. The calculation of these probabilities, expressed as cross sections of the various nuclear interaction processes, is the purpose of this paragraph.

The interaction potential is zero for large distances, or it is reduced to a long distance potential -as the Coulombian one. These potentials are independent of the mutual orientation of spins s  and I  . It results the degeneration on the spin projections, and therefore the

where c y is the amplitude of the spherical incident respectively emerging waves. ' ' ' ' ' ' c

Chapter 1

Introduction

The fission of the minor actinides (Np, Am, Cm, Bk) is a very attractive scientific subject. These isotopes are responsible for the radiotoxicity of the nuclear radioactive waste for more than 1000 years after the fuel has been irradiated; some of them even have a potential terrorist use. The 237 Np isotope is especially important, as it constitutes almost 50% of the minor actinides produced in reactors.

These elements can be destroyed through the fission process in certain reactors or in accelerator -driven systems (ADS) incinerators. The idea behind using ADS for waste incineration is to destroy the plutonium and the transuranic elements through fission and to transmute the undesirable fission products in stable reaction products (or which have a short lifetime). Although this doesn't eliminate the need of storage in repositories, the use of this technique has the major advantage that the amount of waste to be stored decreases by over an order of magnitude. The almost complete elimination of transuranic elements and of some fission products has also the following effects: it substantially decreases the risk of environmental contamination, it allows a better solution of the waste heat problem and it decreases the time during which the deposits are highly radioactive by a factor of 100. These advantages are reflected not only in the increase of the storage capacity of existing deposits, but especially in the simplification of the construction of radiation protection barriers.

The high uncertainty or the complete lack of experimental data for light and minor actinides due to their high radioactivity or the short lifetime of the targets, requires an evaluation based on reaction models with a good predictive power. This is the objective of this work, which fits well in the international efforts (of which it is worth to mention the experiments carried out at IRMM or those at CERN within the program n_TOF). Different models and codes will be used, and the results will be tested by comparison with the experimental data. Fig. 1.1 and 1.2 show the variation of the fission cross-section σ(n,f) with the kinetic energy of the incident neutron (taken from the ENDF-B/VII evaluation) in a wide energy range for the reactions 235 U(n,f) and 239 Pu(n,f) on one side, and for the reactions 237 Np(n,f) and 241 Am(n,f), on the other.

All actinides undergo fission by the absorption of neutrons of any energy. The nuclei of the first case, like 235 U and 239 Pu, have a large fission cross section after the absorption of thermal neutrons. These nuclei are called "fissile" nuclei.

Even if they are not the only ones to undergo fission in reactors, since other nuclei can be fissioned by fast neutrons only, these fissile materials co nstitute the essential contribution to the production of nuclear energy, even in fast neutron reactors.

As it can be seen in shown in Figure 1.1, the fission cross section of fissile nuclei have a resonant character in the energy range 0.1 eV -1 keV; with the increase of the energy this character fades, leading to a slow energy dependence of the fission cross section for incident energies above 100 keV.

The other type of nuclei, like 237 Np and 241 Am fission mostly with fast neutrons, which exceed a threshold energy. This type of nuclei are called "fertile". The compound nuclei of the fertile nuclei have the values of the binding energy higher than the fission barrier, while for the fissile nuclei the binding energy is lower than the binding fission barrier. In this work the neutron induced cross section for both types of nuclei will be evaluated, focusing on the fission cross section, through the following mechanisms: The direct interaction, the compound nucleus and the intermediate mechanism. The direct interaction will be treated by the coupled channel method using the ECIS code [START_REF] Raynal | Notes on ECIS94[END_REF], while the compound nucleus mechanism will be treated by the statistical model, using the codes STATIS (which includes a refined model of the fission process) and GNASH.

Calculations with the code TALYS, in which the direct, statistical and pre-equilibrium mechanisms are included, will be also done, in order to validate this code.

ECIS is a code which uses a sequential iteration method for solving the coupled differential equations arising in nuclear model calculations. It also performs parameter searches to fit calculated results to experimental data. It can be used for a range of structure models, e.g. first or second order harmonic or anharmonic vibrational model, symmetric or asymmetric rotational model, with a similar range of interaction potentials. It includes spinorbit deformation.

The ECIS method is designed to solve the sets of coupled differential equations when the coupling terms are not too strong. The iteration technique searches for the one required solution among the many which are mathematically possible.

The method supposes some ordering of the channels: first the ground state, then the state most strongly coupled to it. All channels must be coupled to some preceding one. The result of each iteration depends on this chosen order. If there is more than one equation related to the ground state the whole calculation must be repeated. The efficiency of the method is proportional to the ratio of the total number of equations to the number of those related to the ground state (Raynal in package NEA-0850/18 in OECD-NEA-DB-CPS).

The code STATIS has been developed by the group of prof. Vladuca at the University of Bucharest [START_REF] Sin | [END_REF]Vladuca et al, 1997 a,b,c). STATIS calculates all the compound nucleus cross section in the frame of the generalized HRTW model. As a function of the excitation energy, STATIS adopts automatically the appropriate form of the decay probabilities allowing a smooth passage from the deep subbarrier excitation energies up to overbarrier energies. The code also includes the effect of the vibrational class II states on the decay-probabilities. Another effect which is also taken into account in STATIS is the gammacascade effect, meaning that after the first emission of γ-rays in the second well, the excitation energy of the nucleus may be released not only through sequential γ -emission, but also through neutron emission (n,γn'), if the excitation energies of the compound states populated by primary gamma-rays are higher than the neutron binding energy, or fission (n, γf), for that part of the γ-ray spectrum which populates the compound nucleus in states lying above the corresponding fission barrier). Therefore, when the neutron capture cross section is calculated in STATIS, the competitive processes (n,γn'), (n, γf) and (n,γγ) are taken into account.

Because of the complexity of the model, STATIS considers only one compound nucleus, thus the particle emission's cross sections can be calculated only for the energy range where the first compound nucleus is formed.

The GNASH code [START_REF] Young | Comprehensive Nuclear Model Calculations: Theory and Use of the GNASH Code[END_REF] provides a flexible method by which reaction and level cross sections, isomer ratios, and emission spectra (neutron, gamma-ray, and charged-particle) resulting from particle-and photon-induced reactions can be calculated. The first version of the GNASH nuclear theory code was completed in 1974. The code has been developed continually since that time and has been often used to supply various types of nuclear data for applications.

GNASH uses the Hauser-Feshbach theory to calculate complicated sequences of reactions and includes a pre-equilibrium correction for binary tertiary channels. Gamma-ray competition is considered in detail for the discrete and continuum spectra, for each decaying compound nucleus.

All particle transmission coefficients are introduced into the GNASH calculations from an external input file (tape10) that is externally obtained, from either spherical or coupled-channel optical model calculations.

Concerning the gamma channel treatment, three model choices for the gamma strength function are included in GNASH: the Weisskopf approximation, the Brink-Axel standard Lorentzian and the Kopecky-Uhl generalized Lorentzian.

For the energy-dependent level densities are also three built-in models: the Gilbert-Cameron model, the backshifted Fermi-gas model and the Ignatyuk form of the Fermi-gas model.

For actinide studies a version of the GNASH code was used, which contains important improvements brought by the research group of the University of Bucharest. This version includes a part of the refined fission model of STATIS. At each barrier, transition states occur that are characterized by the energy above the barrier, as well as by the spin and parity JΠ. At higher energies above the barrier, the discrete transition states are replaced by a continuum of such states, which is described by a density model including enhancement factors relative to the ground state deformation. The fission transmission coefficients are determined through use of the Hill-Wheeler expression for penetration through a parabolic barrier.

The version of the GNASH code that has been used is referred to as the statistical Hauser-Feshbach plus pre-equilibrium version with full angular momentum conservation and with the option to incorporate quantum mechanical pre-equilibrium spectra. The code can be used for calculations at energies between 0.1 keV and 200 MeV.

TALYS [START_REF] Koning | TALYS: Comprehensive nuclear reaction modeling[END_REF] is also a code for the simulation of nuclear reactions.

Many nuclear models are included to cover all main reaction mechanisms encountered in light particle-induced nuclear reactions, similar as in GNASH. TALYS provides a complete description of all reaction channels and observables. It is a versatile tool to generate nuclear data for a variety of applications, as the description of the reaction mechanisms covers a wide energy range (0.001-200 MeV) and mass number range (12 < A < 339).

In TALYS many of the latest nuclear models for direct, compound, pre-equilibrium and fission reactions are implemented. The optical model and coupled-channels ECIS-code is completely integrated; recent optical model parameterisations for many nuclei, both phenomenological (optionally including dispersion relations) and microscopical are also incorporated. The major advantage of the code is the automatic reference to nuclear structure parameters as masses, discrete levels, resonances, level density parameters, deformation parameters, fission barrier and gamma-ray parameters, generally from the IAEA Reference Input Parameter Library. Various width fluctuation models are implemented for binary compound reactions and, at higher energies, multiple Hauser-Feshbach emission take place until all reaction channels are closed.

Various phenomenological and microscopic level density models are implemented: constant temperature + Fermi gas model, back-shifted Fermi gas model, generalised superfluid model, and microscopic level densities from Goriely's table or Hilaire's table.

Different tables with fission barrier parameters are available.

The results obtained by using these four above presented codes will be shown in this work, in comparison with experimental data, where they are available, otherwise with the recent evaluations from the international evaluated nuclear data files.

The thesis is structured as follows:

The direct interaction mechanism is presented succinctly in chapter 2, and the results of the ECIS calculations with three different optical model potentials are given.

Chapter 3 describes the compound nucleus formalism treated by statistical model. The Hauser -Feshbach theory is presented considering the width fluctuations given by the statistical HRTW and the Moldauer model. Here is also the treatment of the radiative capture channel given, and the parameterization of the level density at the ground state deformation is presented. Level density results are shown for all nuclei involved in the studied reactions.

Chapter 4 is dedicated to the study of the fission channel. The double fission barrier is analyzed: the barrier parameters, the different symmetries at the saddle points are described, as well as the transmission coefficients which correspond to the discrete and continuum spectra (including the level density function on the fission path and the corresponding enhancement factors) of the transitional states. The results regarding the neutron induced cross sections together with related comments are given here.

The conclusions are given in chapter 5.

Chapter 2

Direct interaction mechanism treatment for permanent deformed nuclei (actinides)

The formalism of the scattering matrix S

The main purpose of a reaction model is to describe the processes generated during the nuclear interaction of the projectile particle with the target nucleus.

In nuclear reactions result from the general principles of quantum mechanics and the symmetry properties of the nuclear interaction a series of relationships that are independent of the specific details of the reaction. All these relationships, resulted from the symmetry properties of space, time and nuclear interactions, are incorporated in the formalism of the collision matrix. The properties of this matrix are valid for any nuclear reaction. The determination of the collision matrix for a specific reaction, requires the knowledge of the reaction's dynamics -the treatment of the nucleon-nucleus interaction.

The nuclear reactions define the various processes that are generated by the nuclear collisions between the projectile particle a and the nuclear system represented by the target nucleus A. As products of the nuclear interaction result the emerging particle b and the residual nucleus B. Usually a and b are nucleons (neutrons or protons) or light nuclei like deuterons,  particles etc, but can also be nuclei as 12 C, 16 O, etc generating in this case the socalled heavy ion reactions. The result of a nuclear reaction has not necessarily to be composed of only two systems b and B -it can result in only one or even more. In the following, it is assumed that the occuring reaction is A(a, b)B.

