/dR)+BCC(dC/dR)2+2BCR2(dC/dR)(dR2/dR)+BR2R2 (dR2/dR)2
Where
BRR, BCC, BR2R2 are the diagonal components along the 3 independent degrees of freedom
BRC, BRR2, BCR2=nondiagonal components
It is very difficult to miniminize the action integral in a 3 dimensional configuration space, so we must
reduce the number of collective variables.
Microscopic approaches to fission show that the second saddle point of the fissioning nucleus is already
asymmetric at a value compatible with the observed final mass ratio (NPA 02 (1983) 213c, J.P. Bocquet R.
Brissot). So, to reduce the number of generalized coordinates to only two in order to make our problem
tractable, we considered that the mass asymmetry is developed between the first and second potential well
linearly. If the mass asymmetry is formed from the second well up to scission, very large effective mass in
the asymmetry generalized coordinates hinder the process of changing the mass asymmetry. So this
approximation reduces the number of generalized coordinates and offer the possibility to minimize the
action integral versus the neck and the elongation.
So the ratio R1/R2 varies linearly from 1 to the value associated to the final partition (that means the volume
of the light fragment is the final one) value. In order to miniminize the action integral, the effective mass
can be now split in 3 parts:
Belongation=BRR+2BRR2(dR2/dR)+BR2R2(dR2/dR)2
Bneck=BCC
Boffdiagonal=BRC+BCR2(dR2/dC)
Where the derivative dR2/dR is known by the variation imposed to R2.
and B=Belongation+Bneck (dC/dR)2+2Boffdiagonal dC/dR
The effective masses for the sim channel Belongation/μ, log(Bneck/μ) and log(Boffdiagonal/μ) are represented in
the following, the reduced mass being μ.
To minimize the least action integral, the WKB functional is approximated with a spline function in the
configuration space defined by a set of (Ri,Ci) values. Ri are apriori fixed and Ci are allowed to vary (i=32).
The integral is computed with 100 mesh points Gauss Legendre quadrature. The integral is minimized
numerically by varying the Ci values. For different exit points of the barrier, several local minimum are
obtained, but in the following we kept only the best result for each asymmetry.
We studied the 236U fission for three mass partitions: 46118Pd+118Pd (symmetric or sim channel),
102
134
Te (102 channel), 3486Se+150Ce (86 channel).
40 Zr+
The effective mass BELONGATION/μ is plotted here for a symmetric split of 236U. Very low values of
BELONGATION (that means a higher probability to penetrate the barrier) are found for small C values (swollen
shapes).
The effective mass log(BNECK/μ) is plotted here. Very large values are found in the region of swollen
shapes.
The same behavior can be found for the effective mass log(BOFFDIAGONAL/μ)
The large values of BNECK and BOFFDIAGONAL suggest that the system evolves with a fixed value of the neck
generalized coordinate during the tunneling.
The potential barriers in the (C,R) plane are plotted for the three channels. The trajectory of minimal action
is displayed with a red line. The trajectory starts from the ground state, reach the region of the second
potential well and change suddenly the trajectory to penetrate the second potential well. The exit point of
the barrier is consistent with the Brosa assumption of a very flat neck, i.e., the shape resembles very much
with the Brosa prescission shapes. However, in this description we do not have a dent which introduce a of
the system, because the system evolves in such manner that sudden change of the trajectory are not possible
due to the inertia. In the present work, the trajectory depends everywhere on the historical path of the
system. Moreover, the condition for instability invoked by Brosa is not fulfilled, the semilength l becomes
larger than 2 times the compound radius rcn, while the instability condition assesses the value 2.4 rcn
(l>2.4rcn). For the sim channel it was computed l=13.3 and rcn=7.16. (The figures are ordered in the
following way: sim, 102 and 86 channel)
For the three studied channel the trajectories resemble. The shapes in the configuration space are displayed
in the next transparencies.
The potential barrier for the three channels are plotted here for a comparison. The barrier for symmetric
split is larger than for asymmetric split, the barrier for the 102-light fragment mass is smaller, but its second
well have an intermediate value between that of sym and 86.