In the following it is assumed that the principles of the probability conservation, the principle of the invariance to the inversion of space and time, and the causality principle are true for nuclear reactions. These principles are reflected in the unitarity and the symmetry of the collision matrix.

In order to define the collision matrix it will be assumed in the following that the studied reaction is between a electrical neutral projectile and a target nucleus formed of A nucleons. The generalization to the case of a charged projectile is immediate, so this restriction does not limit the generality of the S matrix defined below.

The Hamiltonian of the target, respectively of the projectile, is

) ( ) , , , ( 2 1 A I A A I A q H r r r H A A      , respectively ) ( a s a q H a . The eigenfunctions ) ( A I A q A A   and ) ( a s a q a a  
, as well as the eigenvalues of the energies A  and a  are defined by

) ( ) ( ) ( A I A A A I A A I A q q q H A A A A A       (2.1) ) ( ) ( ) ( a s a a a s a a s a q q q H a a a a a       where     A A r r r q     , , , 2 1 
are the internal coordinates, i r  is the position vector of the nucleus i to the center of mass of the target nucleus A, and similarly for the projectile a, I A and s A are the spins of the target and projectile, and  A and  a are their parities.

These relations describe the internal states of the target nucleus A and of the projectile a respectively. Let ) , , , , ( A a A a aA aA I s q q r V  be the potential of the projectile-target interaction, where aA r  are the coordinates of the projectile in the system's center of mass. The total Hamiltonian of the system becomes:

aA aA A a A a aA aA aA aA a s a A I A aA V K I s q q r V r T q H q H H a A       ) , , , , ( ) ( ) ( ) (  
(2.2) where ) ( aA r T  is the kinetic energy operator, which describes the relative motion of the projectile relatively to the target nucleus, in the system of the center of mass. K aA is the Hamiltonian of the projectile-target system, when they are sufficiently far apart of each other so that the nuclear interaction is negligible.

The nuclear process is described in the nonrelativistic approximation by Schrödinger's equation:

aA aA aA E H    ;       A a E (2.3)
where  aA is the total wave function of the system,  is the energy that characterizes the relative motion and E is the total energy of the system. The function  aA is the eigenfunction of the Hamiltonian K aA , which satisfies the equation:

aA aA aA E K    (2.4)
Given the meaning of the Hamiltonian K aA , it follows that  aA , for aA aA r k    »1 is defined by [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF]:

aA r k i a s a A I A aA v e q q aA aA a a A A          ) ( ) ( (2.5) where ) exp( aA aA r k i   
is the eigenfunction of the operator ) ( aA aA r T  which describes the relative motion of the projectile towards the target nucleus, v aA is their relative velocity and

T k aA aA 2 2 2   
where  aA is the reduced mass of the system formed by a and A, and T is the kinetic energy of the system in the central mass center, and is defined by [START_REF] Vladuca | Elemente de fizica nucleara[END_REF] When the projectile a approaches the target nucleus A, the interaction between them takes place through the potential V aA , which leads to a strong distortion of the function  aA and, finally, to the creation of various reaction products, denoted generically B + b. The hamiltonian of the system b, B, by analogy with the system a,A will be:

bB bB B b B b bB bB bB bB b s b B I B bB V K I s q q r V r T q H q H H b B       ) , , , , ( ) ( ) ( ) (   (2.7)
Similarly, the eigenfunctions of the Hamiltonian K bB are:

) , ( ) ( ) ( bB bB bB b s b B I B bB r k q q b b B B          (2.8)
In order to simplify the notations, the pair

a,A is denoted     A a,  
, forming the "initial partition" and the pair b,B is denoted

    B b, '   , forming the "final partition". The notation       b B a A q q q q q , ,
   will be also used for the case referring to the conservation of the number of nucleons (the energies involved in these reactions are insufficient for particle generation processes).

The theory of nuclear reactions based on the S matrix formalism enables the calculation of transition probabilities from the initial states, corresponding to the time t =  when the projectile and the target are described by K  , to the final states t = +, when the reaction products are defined by K ' . So, in both, the initial and the final state, the wave functions are similar, ie the eigenfunction of the operator ) ( ' function    can be represented by the "spin channel" function for the spin channel s with the projection  [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF]:

q v r r k O Y i s l lm l m sl    
It can be deduced that the total function  corresponding to the time t = + consists, as the function , of incident and emerging waves, c J respectively c O , except that the amplitude of the emerging wave is modified following the interaction:

             c c c c c s s J y O x (2.13)
The scattering matrix establishes the link between the incident and emerging wave amplitudes according to the relation:

   ' ' ' c c cc c S y x (2.14)
The elements ' cc S forming the scattering S matrix also determine the asymptotic behavior of the wave function after the interaction. In terms of general formalism, the relation (2.14) is described with ket vectors:

     ' ' ' b a b b b S b b S a a (2.16)
If the scalar product of vectors c and a is performed and it is considered that these vectors form an orthonormal basis:

       ' ' ' ' ) ( cc c c c c c c S O y or              ' ' ' ' lmc c c c c c c s s S O y (2.18)
The physical significance of these relations is that in the stationary treatment of nuclear interactions, the function describing the system after the interaction, when the emerging particles are far enough from each other, consists of the superposition of incident and scattered waves.

More specifically, the relation (2.18) can be written as:

      s s r ik s s s s q v r e A q r q r   (2.19) where ) ( ' , ' ' '      The function ) , ( q r s    r s s J d dS dt dS dN d d r s s s s (2.22) where ) , ' ' ' ( '         d A I s d ss s s (2.24)
It is useful to pass from the coupling scheme  

m sl  to the coupling scheme   slJM 
, where the total angular momentum J of the pair , which is preserved, is:

' ' l s J l s          (2.25)
In the new representation, the basis vectors c and ' c are expressed by:

            JM slJM JM sl l s sl c 0 0 0 (2.26)           
and the relation between the elements of the matrix S in the two coupling schemes can be determined:

              slJM slJM slJM slJM J y O x (2.29) where         m m sl slJM JM m sl x x         m m sl slJM JM m sl y y (2.30)
The elements of the scattering matrix S become:

     
To obtain the differential cross section of the process    , relation (2.32) is introduced in (2.24), the sum being performed over pairs of indices  

1 1 1 1 1 m l l M J   and   2 2 2 2 2 m l l M J  
, and then the sums over the magnetic quantum numbers

                  L L L P B s I d d ) ( ) , ( ) 1 2 )( 1 2 ( 4 ) ( 2 ,  (2.34) with                        2 1 2 1 2 1 2 2 2 1 1 1 , * , 2 2 1 1 2 2 1 1 ) ; ( ) ; ( ) 1 ( ) , ( l l l l s s J J J sl l s J sl l s s s L L s J l J l Z sL J l J l Z B (2.35)
where the notation Z stands for the Biedenharn coefficients, defined by the Racah coefficients

W as follows: ) ; ( 0 00 ) ; ( 2 2 1 1 2 1 2 2 1 1 2 2 1 1 sL J l J l W L l l J l J l sL J l J l Z  (2.36)
In this expression the notation 1 2 ˆ  a a has been used.

Because after the summing over the magnetic quantum numbers the only remaining nonzero terms are the ones with 0  M , it follows that the differential cross section

       d d ) (
, depends only on the angle  , meaning that it shows an azimuthal symmetry to the incident beam.

The integral cross section of the process    is obtained by integrating the expression (2.34):

                                  l l s Js J sl l s J g B I s d d d 2 , ' ' 2 0 2 , , ) , ( ) 1 2 )( 1 2 ( ) (   (2.37)
where J g  is the spin factor defined as:

) 1 2 )( 1 2 ( 1 2        I s J g J .
(2.38)

In the obtained relations the Coulombian interaction has not been taken into account. If both reaction partners a and A are electrically charge, the differences from the above written start from the fact that the interaction potential projectile -target ) , , , , (

A a A a aA aA I s q q r V  from (2.2)
is not strictly nuclear anymore, but has also incorporated a term that describes the Coulombian interaction. This term is the potential of a uniformly charged a sphere of radius

3 / 1 A r R c c 
, defined by the relations:

                   c A a c c c A a c R r r e Z Z K R r R r R e Z Z K r V for for 3 2 ) ( 2 2 2 2 ; with 0 4 1   K ; fm MeV 44 . 1 2   Ke (2.39)
where e Z a is the charge of the projectile, and e Z A is the charge of the target nucleus.

In this case, the difference consists in the fact that the spherical Bessel functions

) (   r k j l ,
resulting from development of the plane wave

  r k i e  
in spherical waves, are replaced by

regular Coulomb functions ) (   r k F l
, which have the following asymptotic behavior for r    [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF]:

             e Z Z K A a 2 2  (2.41)
and c l  is the Coulombian phases of the partial wave l ) 1 ( arg

       i l c l (2.42)
These changes have direct consequences for the scattering amplitude of (2.20) or (2.32) by adding a non-zero amplitude only for the Coulomb elastic scattering  and a phase in the expression of

      J sl l s , from (2.33)         where the Coulombian amplitude ) ( c f is defined as   c c i i k f 0 2 2 2 ) 2 / ln(sin exp 2 sin 2 ) (              (2.44)
In this case, the differential cross section (2.34) will contain three terms: a term corresponding to the nuclear interaction, one to the elastic Coulombian scattering and another one for the interference between these processes. The results presented above show that the generalization to the case where both reaction partners are charged is immediate, suggesting that the neglect of the Coulomb interaction does not limit the generality of obtained relations.

Often in nuclear physics, instead of the spin coupling (or l s  coupling), the j j  coupling is used. This coupling is defined by the relations:

j l s       ; J I j       (2.45)
In this coupling, the differential cross section for the process    is [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF])

                  L L L P A s I d d ) ( ) , ( ) 1 2 )( 1 2 ( 4 ) ( 2 ,  (2.46) with                                          ) 1 ( ) , ( J j l j l J j l j l l l l l j j j j J J s s I I L L I j J j J W L I j J j J W L s j l j l Z L s j l j l Z J J A (2.47)
From this expression the cross section of the process    can be obtaind through integration:

                j j l Jl J lj j l J g 2 , 2 ,  (2.48)
For the case of the elastic process    :

                             J lj j l J lj j l J lj lj J S S g 2 , , 2 , Re 2 1  (2.49)
A particular case of the elastic process, called "shape elastic" (which is the only coherent process) is the one in which all quantum numbers remain unchanged (the process in which the mutual orientation of the spins of the projectile and the target nucleus is the same before and after the scattering, so the probability of "spin-flip" is zero

   , l l   , j j   )                         J lj J lj lj J lj lj J se S S g 2 , , 2 , Re 2 1  (2.50)
The cross sections of the non-elastic processes are expressed by:

                   J j j l l J lj j l J S g 2 , 2 ) ( ,  (2.51)
Using this expression and using the unitarity property (2.17) of the matrix S, we can express "reaction cross section", defined as the cross section of all non-elastic processes, ie the sum of all possible processes   

, except the elastic one

) (    :                                  J lj j l J lj j l J r S g 2 , 2 ) ( , 1  (2.52)
The expression of the total cross section of all processes generated by the partition  is:

                            J lj J lj lj J r t S g , 2 , , Re 1 2  (2.53)
From the previous relations it is visible that the cross section of the shape elastic process se    , and the total cross section t   are expressed only in terms of the diagonal elements of the matrix S.

The optical model

Knowledge of accurate cross sections of a number of neutron-induced reactions (e.g. total, elastic and inelastic scatterings, capture, fission, and so on) on actinides is crucially for designing various reactor systems. Especially for accelerator-driven system (ADS) of radioactive wastes transmutation and energy generation, the nuclear reaction data are needed for neutrons up to several hundred MeV. Also proton induced reactions on actinides are important in various applications.

The optical model is one of the fundamental theoretical tools which provides the basis of nuclear reaction data analysis and nuclear data evaluations for applied purposes. It suggests a convenient approach for predicting various cross sections.

Reaction mechanisms make different hypotheses on the dynamics of nuclear reactions.

The reaction mechanisms are closely correlated with the nuclear structure models and reflect the "extreme" assumptions which they are based on. For example, the direct interaction mechanism as well as the intermediary mechanism involve a weaker interaction between the projectile and the target nucleons; although entering the nuclear interaction potential, the nucleons of the projectile can move without forming a compound nucleus; a "compound system" is formed, in which the behaviour of the projectile nucleons can be individualized.

The weak interaction between the projectile and the target nucleons can be understood if we assume that the target nucleus is correctly described by the nuclear shell model: the target nucleons are positioned on single-particle levels, consistent with Pauli's principle. In this case only those interactions between the projectile and target nucleons are possible, which lead to the occupation of free single-particle levels. Because the projectile and target nucleons interact relatively weak, the number of interactions between the nucleons is small. Taking into account all these, it can be considered that the direct interaction model is an "extension" of the nuclear shell model in the positive energy range.

Among the models which supply nuclear data for applications, the direct interaction mechanism, which can be described by the optical model, provides the basis for many analyses. Besides the calculation of reactions, direct elastic and inelastic scattering and total cross sections, the direct models supply the neutron transmission coefficients necessary for the statistic models used in data evaluation. For permanent deformed nuclei, as actinides, the dependence between different channels being quite strong, the direct calculations have to be performed in the frame of the coupled channel model.

The importance of optical model parameterizations is made even more apparent by the worldwide diminution of experimental facilities for low-energy nuclear physics measurements and the consequent increased reliance on theoretical methods for providing nuclear data for applications. Therefore the preservation of past work aimed at describing experimental results with optical model potentials is vital for the future development of nuclear databases.

Additionally, the availability and use of microscopic optical model codes is important for predicting data for target nuclei far from the line of stability, where phenomenological models might not be valid.

The central assumption underlying the optical model is that the complicated interaction between an incident particle and a nucleus can be represented by a complex meanfield potential, which divides the reaction flux into a part covering shape elastic scattering and a part describing all competing non-elastic channels. Solving the Schrödinger equation numerically with this complex potential yields a wealth of valuable information. First, it returns a prediction for the basic observables, namely the elastic angular distribution and polarization, the reaction and total cross section and, for low energies, the s, p-wave strength functions and the potential scattering radius R'. The essential value of a good optical model is that it can reliably predict these quantities for energies and nuclides for which no measurements exist. Also, the quality of the not directly observable quantities that are provided by the optical model has an equally important impact on the evaluation of the various reaction channels. Well-known examples are transmission coefficients, for compound nucleus and multi-step compound decay, and the distorted wave functions that are used for direct inelastic reactions and for transitions to the continuum that describe statistical multistep direct reactions. Also, the reaction cross sections that are calculated with the optical model are crucial for the semi-classical pre-equilibrium models.

Because actinides are expected to be well deformed rigid rotors, where low-lying collective levels are strongly excited in nucleon inelastic scattering, the deformed nuclear optical potential arises from deformed nuclear shapes

               2,4,6 λ λ0 λ0 0 , θ Y β 1 R ) , θ R(   (2.54)
where Y λ0 means spherical harmonics,   , θ are angular coordinates in the body-fixed system and 1/3 0 A r R 

(the corresponding radius for the spherical shape).

In the optical potential parameterization a standard Wood-Saxon form factor is considered

      1 i i i i WS /a R r exp 1 a , R r, f     (2.55)
taking into account the deformed nuclear shapes defined by (2.54). This factor takes into account the fact that the nuclear interaction decreases fast with the distance to the nucleus, and inside the nucleus it's practically uniform (due to the nuclear saturation forces a nucleon placed inside the nucleus "feels" only the nucleons from his close proximity).

A deformed optical model parameterisation is usually given by the following equation [START_REF] Koning | Local and global nucleon optical models from 1 keV to 200 MeV[END_REF]:

V(r,E) = -V V (r,E) -iW V (r,E) -iW D (r,E) + V SO (r,E)   l σ ˆ + iW SO (r,E)   l σ ˆ + V Coul (r) (2.56)
where V V,SO,Coul and W V,D,SO are the real and imaginary terms of the volume (V), surface (S), spin-orbit (SO) and coulombian (Coul) potentials, respectively. E is the laboratory energy of the incident particle in MeV. All components are separated in energy-dependent well depths and energy-independent radial parts:

V V (r, E) = ) a , R (r, (E) V V V V f W V (r, E) = ) a , R (r, (E) W V V V f W D (r, E) = ) a , R (r, dr d (E) W a 4 - D D D D f V SO (r, E) = ) a , R (r, dr d r 1 c m (E) V SO SO 2 π SO f          W SO (r, E) = ) a , R (r, dr d r 1 c m (E) W SO SO 2 π SO f          where   i i a , R r, f
the Wood -Saxon form defined in (2.55) is.

In the dispersion relations treatment, the real (V) and imaginary (W) volume potentials are connected by a dispersion relation

) ( ) ( ) ( E V E V E V HF    (2.57)           E d E E E W P E V ) ( ) (  (2.58)
where P denotes the principal value of the integral and V HF (E) the Hartree -Fock contribution to the mean field.

The volume imaginary potential at the energy E in the center of mass system is parameterized as:

    2 v 2 2 v v B E E E E C W F F     (2.59)
where C v and B v are constants.

The surface imaginary potential is assumed to be similar, but multiplied by a damping factor

        F F F F E E r E E c B E E E E C W         exp 2 s 2 2 s s (2.60)
C s and B s are parameters similar to those of the volume imaginary potential, the parameter c introduces an exponential decrease of the surface potential and r is a non-locality range parameter.

The real and imaginary volume terms share the same geometry parameters, the radius r V and the diffuseness a V , and likewise the real and imaginary spin -orbit terms share the same r so and a so parameters. The geometries of the real and imaginary volume terms of the optical model potential are usually assumed to be energy-independent.

Phenomenological deformed potentials for actinides

In the following three deformed optical model parameterisations recently developed will be presented and used: a parameterisation proposed by R. Capote (Capote et al., 2005a(Capote et al., ,b, 2008)), one by E. Souhovitskiĩ (Soukhovitskiĩ et al., 2004) and one by P.Romain from the Bruyères-le-Châtel group (Morillon and Romain, 2004).

The detailed parameterisations for these potentials are given in detail in Annex 1.

Calculations using these three optical model potentials are exemplified in this work for the following cases: n+ 241 Am, n+ 237 Np, n+ 235 U and n+ 233 U. The optical model calculations have been performed using the code ECIS [START_REF] Raynal | Notes on ECIS94[END_REF].

An example of depths of the three mentioned potentials is given for the reaction n+ 237 Np is given in Fig. 2.1. It is visible that the values of the imaginary volume depth, the real and imaginary spin-orbit depths, plotted with red, cyan and magenta lines, are very close to each other for all three potentials. Only the real potential depths, plotted with black lines, and the imaginary surface depths, plotted with blue lines, show differences. The effect of the number of coupled levels is ilustrated in Fig. 2.2 for the case of the BRC potential applied to n+ 237 Np. Calculations of the total cross section are done by coupling 3 levels of the fundamental rotational band (plotted with green), 4 levels (plotted woth red)

and 5 levels (plotted with blue). The effect of the number of coupled levels is visible in the low energy part and it consist in the decrease of the total cross section with the increase of the neutron energy, and at higher energies, at about 3 -4 MeV, the total cross section gets saturated.

The importance of the choice of the deformation parameters β 2 and β 4 is exemplified in Fig. 2.3 for the total cross section of the BRC potential and the reaction n+ 237 Np. Total cross-section (MeV) En (MeV) Fig. 2.2. The total cross sections of n+ 237 Np resulting from coupling different numbers of levels, using the BRC potential. The results of the total cross section obtained by coupling 3 levels are plotted with green, by coupling 4 levels with red and the cross section obtained by coupling 5 levels with blue. Total cross-section (MeV) En (MeV) Fig. 2.3. The total cross section of n+ 237 Np resulting from using different deformation parameters in the BRC potential. With a green line is the total cross section plotted which has been obtained by using the deformation parameters ß 2 =0.207 and ß 2 =0.070, with red the total cross section obtained using the deformation parameters ß 2 =0.215 and ß 2 =0.070 and with blue the one obtained by using by using ß 2 =0.207 and ß 2 =0.102.

An increase in the β 2 parameters results in a decrease of the total cross section between 1keV and 1 MeV and an increase in the energy range 1 -6 MeV. An increase in the β 4 parameter plays a role at small energies, up to 0.1 MeV, lowering the total cross section.

The agreement between the coupled channel calculations using the studied parameterizarions and the experimental data concerning the total cross-section has been studied for all three cases in two incident neutron energy (E n ) ranges: E n from 0.001 up to 1

MeV, and for E n from 1 to 30 MeV.

Results for the reaction n+ 237 Np

In the case of n+  Spin-orbit parameters:

V so = 5.75 λ so = 0.005 W so = -3.1 B so = 160  Geometric parameters: r v = 1.255 a v = 0.58 r s = 1.15 a s = 0.601 r so = 1.15 a so = 0.59

The deformation parameters given in Table 2.2. were used. The deformation parameters given in Table 2.4 were used.

Chapter 3

The compound nucleus mechanism treated by statistical model.

The compound nucleus mechanism

A scenario for the development of a nuclear reaction is that the projectile, once inside the interaction area of the target, suffers a collision with one of the target nucleons. The transfer of energy to the nucleon is great enough that the projectile gets to a bound single particle level and loses its individuality. All A nucleons of the system are on bound single particle states.

The eigenvalues of the total Hamiltonian 0 H are denoted by   and the eigenvalues of the

energies by          0 H        (3.1)
Later, the energy gets redistributed on a nucleon or a group of nucleons, and different processes can take place, for example elastic or inelastic scattering, transmutation; the formation of the compound nucleus implies a long time on the nuclear scale ( The compound nucleus resonances are "isolated", if their average width   is smaller than the average level spacing  D :

    D (3.2)
The average level spacing  D for a given nucleus is defined by

) ( 1 *     D (3.3) where ) ( *  
is the level density of the nucleus corresponding to the excitation energy , where the compound nucleus resonances overlap. A "continuum" in which the states lose their individuality is formed, generating in the excitation functions a fluctuating energy dependence, which, in average, varies slowly with the energy -dependence which is typical for the direct reactions [START_REF] Vladuca | Elemente de fizica nucleara[END_REF]. Thus, it is impossible to say whether the slow variation with the energy of the experimental cross sections corresponds to the direct reactions, or is the mediated effect of the overlapping compound nucleus states, especially as for these energies the compound nucleus reactions strongly compete with the direct reactions.