Now, having in mind that the penetrabilities are proportional with the probabilities to produce a mass
partition, that means with the experimental yields, we can compare our results with the experimental values
of the compilation. The experimental yields are 1.310 -2 for symmetry, 6 for 102 and 2 for 86 (normalized to
200%) (Compilation of T.R. England and B.F. Rider)-Top of Figure
Unfortunately, the theoretical trends of the penetrabilities (renormalized for the sim value) are very
different from those given by experimental values (second plot in the figure):
If we neglect the effective mass, the effective mass being considered 1 (third plot in the figure), we obtain a
qualitative agreement (this agreement was invoked already to obtain the optimization of a neutron rich
nuclei source based on fission in M. Mirea, O. Bajeat, A.C. Mueller et al Europ. Phys. J. A 11, 59 (2001)).
But it is not possible to neglect the effective mass. So the experimental distribution must be directed or
ruled by other effects apart the ingredients used as deformation and effective masses. This effect can be the
dissipation.
The coupling of collective degrees of freedom with the microscopic ones causes dissipation and a
modification of the adiabatic potential [A. Weidenmuller, NPA 502 (1989) 387c). The term dissipation
usually refers to exchange of energy (or angular and linear momentum) by all kind of dumping from
collective motion to intrinsic heat, whereas friction is a special dumping process associated with frictional
forces. The collective degrees relaxation times (10-21-10-20) are two or three orders of magnitude longer
than the typical relaxation time of single particle degrees of freedom.
We compute now the dissipation along the trajectory of minimal action. We use again the HFB equation of
motion along the trajectories with a velocity of passage through the barrier given by our fit obtained for
cluster decay. The excitation energy is the difference between the energy given by solving the system of
equations and the energy of the ground state at any deformation.
We found a very exciting phenomena. The dissipated energy is larger for the sim channel and decreases for
the very asymmetric channel 86.
If the velocity is varied, it can be seen that for larger velocities of passage under the barrier, the dissipated
energy increases (example for 86 channel).
Another interesting phenomena, is the fact that the dissipated energy is practically zero up to the second
well and begins to increase when the second barrier is penetrated. So, dissipation appears only when the
barrier is penetrated.
The effect of the dissipated energy is the following: the barrier is increases with the amount of dissipated
energy as remarked in the case of cluster decay and secondly, the dissipation causes a modification of the
Strutinsky effect because the shell effects varies with the heat in a statistic way. This statistical
modification is possible due to very large difference in relaxation times of collective and intrinsic degrees
of freedom.
So the barrier for sim channel is much more increased than that for 86 channel.
So we plotted here the level diagram for the three channels (the 50 levels around the Fermi energy-red line)
used to calculate the dissipation. The dissipation is practically zero up to the second well in all the cases.
The dissipated energy becomes to increases only when the second barrier is penetrated and only when the
neck becomes to vary rapidly (as see on previous transparencies).
It was remarked that close to mass symmetry (F. Gonnenwein, pg 12. XVIIIth International Symposium on
Nuclear Physics, Physics and Chemistry of Fission Castle Gaussig, Dresden, GDR Nov. 21-25, 1988, Ed.
H. Marten and D. Seeliger) there is no measurable odd-even effect in the charge yields δ (difference
between the even and odd total charge yields normalized to their sum). That means the dissipated energy
must be superior to 14 MeV while the average value is approximately 5 MeV (experimental statistic). The
increased number of evaporated neutrons in symmetric mass division and the kinetic energy dip was
usually explained as a large deformation of the prescission shape in this symmetric decay (small TKE) but
in fact can also be an effect due to rearrangement of orbitals. Calculating again the yields within the
dissipation included we find a good agreement with the experimental data (lower part of the figure with the
yields).
As a conclusion, the experimental distributions can be explained not only as a guided evolution of the
nuclear system managed by potential energy and inertia, but also by the effect of dissipation. The
dissipation is ruled by the dynamics of the system and by the reorganization of levels during the fission
process.