In the dynamic models, besides the actual direct reactions, the averaged effect of the compound nucleus states is also included, and the compound nucleus models contain only a part of the reactions which are taking place through the compound nucleus mechanism. Thus, theoretically, the dynamic models are defined by expressing the collision matrix S for the process c c   as a sum of the averaged matrix elements and a fluctuating part fl c c S  :

fl c c c c c c S S S      (3.4)
The averaging range I should be chosen so that 0

  I fl c c

S

, although the terms [START_REF] Vladuca | Elemente de fizica nucleara[END_REF].

I fl c c fl c c S S * 4 3 2 1 are different from zero
To obtain an averaged form of the collision matrix, the starting point is an expression of the S matrix obtained in the theory of resonant nuclear reactions. The collision matrix in the "multilevel approximation" is:

                              2 / ) ( , i E i S c c c c i c c c c e (3.5)
where c  and c  are "scattering phases", with a slow kinetic energy dependence; c   and c   are the partial decay widths of the state , which fluctuate rapidly with the energy at which the reaction is taking place. Between the partial and the total decay widths exists the following relation:

        c c (3.6)
In order to average this expression of the matrix S, a "rectangular" weighting function is used:

               2 for 0 2 for 1 ) , ( I E E I E E I E E R (3.7) In the region in which E > . cont E the resonances overlap, so that        d D  1 (3.8)
Averaging the second term of (3.5), the following is obtained:

  I c c i c c i c c i c c i r c c D d I D I E d E i E I i S c c c c c c c c e e e e                                                                             ) ( ) ( ) ( 2 2 ) ( 1 4 / ) ( 2 /
where the integration limits have been extended to   (the integration interval is considerably larger than the widths   ; the compound nucleus resonances have a Lorentzian shape, and decrease rapidly to zero for values of E very different of   ) (Vladuca, 1990).

The averaged elements of the matrix S have the following expression:

                        c c c c i c c D S c c e ) (
(3.9)

The cross sections obtained experimentally can be decomposed as

fl c c ID c c c c         (3.10)
The equations (3.4) and (3.10) are the basis of the direct interaction and the compound nucleus models. The direct interaction mechanism is defined by The dynamic fluctuation models try to evaluate the second term of expression (3.10) using statistical methods; for this reason they are also called statistical models.

Generally, when it is necessary to evaluate the physical characteristics of complex systems, it is appealing to rely on statistical methods in order to avoid laborious, sometimes impossible, calculations, which would result from solving the equations of motion for all elements of the given system. Additionally, it often happens that the available information about the internal structure of the system and the interactions between its constituent elements are very poor (the atomic nucleus is a typical example), in which an a-priori statistical treatment is necessary in order to predict accurately the average physical parameters.

The application of statistical methods, in which the atomic nucleus is regarded as a statistical ensemble consisting of protons and neutrons, have to be considered with caution for the following reasons: the nucleus is composed of a relatively small number of nucleons, which interact strongly with each other; the nucleus is a closed system with a finite size, which's properties are strongly influenced by the adding or removing of a nucleon [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF]. Despite these remarks, the statistical interpretation of the nucleus has however proven to be very successful in the quantitative description of many nuclear properties such as the level density, the widths of these levels, the average cross sections of the reactions which result by the formation of the compound nucleus. The fission process and the radiative capture take place exclusively through the compound nucleus mechanism, which shows once more the importance of the statistical models.

The Hauser-Feshbach theory

In order to obtain the contribution of the compound nucleus mechanism in the S-matrix formalism, it is necessary to calculate the term

* fl cd fl ab S S
, where the indexes a, b, etc stand for the quantum numbers  

    1 1 1 J j l ,     1 1 1 J j l
, etc. As it is very difficult to calculate this term, some hypotheses have been made [START_REF] Vladuca | Elemente de fizica nucleara[END_REF]. First of all, the S -matrix is considered to be unitary, with the conservation of the flux

ac b cb ab S S    * (3.11)
The S-matrix is also time invariant.

In the concept developed by Bohr, the decay of the compound nucleus is independent of its formation. This hypothesis, which removes any correlation between the input and output channels, is expressed mathematically as Given that the S-matrix is unitary and symmetric, the matrix P is Hermitian. In addition, comparing the relations (3.11) and (3.14), it can be concluded that while the Smatrix is unitary, the averaged matrix S isn't, and the matrix P is just a measure of "unitarity deficit" of the averaged matrix S . For an evaluation of these expressions, a direct dynamic model, that provides the elements of the matrix P, has to be used. The direct interaction model and the compound nucleus model have to be always used together, in order to achieve a good evaluation of the cross sections. In particular, the statistical model Hauser -Feshbach can be used in tandem with the optical model. Considering this fact, a further assumption is introduced: the only direct process is the elastic scattering In these conditions, the transmission matrix P becomes diagonal, and the elements ac P can be written as

  ac a ac aa ac T S P      2 1 (3.21)
where a T are the optical transmission coefficients. The expression (3.17 which is the formula derived by Hauser and Feshbach (Hauser and Feshbach, 1952). In this case, the above relation gives both, the differential and the integrated section.

The Moldauer statistical model

Even if the statistical hypothesis can be accepted as being true, the hypothesis that the decay mode of the compound nucleus is independent of the way of the formation, remains debatable. Even more questionable is the most restrictive of the assumptions above, that the only direct process is the elastic scattering one, which eliminates direct reactions. These reasons impose a reformulation of the statistical model, starting from the statistical matrix-S [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF]. The matrix S must satisfy the following conditions: it has to have such a form, that the kinematic effects, which lead to slow variations with the energy (except for threshold effects), are well separated from the dynamic effects, so that the mediation is simple; it has to be unitary, symmetric and analytic; the statistical parameters defining the S-matrix are known (and they must result from the intrinsic properties of the physical system).

The most common expression of the S-matrix in an interval E  I, where I is the averaging interval, and for a fixed value  J for the spin and parity of the compound nucleus, which satisfies the first condition previously mentioned is [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF])

p ab ab b a ab J ab S S i E E g g i S S               0 0 2 (3.23) with      N g a a 2 ; 1   N ;       a a (3.24)
where E is the energy of the system,

    2 i E
is the "complex resonance energy" for the state  of the compound nucleus, the sum over  includes all resonances in the averaging range E (E »   ); the remaining resonances are included in the background matrix 0 ab S , which is considered symmetric and energy independent (this component of the matrix S is associated to the direct processes, but without fully describing them). This form of the Smatrix is very similar to the equation (3.5), except that the partial decay widths a   of the state  have been replaced with partial amplitudes a g  of the state  of the compound nucleus, which are characterized by the spin and parity  J , for the channel a defined by the quantum numbers   lj  in the j-j coupling.

The averaging S , over the energy, is performed according to

         E d E E E S E S ) , ( ) ( ) ( (3.25) where ) , ( E E   is a weight function. For the interval I ) ( ) ( 0 iI E S g g D S E S ab b a ab I ab         (3.26)
If 0 ab S is symmetric, it follows from expression (3.23) that the matrix S is also symmetric. The unitarity of the matrix S can be extended over complex values of the energy E as follows:

ab c cb ac E S E S    ) ( ) ( * * (3.27) or   ab ab iI E S iI E S ) ( ) ( 1 *     (3.28)
It can be proven that, similarly to equation (3.26), the following equality takes place

       b a ab ab g g D S iI E S 0 ) ( (3.29) Combining (3.26) and (3.29) it results         b a ab ab g g D iI E S iI E S 2 ) ( ) ( (3.30) As I ab ab S iI E S   ) (
, it results from the previous relation, and taking into account (3.28), that

             b a ab ab g g D S S 2 1 * (3.31)
This relation, which is equivalent to equation (3.30), is called the "Moldauer sum rule" and it is the test of unitarity and analyticity for the S-matrix, as it resulted from the unitarity and analyticity condition expressed in (3.27). Consequently, the matrix S defined by (3.23) must satisfy the sum rule, because any deviation from this relation automatically is a violation of the unitarity.

In obtaining the sum rule no statistical assumption on the S-matrix has been made.

Porter and Thomas (Porter and Thomas, 1956) have shown that the neutron decay widths (the observable

a a    
) follow the 2  distribution. Trying to make the distribution of this observable to be consistent with the experimental data, it has been observed that there is a close correlation between the number of degrees of freedom of the 2  distribution, noted , and the number of possible decay channels.

Having defined the statistical matrix S, it's properties will be used in order to evaluate the mediated bilinear product

                                      2 2 * * * * 2 2 * ) ( 2 ) ( 2 ) ( 2 I E E g Ig iI E E g g i i E E g g g g I E E I i S S d c b a d c b a fl cd fl ab (3.34)
For the first term of equation (3.24) the following two cases will be considered:

i)    ;                       * * * * 2 2 2 ) ( 2 d c b a d c b a g g g g D i g g g g I E E I i ii)    ;                          ) ( 2 ) ( 2 * * 2 2 i E E g g g g I E E I D dE i d c b a                                              D g g g g D i E E g g g g D i d c b a d c b a 2 2 ) ( 2 2 * * 2 2 * * where                     i R d D i D ) ( 2 (3.35) ) ( 2  R
is the Dyson correlation function [START_REF] Dyson | [END_REF], which expresses the probability that any level  E is found at the distance  from the level  E :

        x dt t S dx x dS x S DR ) ( ) ( ) ( 1 ) ( 2 2 (3.36) x x x S sin ) (  ; y D x     
If there is no correlation between the levels,

M g g g g D g g g g D D g g g g D g g g g D S S                                                    * * * * 2 2 * * 2 2 * * * 2 2 2 2 2 (3.37)
where abcd M stands for the "level-level correlations term", considering not only the repulsion between levels, but also the correlations between a g  and b g  for   .

To obtain the term 2 fl ab

S

, firstly obtained by Moldauer (Moldauer, 1961(Moldauer, , 1963)), which is necessary to evaluate the integrated cross sections, the conditions a  c, b  d are applied to the previous relationship: In order to evaluate the expression (3.40), it would be desirable to be able to neglect ab M . In the following it will be shown in which conditions this is possible.

ab b a M fl ab M g g D S         2 2 2 2 (3.38) Introducing the notations 2 2 a a g N D       and      
To this end, the event where the direct reactions are excluded, will be consider, case in which (3.20) is valid and implies the diagonality of the background matrix 0 ab S and

ab a b a g g g       2 (3.42)
This relation shows that the reduced amplitudes a g  from different channels are not correlated. In this case, the coefficients are reduced to optical transmission coefficients a T and the sum rule (3.31) becomes This 2  distribution with  = 1 has resulted as a consequence of the unitarity condition for the situation where the direct reactions and level-level correlations contained within ) is distributed according to the 2  law, it can be proven that the Moldauer correction term ab W becomes [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF])

) 2 exp( ) 1 ( 2 2 / 1 2 a a a a i T T g D       (3.43)
                0 2 / 2 1 c c b a ab b ab c f f f dt W (3.49) with     c c a a a T T t f 2 1 .
The degrees of freedom are dependent on the characteristics of each channel. It can be shown [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF]) that a  varies from 1 (for poor absorption and a small value for Because the integral in (3.35) is extended to +  may hinder the numerical calculations, in practice the following change of variable is made

t y   1
1 and thus the infinite integration limit of (3.49) turns into a finite one. The following expression is obtained

                          1 0 2 2 ) 1 ( 2 ) 1 ( 2 ) 1 ( 2 2 1 dy y T y y T y y T y y T W b a c f c c ab b ab         (3.50)
This integral can be calculated analytically for the particular case of only two open channels a and b and if the 2  distribution of the resonance widths of the compound nucleus has one or two degrees of freedom. Summarizing the results it follows that

             b a W b a W ab ab pt. ) 3 1 ( pt. 1 5 . 0 1 1                b a W b a W ab ab pt. ) 2 1 ( pt. 1 3 2 2 2 (3.51)
This relations show the importance of the correction of the level fluctuations widths. This correction, regardless of the value of , favors the re-emission in the incident (elastic) channel (or channels), disfavoring (as a consequence of the flux conservation) the other "nonelastic" channels. It follows that the compound nucleus doesn't completely "forget" the formation way (the hypothesis of Age Bohr), and favors it in the decaying process. Thus, with this correction, one of the basic assumptions of traditional statistical model is contradicted: the independence of the decay mode of the formation mode. It appears however, that if there is a large number of open channels, the Moldauer correction factor for the reaction channels is practically equal to the unity, regardless of the value of  [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF]. This explains the success of the standard HF formula (3.22) in compound nucleus' interpretation of the nuclear reactions.

The statistical treatment in the HRTW version

One of the major shortcomings of the ep (3.50) is a rigid choice for the values of  to be 1 or 2, without considering its dependence of the reaction channel [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF]. Tepel (Tepel et al., 1974) solved this problem by factorizing the average where, contrary to Bohr's hypothesis, the enhancement factor a G of the elastic scattering cross section is introduced, which takes into account the fact that the partial widths for the incident and emergent channel are identical and therefore correlated [START_REF] Berinde | Reactii nucleare neutronice in reactor[END_REF].

If direct interactions are neglected and the notation

  2 1    c c a a V V (3.54)
is introduced, for optical transmission coefficients a T the following relation is obtained (Hofmann et al, 1975)

) 1 ( 2 2       a c c a a b fl ab a G V V V S T (3.55)
For the averaged elements 2 fl ab

S

Tepel (Tepel et al, 1974) obtains

  ) 1 ( 1 2      a ab c c b a T fl ab G V V V S (3.56)
In order to correctly determine the correction a G and the terms a V , the S-matrix should be generated numerically, meaning that the statistics of the parameters that define this matrix should be well known. However, the unitarity of the matrix S imposes further correlations to the parameters and their statistics becomes so complicated that a numerical simulation of the statistical matrix S becomes very difficult, if not impossible. To solve this problem, the K

matrix is introduced           E E K K b a ab ab 0 (3.57)
which is in the following relation to the matrix S

1 ) 1 )( 1 (     iK iK S (3.58)
The advantage of the matrix K consists in the fact that the unitarity and symmetry conditions of the S-matrix imply that the matrix K is symmetric and real. The matrix K is not unitary and hence the statistics of the parameters of the matrix K is not affected by this condition. It is considered that the terms a   have a normal distribution with the average of zero, and the poles  E are distributed according to Wigner's law:

dy y y dy y P W            2 4 exp 2 ) ( (3.59) D E E y ) ( 1     
These values are randomly generated by numerical methods so that they follow the above mentioned distributions. Knowing these parameters the elements of the matrixes K and S can be calculated numerically with the relations (3.57) and (3.58). The averaging of the matrix S is more difficult, but it can be also made numerically by replacing the energy mean with the average over a large enough set of the numerically generated statistical matrices S.

Knowing all these quantities, the numerical ratio

T fl ab T fl aa S S 2 2
, which generates the values of the enhancement factor a G as a function of a T , can be performed. After fitting these values, a first approximation is obtained, an empirical expression for a G (Tepel et al, 1974)

a a T G    1 1 2 1 (3.60)
Once a G is correctly defined, the equation (3.55) has to be solved in order to determine a V . The equation (3.55) has a unique solution, which is difficult to determine, so an iterative solving is preferred.

If  c c a V V is substituted by  c c a T T , the solution is 1 ) 1 ( ) 1 ( 1                 a c c a a a a G T T T V V (3.61)
The link between the HRTW and the Moldauer models can be studied. It will be considered that only two channels, a and b, are open. In agreement with (3.61) for a V and b V , which correspond to these channels, can be written

A T g T T T T V a a b a a a a     1 B T g T T T T V b b b a b b b     1 (3.62) 1 1     b b a a G g G g
The averaged elements in Tepel's version becomes

ab b a b a b a b a b a T fl ab R T T T T A T B T T T S      ) ( 2 (3.63)
The correction ab R is then

) ( 1 1 1 b a a b a b a b b b a b a a b a ab g g r r r r g T T T T T g T T T T T T T R              (3.64) where a b T T r  ; a a T g   1 2 ; b b T g   1 2 (3.65) If b a T T  (following that r = 1) the Tepel correction becomes a ab g R   2 2 (3.66) If a T « 1 ( 2  a g ) the correction 5 . 0  ab R
is identical to the Moldauer correction

factor for  = 1. If 1  a T it follows that 1  a g and 3 / 2  ab R
, corresponding to the Moldauer correction for  = 2.

The conclusion that can be drawn, is, that for all real cases the Tepel correction is situated between the Moldauer correction corresponding to one degree of freedom, and the one corresponding to two degrees of freedom, being closer to the physical reality.

The relations (3.61) and (3.61) show that the main error source is given by the precision of the determination of a G . For this reason, Hofman repeated the calculations (Hofman et al, 1975), increasing significantly the number of levels  and the number of numerically generated samples, obtaining a improved fit formula for a G V is insufficient, especially for more than 10  20 channels. For a V higher order iterations have to be used

  2 1 5 . 1 3 . 0 ) ( 2 1 2 1                c c a T T a a T T T T G c c
1 ) 1 ( ) 1 ( ) ( ) 1 ( 1                  a c m c m a a m a G V V T V (3.68)
where m, the number of iterations, is chosen so that the compound nucleus cross sections calculated with the a V coefficients, obtained in a higher iteration, do not differ significantly from the cross sections obtained in the m-th iteration.

In order to determine the differential cross sections, the mean

i i X X S S     c a  (3.70)
where the real phases a  result from

) 2 exp( ) 1 ( 2 / 1 a a aa i T S    (3.71) and ) 4 1 ( 2 a a a a y y y X    (3.72)     1 2 / 1 2 / 1 15 . 0 1 ) 1 (       c c a c c a a T T V V y (3.73)
Thus, the function (3.69) is completely determined.

The particle transmission coefficients in the extended HRTW version

In the statistical model calculations the necessity to evaluate the term  c c T appears.

In this term the sum extends over all open reaction channels, which are compatible with the selection rules for the spin and parity. The neglect of channels reflects in the over evaluation of cross sections. Each of the competitive processes (elastic and inelastic scattering, fission and radiative capture) will be characterized of corresponding transmission coefficients.

If the only direct process is the elastic one, the transmission coefficients are calculated using eq. (3.21), after having first performed an optical model calculation. In the S matrix formalism the transmission coefficients are given by 1 ) (

j l nlj S E T   (3.74)
But the scattering of the particles on deformed nuclei, where the rotational or vibrational excited states are strongly coupled, isn't well described by the optical model. In this case the coupled channel method has to be applied. To calculate the transmission coefficients the following averaging procedure can be used (Lagrange et al, 1983)

            j I j I J J nlj nlj E T j I J E T ) ( ) 1 2 )( 1 2 ( 1 2 ) ( (3.75) where          j l J nlj j l n J nlj S E T 2 , 1 ) ( (3.76)
Thus the problem is reduced to a statistical model calculation associated to the optical model, in which the direct nonelastic processes are neglected (in fact they are taken into account through the above mentioned mediation process).

With these coefficients, the contribution of the particle channels to the sum  c c T can be calculated. This term will be denoted part T , and noting that, for the calculation of any cross sections, the element  c c T is included in a sum after the compound nucleus' spin, it can be stated that the  c c

T and part T depend on the quantum numbers J and . This means that the sum over the transmission coefficients is performed by taking into account the conservation laws of the total angular momentum and of the parity.

The term part T is calculated by summing all transmission coefficients which correspond to the opened inelastic channels. As these coefficients depend on the particle's energy in the mass center, it is necessary to know the energy of the residual nucleus' level and also some of its properties (spin and parity) to be able to apply the conservation laws.

If the excitation energy of the compound nucleus is not too high, only the inelastic scattering processes on discrete states can take place. The properties of these states can be obtained from nuclear databases (RIPL3) or from estimations provided by nuclear structure models. As the excitation energy increases, the levels get closer to each other while their width increases. Thus, above a certain excitation energy, the levels can't be separated anymore, so that the description of their properties is done using the level density function, which will be presented in chapter 3.4.

Thus the coefficient ) , (  J E T part will be expressed as the sum of two terms

) , ( ) , ( ) , (      J E T J E T J E T cont part dis part part (3.77)
The first term describes the particle emission on discrete low lying energy levels i of the target nucleus The second term of the equation (3.77) is given by the contribution of the inelastic scattering processes after which the residual nucleus is found in an excited state, which lies above the last discrete level

       i lj l i i lj n dis part J E T J E T i ) ) 1 ( , ( ) ( ) , ( (
                            j J j J I m M M E E lj l I I T n T lj n cont part n T T c i d I J M m M E T J E T ) 1 ( , ( ) , ( , ) , ( (3.80)

The generalized HRTW version of the statistical model

Using the previously defined notations, eq. (3.56) can be rewritten for the averaged elements 2 fl ab S in Tepel's version (Tepel et al, 1974) more explicitly as

1 ) 1 ( * ) 1 ( ) ( ) 1 ( ) , ( ) , ( ) , ( ) , ( 1 ( 1 ) , ( 1                       lj n m part i m lj n f i lj n m lj n G J E V J E T J E V J E P J E T V i i i (3.81)
where the quantities ) , (

 J E V i lj n i in the m -th iteration are obtained from eq.(3.68) 1 ) 1 ( * * ) 1 ( ) ( ) 1 ( ) , ( ) , ( ) , ( ) , ( 1 ) , ( ) , ( 1                        lj n m part f I i m lj n i lj n i m lj n G J E V J E T J E T J E V J E T J E V i i i (3.82)
where P f is the fission probability, which will be defined in chapter 4.2.2. and the enhancement factor G a is defined by eq.(3.67).

If eq.(3.81) is introduced in the expression of the integrated cross section of the compound nucleus for the process

i n n  , it follows      J n CN n CN n n J E P J E E i i ) , ( ) , ( ) ( ,   (3.83) where ) , (  J E CN n 
is the cross section of the compound nucleus formation in a state characterized by the spin and parity J, after the interaction with incident neutrons of the

energy E       lj lj n J n n NC n J E V g J E ) , ( ) , ( 1 2  and with ) , (  J E P i n
the disintegration probabilities (Vladuca et al, 1997c)

) , ( ) , ( )] 1 ( 1 )[ , ( )) , ( 1 ( ) , ( * ' ' ' ' , ' ' 1 1              J E T J E V G J E V J E P J E P I part j l lj n j l n lj n i j l n f n i i i (3.84) ) , ( ) , ( ) , ( )) , ( 1 ( ) , ( * *            J E T J E V J E T J E P J E P I part I f (3.85)
The only code which calculates the transmission coefficients in the generalized HRTW version of the statistical model, is the STATIS code.

The GOE triple integral

In the previously described methods, a semi-empirical parameterization is used to obtain W ab . The Gaussian Orthogonal Ensemble (GOE) formulation (Verbaarschot et al, 1985), as it is looking to avoid such a parameterization, is a more general expression, where

W ab reads         c ab c c ab g f d d d T W ) , , ( ) , , ( ) , , ( 8 2 1 2 1 1 0 2 1 0 2 0 1             (3.86) where 2 2 2 1 2 1 2 1 2 1 2 1 ) ( ) ( ) 1 )( 1 ( ) 1 ( ) , , (                       f       c c c c c T T T ) 1 )( 1 ( 1 ) , , ( 2 1 2 1                                          ) 1 )( 1 ( ) 1 ( 2 ) 1 )( 1 ( ) 1 ( ) 1 )( 1 ( ) 1 ( ) 1 ( 1 2 1 1 ) 1 ( ) , , ( 2 2 2 2 1 1 1 1 2 2 2 1 1 2 1 b a b a b a ab a a a a ab ab T T T T T T T T T T g                       

The gamma-ray strength function

The neutron transmission coefficients (or any transmission coefficients that characterize a emerging particle in the final state), are provided by the direct interaction models. In contrast, the gamma-ray transmission coefficients are determined by specific means. Thus, starting from the principle of the detailed equilibrium and the interpretation of the reverse photo-absorption process, the gamma-ray transmission coefficient can be expressed as

1 2 ) ( 2 ) (           L XL XL f T (3.87)
where   is the energy of the  radiation, X indicates the electric or magnetic type of the transition, L is the multipolarity and

) (   XL f
is the strength function (given in Section 3.2.1).

The gamma transmission coefficient ) ,

( *   J E T
at the excitation energy * E , and corresponding to the spin and parity J  of the compound nucleus, is given by the following equation:

          * 0 ) ( * ) , ( ) , , 1 ( ) ( ) , , 1 ( ) ( ) , ( E B K K n XL j j XL n J B j J f T d j J f T J E T             (3.88)
in which the first term refers tot the discrete spectrum and the second to the continuum spectrum described by the level density. f(1,J,j') is a factor or 1 or 0 to force compliance with multipole radiation selection rules.

In most of the cases, and also in the STATIS code, it has been considered that the energy spectrum of the compound nucleus is completely continuous, which is a very good approximation, since its excitation energy * E is at least equal to the neutron separation

energy n B .
The constant

XL K entering the expression of ) (   XL f
can be obtained by normalization to the s-or/and p-waves resonance data.

In all codes (STATIS, GNASH, TALYS) the option exists to normalize the gammaray strength function. At sufficiently low incident neutron energies, the average radiative capture width Γ 0 is due entirely to the s-wave interaction, and it is Γ 0 at the neutron binding energy B n that is often used to normalize gamma-ray transmission coefficients

                         J XL L J L J J E B K K n Xl l k Xl K n J B j J f T d j J f T S * 0 0 0 ) , ( ) , , 1 ( ) ( ) , , 1 ( ) ( 2            
where the JΠ sum is over the possible compound nucleus states that can be formed with swave incident particles, the Xl sum is over the multipole radiations included in the calculation and the j' sum is over the spins in the final state.

Very commonly used models for the calculation of the gamma-ray strength function

) (   XL f
included in the codes used nowadays (i.e. STATIS, GNASH, TALYS, EMPIRE, etc) are the Brink-Axel (Brink, 1957 and[START_REF] Axel | [END_REF] and the Kopecky-Uhl (Kopecky and Uhl, 1990).

The Brink -Axel model

In this model, also called the giant resonance model (GDR), the shape of the resonance is described by a standard Lorentzian

2 2 2 2 2 2 0 ) ( ) (                  E K f XL XL (3.89)
where 0  ,  E and   are the strength, energy and width of the giant resonance. Currently they can be retrieved from statistics and databases (for example the RIPL3 database).

For deformed targets usually an expression with two giant resonances is taken

                                        2 2 2 2 2 2 2 2 2 02 2 1 2 2 2 1 2 2 1 01 ) ( ) ( ) ( E E K f XL XL (3.90)
The resonance parameters for the electric dipole transitions (E1), are given by a comprehensive study by D'Arigo [START_REF] D'arigo | Proceedings of the International Conference "Large-Scale Collective Motion of Atomic Nuclei[END_REF] in which the following global expressions for all nuclei have been derived

) (MeV 50 232 . 0 2    A E     946 . 0 ln 1 2 E E (MeV) ) 263 . 0 283 . 0 ( 1 1       E ) (MeV ) 14 . 0 35 . 0 ( 2 2       E (3.91) (mb) 48 . 3 1 01     A ) (mb 464 . 1 2 02     A

The Kopecky -Uhl model

In this model, the standard Lorentzian from the Brink-Axel model is replaced by a generalized Lorentzian (Kopecky and Uhl, 1990), and the strength function becomes

                                  5 2 2 2 2 2 2 2 0 4 7 . 0 ) ( ) ( ) , ( E T E K T f XL XL (3.92)
where T is the nuclear temperature

) ( n n n S a B E T        (3.93)
where B n is the neutron binding energy from the compound nucleus that is getting de-excited, E n the incident neutron energy, Δ is the pairing energy and a the level density parameter at S n .

) (     is a energy-dependent damping width given by 2 2 2 2 4 ) (             E T (3.94)

Phenomenological level density models at the ground state deformation

In statistical models for predicting cross sections, nuclear level densities are used at

excitation energies where discrete level information is not available or incomplete. Various phenomenological level density models have been developed, recalled by Egidy and Bucurescu in (Egidy and Bucurescu, 2005). For the case of the actinides the Gilbert -Cameron composite formula, which will be described below, is giving the best results.

The composite level density function of Gilbert-Cameron

A function which is often used to describe the level density for nuclei at the ground state deformation is the composite function of [START_REF] Gilbert | [END_REF]. For low excitation energies experimental evidence has shown that the level density can be well reproduced by a constant temperature expression, while for higher energies, the Fermi gas model is more suitable

) ( ) ( ) ( ) , ( * *        J E J E (3.95)                            M FG M T E E E a E a E E E T E E T E E * 4 5 * 4 1 * * * 0 * * * ) ( 2 12 ) ( 2 exp ) ( exp 1 ) ( ) (     (3.96)           2 2 2 2 ) 2 / 1 ( exp 2 1 2 ) (    J J J (3.97) 2 1 ) ( ) ( ) (            (3.98)
where T is the nuclear temperature, E 0 the "back-shift" energy, E M the matching energy, Δ the pairing energy and σ 2 is the spin cut-off parameter, which represents the width of the angular momentum distribution.

Methods to determine the level density parameter a

Because, in general, the binding energy of the neutron B n is greater than the matching energy, at this energy the Fermi gas model can be used. In this model the level density parameter a can be obtained from D 0 , the average s-wave level spacing at B n , which is usually obtained from the available experimental set of s-wave resonances.

                0 ) 2 1 , ( 2 1 0 ) 2 1 , ( ) 2 1 , ( 2
1 0 0 0 0 1 0 I B I I B I B D n FG n FG n Fg    (3.99)
where 0 I is the spin of the target nucleus. This is a transcendental equation, from which the a parameter can be extracted by an iterative search procedure.

For the spin factor usually the following expression is used

) ( 0888 . 0 * 3 2 2     E a A (3.100)
If the experimental value of the average distance between the s-wave neutron resonances is not available, the level density parameter a can be calculated using a systematic given by Gilbert and Cameron:

        nuclei) (deformed 120 . 0 00917 . 0 nuclei) (spherical 142 . 0 00917 . 0 / S S A a (3.101) where ) ( ) ( N S Z S S  
is the shell correction energy.

Methods to determine the level density parameters at the ground state deformation

The expressions for T  and FG  from equation (3.96) have to be matched at a

matching energy E M , where they and their derivatives, are identical. First, continuity requires that

) ( ) ( M FG M T E E    (3.102)
This leads directly to

)) ( ln( 0      M FG M E T T E E  (3.103)
Second, continuity of the derivatives requires that  gives the best description of the observed discrete states. The temperature T will have the following equation:

) ( 2 3 1       M M E E a T (3.109)
The problem that can occur when using N high and N low as input parameters, is to obtain a false solution of the equation, as shown in Fig. 3.1: instead of obtaining the parameters which describe the red curve, the blue curve can be computed. In order to solve this problem, other input parameters can be utilised. One possibility is to use T and N low . In this case, solving the above listed equations, the following parameterisations will be obtained: For the isotope 242 Am the value of the a parameter has been determined from the value of <D 0 >=0.58eV, taken from the database of Mughabghab [START_REF] Mughabghab | Atlas of Neutron Resonances: Resonance Parameters and Thermal Cross Sections, Z=1-100[END_REF]. For the other isotopes, 241 -239 Am there is no value given for the average s-wave level spacing at the neutron binding energy, consequently the systematics of Gilbert and Cameron, presented in equation (3.101) has been used. The parameters at the fundamental deformation for the americium isotopes are given in Table 3.1. Looking at the values of the temperatures used to fit the experimental level cumulative numbers, a systematic behaviour can be observed for the odd-odd and the odd-even americium nuclei. In case of the odd-odd nuclei, 242 Am and 240 Am, the temperature is about 0.400 -0.405 MeV, lower than in the case of the odd-even nuclei, 241 Am and 239 Am, where the temperature is about 0.420 MeV. For the isotope 238 Np the value of the a parameter has been determined from the value of <D 0 >=0.57eV, taken from the Mughabghab database [START_REF] Mughabghab | Atlas of Neutron Resonances: Resonance Parameters and Thermal Cross Sections, Z=1-100[END_REF]. For the other isotopes, 237 -235 Np there is no value given for the average s-wave level spacing at the neutron binding energy, consequently the systematics of Gilbert and Cameron, given in eq.(3.101), has been used. The parameters at the fundamental deformation for the neptunium isotopes are given in Table 3.2. For the isotopes 236 U -233 U the values of the a parameter have been determined from the values of <D 0 >, as found in the Mughabghab database [START_REF] Mughabghab | Atlas of Neutron Resonances: Resonance Parameters and Thermal Cross Sections, Z=1-100[END_REF]. For the other isotopes, -231 U, where there is no value given for the average s-wave level spacing at the neutron binding energy, the systematics of Gilbert and Cameron, given in eq.(3.101), has been used. The parameters at the fundamental deformation for the uranium isotopes are given in Table 3.3. Looking at the values of the temperatures used to fit the experimental level cumulative numbers, here too can a systematic behaviour be observed for the even-even and the even-odd uranium nuclei. In case of the even-even nuclei, 236 U, 234 U and 232 U, the temperature is about 0.423 -0.425 MeV, higher than in the case of the even-odd nuclei, 235 U, 233 U and 231 U, where the temperature lies by 0.420 MeV. Two of the most important concepts for fission cross section calculations are: the double-or even triple-humped barriers of the actinides and the role of transition states (meaning the excited states of the nucleus at the deformations corresponding to the maxima of the fission barriers (saddle points)).

     ) ) 6 ( 3 ( 2 aT aT aT T E M (3.110) )) ( ln( 0     M FG M E T T E E 
In order to be able to decide if two or three barriers have to be used, M.J. Lopez et al.

proposed (M.J. Lopez et al., 2005 and references therein, Britt) a method to compare the slopes of the near-barrier experimental cross-sections for several actinides. The slopes follow different behaviors depending if the predicted fission barrier has a third minimum or not. They divided the fission cross-section by the value of the plateau of the first fission chance, and looked at the slope of the normalized cross-section until it reaches the plateau. The slopes of the near-barrier experimental cross-sections are higher for compound nuclei that are known to have a double-humped fission barrier (as for example 240-243 Pu) than for nuclei for which a third well in the fission barrier is experimentally well known (as for example 232 Th).

The experimental near-barrier cross-section slopes for the reactions n + 237 Np and n + 241 Am, plotted in Fig. 4.1., have been compared to the slope of n + 232 Th. The results are given in Table 4.1. The most used parameterization for the double-humped fission barrier, consists in three parabolas smoothly connected. The g.s. barrier can be expressed as:

2 2 ) ( 2 1 ) ( f f f q q V q V     (4.1)
where the plus sign stays for f = 2 (isomeric well) and the minus sign for f = 1, 3 (first (A) and second (B) barrier). The inertial mass parameter µ is usually considered constant for all values of the deformation parameter q (although the STATIS code allows also the use of different On each intrinsic or vibrational state are rotational bands built. Each rotational level built in such a way, is associated a double barrier with:

                                             2 1 ) 1 ( 1 ) 1 ( ) 1 ( 2 ) ( ) ( 2 1 2 1 , 2 I a K K J J K V KJ E I K f f f f    (4.2) where ) ( K f K 
are the rotational band head energies with respect to the fundamental barrier peaks, and f ) ( represent the inertia moments corresponding to the deformations. The spin and parity are JΠ = K, K+1, K+2,… for K half-integer or an integer ≥ 2 0 + , 2 + , 4 + , 6 + for K = 0 + 1 -, 3 -, 5 - for K = 0 - If K = ½, then the Coriolis decoupling factor a must also be taken into account.

The fission barriers associated to the transition states keep their individuality up to some excitation energies. Above these energies, they are described using density functions.

So the parameters that define the fission barriers of a nucleus and that are involved in the penetrability calculation are f V ,

f   , f ) 2 / ( 2   , ) ( K f K  (f = 1,2,3
) and the density function parameters, that describe the continuous spectrum of transition states.

Determination of the shape symmetry at the saddle point deformations

According to calculations that take into account the shell corrections, Pashkevich (Pashkevich, 1969) and later Howard and Möller (Howard and Möller, 1980) anticipated that, for the first saddle point, in the reaction chain exists one isotope, characterized by the mass number A tr , starting from which, the symmetry of all following isotopes changes. In other words, that all isotopes with A > A tr have in the first saddle point axial asymmetry and mass symmetry (AS  AAMS), while isotopes with A < A tr have in the first saddle point both symmetries -mass and axial (SS  ASMS). For a given isotope chain exists an isotope with the mass A tr for which the symmetry is changing for A > A tr . The fissioning nuclei with AAMS at the first saddle have V A > V B while the nuclei with ASMS have V A < V B .

To find the A tr transition value for each isotope chain for which a change of symmetry takes place, a map of the main actinide nuclei, that can fission in different nuclear reactions, has been build by Vladuca et al. (G. Vladuca et al., 2006). In this map the symmetry

parameter A Z N   
has been represented with respect to the fissility parameter X F , defined as

) 7826 . 1 1 ( 883 . 50 / 2 2    A Z X F (4.3)
In this figure all the existing isotopes of the actinide nuclei from Z = 89 (Ac) up to Z = 96 (Cm) have been taken into account. The two zones, labeled in the figure SS (axially and mass symmetry for the inner barrier) and AS (axially asymmetry and mass symmetry for the inner barrier) are indicated.

One point through which the line separating the two zones passes is between the two uranium isotopes with the mass numbers A = 236 and A = 237, in agreement with the theoretical predictions of Howard and Möller (Howard and Möller, 1980). They state that the inner barrier of uranium nuclei with A ≤ 236 is axially symmetric, and the inner barrier height V A is smaller than the height of the outer barrier V B . This fact can be also seen in the databases RIPL-2,3.

In order to establish a line, another point is needed. For the isotopes of plutonium the theoretical calculations of Howard and Möller predict that the change of symmetry occurs for the nucleus 236 Pu. This means that 235 Pu is in the SS zone, while 237 Pu lies in the AS zone.

This assumption is in good agreement with the results from neutron induced reaction data analysis by Vladuca et al. (Vladuca et al, 1997[START_REF] Vladuca | [END_REF], but is not in accord with the RIPL 2,3 databases, which the transition is taking place between 239 Pu and 238 Pu. The disagreement between Vladuca et al and the RIPL databases is not surprising because the systematic of RIPL is based on neutron induced cross section calculations using a simplified model for fission (with completely decoupled barriers where

B A B A f T T T T T  
) while the model of Vladuca et al. is much more refined.

Based on these two points of transition, the line that separates the two areas with different symmetries can be drawn. For all the other nuclei, the isotope with the mass number for which the change of symmetry occurs can be determined.

In the case of neptunium, the transition is taking place at A = 236. Therefore the nuclei with A ≥ 237 are axial asymmetric and mass symmetric, while all neptunium isotopes with A ≤ 236 have mass and axial symmetry at the inner barrier. This result is between the results of RIPL-2,3 ( where the transition is taking place at A = 235) and the results published by Maslov (Maslov ,1995) (where 238 Np is considered axially asymmetric and 235 Np, 236 Np and 237 Np possess both symmetries).

In the case of americium, the transition point is at A = 234. Thus the nuclei with A ≥ 235 have AS symmetry, while the isotopes with A ≤ 234 are SS symmetric at the inner barrier.

According to this chart, all actinium ( 206-236 Ac), thorium ( 209-237 Th) and protactinium ( 212-237 Pa) nuclei are located in the SS zone, while the curium isotopes (

Transmission coefficients for fission

The Hill-Wheeler formula

Within the framework of the Bohr channel theory, and the assumption that the fission barrier is single-humped as predicted by liquid drop model calculations and that the shape of fission barrier is well approximated by a parabola, the probability of tunneling through a barrier with height V f and width

f  
for a compound nucleus with excitation energy E* reads

            f f f V E E T    * * 2 exp 1 1 ) ( (4.4)
For a transition state with excitation energy ε above the top of the barrier

             f f f V E E T      * * 2 exp 1 1 ) , ( (4.5)
For a compound nucleus with excitation energy E*, spin J, and parity Π, the total fission transmission coefficient is the sum of the individual transmissions coefficients for each barrier through which the nucleus may tunnel, and is thus

                                cont E f f i f f f V E J V E J i f J E T          d 2 exp 1 1 ) , , ( 2
       J E T J E T J E T J E T J E T B A B A f (4.7)

Statistical model with sub-barrier effects of Vladuca et al.

To define the direct transmission coefficient and the absorption coefficient in the isomeric well, the equation used to calculate the penetrability of the double potential barrier obtained by Fröman in the JWKB approximation is [START_REF] Fröman | JWKB Approximation[END_REF])

    2 2 2 2 1 2 2 2 2 1 sin )) 1 )( 1 (( 1 cos )) 1 )( 1 (( 1           B A B A B A f T T T T T T T (4.8)
where in the JWKB approximation the expressions for the transmission coefficient through the internal T A and external T B barrier and the momentum of the isomeric well  2 are:

) 2 exp( 1 1 1    A T (4.9) ) 2 exp( 1 1 3    B T (4.10)
For energies lower than the fission barrier peak:

) 2 exp( 1 1 1     A T (4.11) ) 2 exp( 1 1 3     B T (4.12)
For energies higher than the fission barrier peak, the quantities  are represented by:

) ( )] ( [ 2 ) ( 2 2 1 q iK q V E q K      (4.13)    2 1 ) ( 2 1 a a dq q K ; dq q K a a    3 2 ) ( 1 2 ;    4 3 ) ( 2 3 a a dq q K ; (4.14)
where a 1 , a 2 , a 3 , a 4 are the "turning points" for the energy E. When the intercept point a 1 , respectively a 3 , is above the coupling point z 1 , respectively z 2 , (Fig. 4.4), it can be observed that the above expressions are identical to the Hill-Wheeler penetration expression. It should be noted that the previously expressed transmission coefficients refer to the fission barrier associated to a transition state characterized by EKJ , which will be self understood.

The absorption in the isomeric well is taken into account by an imaginary potential W(q), which corresponds to the isomeric well as shown in Fig. 4.4, and has a parabolic shape.

For this reason this model is also called "the optical model for fission".

Fig. 4.4. Complex deformation potential

The relations for the direct penetration though the double barrier and the absorption in the secondary well are obtained by adding the imaginary phase i, corresponding to the imaginary potential, to the phase 2  (Bhandari, 1979)   where

           2 2 2 1 2 ) 1 )( 1 ( 2 cos ) 1 )( 1 ( 2 e T T T T e T T T B A B A B A dir (4.15)               
            3 2 2 1 2 1 2 )] ( [ ) ( 2 a a dq q V E q W  (4.17)
From the above equations it can be observed that when 0 the absorption is zero and the expression (4.15) for the direct coefficient is identical with the expression (4.8).

A part of the flux absorbed in the isomeric well reaches fission by penetration of the external barrier in competition with the return into the first well and the radiative transition to the isomeric level [START_REF] Vladuca | Evaluation of Pu-242 Data for the Incident Neutron Energy Range 0.01-6 MeV[END_REF](Vladuca et al, , 1997 a,b,c) a,b,c). Taking into account the isomeric fission, the indirect penetration through the barrier is given by the expression:

                 II B A II II B A B abs ind T T T RT T T T T T T (4.18) 
T are the radiative capture transmission coefficients; R is the branching ratio for fission of the isomeric state and is calculated related to the half-lifes (Vladuca et al, 1997 a,b,c):

iso f iso iso T T T R 2 1 2 1 2 1     (4.19)
The isomer half lives, for fission and for  decay, are represented by the relations:

               II II B B II iso f V V T        2 1 2 exp 2 ln 2 2 1 (4.20)                II II A A iso V V T T        2 1 2 exp 0 2 1 2 1 (4.21)
where 0 2 1  T is the half life for the decay in the first well.

As specified, all transmission coefficients A T , B T , dir T , abs T , f T from the above relations are characterized by EKJ. Although, due to the excitation of other degrees of freedom than the fission ones, in the isomeric well, the value K for the spin projection onto the symmetry axis can be different for the moment in which the energy is concentrated in internal excitations. In the isomeric well, several nuclear forms can be characterized by the same excitation energy, so it results the possibility that no matter through which energy channel (KJ) the second well has been reached, the exterior barrier penetration can be done through another, more favorable, energy channel. This phenomenon is known as "K-mixing" and it has an important effect on the angular distribution of the fission fragments, and on the fission probability.

If the K-mixing effect is missing, the total fission coefficient is (Vladuca et al, 1997 a,b,c):

              J K ind J K dir J K f f KJ E T KJ E T KJ E T J E T ) , ( ) , ( ) , ( ) , ( * * (4.22)                           J K II B A II B abs J K dir KJ E T KJ E T KJ E T J E RT KJ E T KJ E T KJ E T ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , (                         J K II J K B A B abs ind J E T J K E T J K E T KJ E T J K E T KJ E T ) , ( )] , ( ) , ( [ ) , ( ) , ( ) , ( * * * * * *                   ) , ( )] , ( ) , ( [ ) , ( * * * * J E T J K E T J K E T J E RT II J K B A II (4.23)
The total fission coefficient becomes:

                                    ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( * * * * * * * * * J E T J E T J E T J E RT J E T J E T J E T KJ E T KJ E T KJ E T J E T II B A II B abs dir J K ind J K dir J K f f (4.24) where      J K i i KJ E T J E T ) , ( ) , ( * * abs dir B A i , , ,  (4.25)
At high energies, when the level density in the secondary well is high, the damping phenomenon is almost complete. Formally, the imaginary potential strength is increasing in such a manner that all that penetrates the barrier is absorbed (T abs =T A ), the direct element vanishes (T dir =0) and the isomer fission can be neglected. The case of "strong coupling" is when the fission process takes place in two independent steps: interior barrier penetration followed by exterior barrier penetration, therefore the fission coefficient (4.24) becomes the expression of probabilities product, used in the standard statistic models:

                       J K J K B A J K B A J K ind f J K E T J K E T KJ E T J K E T KJ E T J E T )] , ( ) , ( [ ) , ( ) , ( ) , ( ) , ( * * * * * ) , ( ) , ( ) , ( ) , ( * * * *       J E T J E T J E T J E T B A B A (4.26)
If the isomer fission is neglected, the fission probability can be written as (Vladuca et al, 1997c)

a J E T J E V J E T J E T J E V J E T J E V J E T J E T J E P I part dir I part I part dir dir f 1 ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( * * * * * *                      where 2 1 * * 2 2 ) , ( ) , ( coth 2 1                     J E T J E T b b a B A ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( * * * * * *              J E T J E T J E T J E T J E T J E V J E T b B B A abs I part dir
This model has been extended also for the case of triple humped barriers (Sin et al, 2006) and also for multi-modal fission (Vladuca et al, 2002(Vladuca et al, , 2003)).

Head band states

The procedure for determining the single-particle levels in general consists of three steps (Bolsterli et. al, 1972):

(1) specifying the over-all geometrical shape of the nucleus (2) generating single-neutron and single-proton potentials related to this shape, and

(3) solving the Schrodinger equation with these potentials for the single-particle levels.

(1) In fission it is necessary to be able to describe in a continuous way the sequence of shapes of a fissioning nucleus from its ground state, through its saddle and scission configurations, including all asymmetric shapes. During the early stages of fission the symmetric-deformation coordinate y introduced by Hill and Wheeler (Hill and Wheeler, 1953) provides a good approximation of the fission coordinate. This coordinate is defined in terms of the saddle point shapes for an idealized uniformly charged liquid drop.

The saddle-point shape corresponding to a given value of the fissility parameter X F , defined by (4.3), represents a deformation of

F X y   1 (4.27)
To first order, y is related to the coordinates that describe prolate spheroidal and Legendre- (2) Knowing the nuclear shape, the first step in order to determine single-particle levels is to generate single-neutron and single-proton potentials related to that shape.

For the initial ground-state nucleus the nuclear spin-independent part of the singleparticle potential (for either neutrons or protons) is roughly constant in the nuclear interior and rises to zero in a small distance near the nuclear surface. After scission has occurred, the potential has similar features concentrated in each of the separated fragments. This means that during the intermediate stages the potential must remain nearly constant in the interiors of the nascent fragments and approach zero in the neck region.

Such a potential can be obtained by following two steps, proposed by Bolsterli (Bolsterli et al., 1972): i) to establish a uniform sharp-surface "generating potential" whose shape corresponds to the given nuclear shape (the generating potential has the value -V 0 inside the specified surface and 0 outside) ii) a potential with a diffuse surface is then obtained by folding a Yukawa function over the sharp generating potential.

For a spherical shape both, the Woods-Saxon and the Yukawa potential give good results, but for deformed shapes the folded Yukawa potential is more appropriate than the Woods-Saxon potential. The Yukawa potential can be defined as:

         V 3 a r r 3 0 1 r d a r r e 4ππ V ) r ( V      (4.29)
where V 0 is the well depth felt by a nucleon, and a is the range of the Yukawa folding function. The integration is over the volume of the shape, whose magnitude is kept fixed as the shape is deformed. The volume integral of the potential is given by

3 0 0 3 1 R 3 4π V r )d r ( V      (4.30)
where R 0 is the radius of the spherical sharp-surface generating potential.

Besides the spin-independent part of the potential, there is an additional potential, which has to be taken into account, which appears from the interaction between the nucleon spin and orbital angular momentum (Bolsterli et al, 1972). The simplest form for the spinorbit interaction that has the needed symmetry properties for deformed nuclei is p

V σ       .
For V' the spin-independent potential described above in being used. So the spin-orbit term is

     p V σ 2mc λ ) r ( V 1 2 SO            (4.31)
where λ is the spin-orbit interaction strength, and m is the mass of a nucleon.

The Coulomb potential for protons is

     V C r r r d R Ze r V    3 3 0 2 3 4 ) (  (4.32)
For a spherical shape, the integrations can be performed explicitly:

                                           0 0 0 0 0 0 0 0 0 1 , ) exp( sinh cosh , ) sinh( exp 1 1 ) ( R r a r a r a R a R a R V R r a r a r a R a R V r V (4.33)                            0 2 0 2 0 0 2 , , 3 2 1 ) ( R r r Ze R r R r R Ze r V C (4.34)
It is convenient to use the single-particle potentials determined by [START_REF] Myers | [END_REF], but to increase slightly the neutron and proton well depths to better reproduce the over-all positions of the experimental single-particle levels.

In using the results of Myers and Swiatecki, it is convenient to compute first some auxiliary quantities. ε and δ are defined as

0 0 3 1        (4.35)     Z N   (4.36)
where N  the neutron density is, Z  the proton density,  the total density and 0  the equilibrium density of standard nuclear matter. The average values of ε and δ are

3 1 3 5 2 15 . 3 1 0112 . 0 A A Z A Z N      (4.37) 3 4 2 2 3 1 00248 . 0 330 . 0 147 . 0 A Z A       (4.38)
The radius of an equivalent sphere which represents the density distribution is given by the droplet-model expression

) 1 ( 3 1 0     A r R (4.39)
Then, the five single-particle potential parameters are given by (Bolsterli et al, 1972)

MeV ) 7 . 48 5 . 52 (    n V MeV ) 7 . 48 5 . 52 (    p V   R R R 2 0 fm 56 . 0 fm 82 . 0    fm 90 . 0  a 32  
(3) The most fundamental approach for calculating the nuclear potential energy would be to start with a nucleon-nucleon interaction and solve the resulting many-body equations in some approximation. In the macroscopic-microscopic method, the total nuclear potential energy E is given by the sum of a liquid-drop energy E LD , a shell correction ΔE SC , and a

pairing correction ΔE PC E = E LD + ΔE SC + ΔE PC (4.40)
To calculate the liquid-drop energy Bolsterli (Bolsterli et al, 1972) adopts the smooth part of the mass formula presented by [START_REF] Myers | [END_REF])

MeV A A Z N 1.7826 1 17.9439 E 2/3 2 (0) s                  (4.41)
Higher-order terms in the expression for the macroscopic energy, such as compressibility and curvature effects are neglected.

The shell correction arises because of fluctuations in the actual distribution of singleparticle levels relative to a smooth distribution of levels. It is calculated from the singleparticle energies at a given deformation by means of the method developed by Strutinsky (Strutinski, 1966). Because neutrons and protons independently fill their own set of singleparticle orbits, both the shell correction and the pairing correction are given as independent sums of a term for neutrons and an analogous term for protons. The method for calculating the shell correction can be easily described with the aid of Fig. 4.5.

 up to N   dn (n) ε ε dn (n) ε (n) ε ΔE N N n n N stair SC         0 1 0 (4.42)
In Strutinsky's method (Strutinski, 1966) the quantity that is given explicitly is the inverse of the function ) (n  -the average particle number ) ( n as a function of the singleparticle energy. The exact particle number is given by

                     d g d d dn n ) ( ) ( ) ( (4.43) where ) ( g is exact single-particle level density      1 ) ( ) ( n n g     (4.44)
The exact level density ) ( g can be separated into a smoothly varying part ) ( g and a part ) (  g that contains the local fluctuations

) ( ) ( ) (     g g g   (4.45)
In order to do that, Strutinsky expanded the δ function in a series of Hermite polynomials and then separating the terms into a smoothly varying part and a fluctuating part

            0 1 1 ) ( 1 ) ( 1 ) ( 2 m n m m n u n n u H c e u g n      (4.46) where    n n u          odd m even m m c m m m 0 )! 2 / ( 2 ) 1 ( 2 /
and γ is a scaling factor. Since Hermite polynomials of low order oscillate more slowly than those of high order, the first few Hermite polynomials of the equation represent the smoothly varying contribution to ) ( g , and the remaining terms the fluctuating contribution.

The average particle number can now be explicitly evaluated

                        1 1 1 ) ( 1 ) ( 1 2 1 ) ( ) ( 2 n p m n m m u n u H c e u erf d g n n      (4.47)
where p defines the order of the shell correction.

In practice it is more convenient to transform the integration over particle number into an integration over energy

    d g dn (n) ε N      ) ( 0 (4.48)
where  is the Fermi energy of the smooth distribution of levels, and can be solved iteratively from

N n  ) ( (4.49)
The desired integral can then be then evaluated explicitly

                               p m n m n m n n m m u n u n n N u H m u H u H c e e u erf dn n n n 1 2 1 1 0 ) ( ) ( ) ( 2 1 1 2 1 ) ( 1 2 1 ) ( 2 2         (4.50) where    n n u   (4.51)
For his calculations, Bolsterli (Bolsterli et al, 1972) has used the values p = 6 (sixthorder correction) and

3 / 1 0 / 41 A MeV      (4.52)
For a given single-particle potential the shell correction, calculated in this manner, has an accuracy of about 0.5 MeV.

The pairing correction, arises from the short range interaction of correlated pairs of nucleons moving in time-reversed orbits. In calculating the pairing correction for either neutrons or protons, N p pairs of particles are considered, with half of them lying above the sharp Fermi surface and the other half lying below. Then, for a specified pairing strength G, the pairing correlation energy (relative to the energy without pairing) is given in the BCS approximation by

) 1 ( ) ( 2 2 1 1 1 4 2 2 1 1 1 2               p p p p N k N k k N k k N k k k PC v G G v E   (4.53)
If the variation of the 4 k v -term is neglected, the pairing gap Δ and the BQS Fermi energy λ (which is not explicitly needed) are given by the solutions of the equations

                  p N k k k p N 1 2 2 ) ( 1     (4.54)       p N k k G 1 2 2 ) ( 1 2   (4.55)
The occupation probabilities 2 k v are given by

                2 2 2 ) ( 1 2 1     k k k v k = 1, 2, …, N p (4.56)
The pairing correlation energy of an average nucleus is determined from analogous expressions, but with the summations over discrete states ε k , replaced by integrals over the smooth function ) (n  of Fig. 4.5. The resulting integrals can be evaluated explicitly if ) (n  is replaced by a linear function, by making a Taylor expansion about the sharp-Fermi-surface particle number and neglecting quadratic terms. The inverse of the slope of this curve is ) ( g and is obtained in the same way as for the shell correction. The average density of pairs is then given by

                              2 tan 1 2 1 1 2 1 4 1 2 2 p p p PC N G N N E (4.57)
where the pairing strength G is given by

                          2 2 1 ln 1 2 p p N N G (4.58)
and the average pairing gap  is chosen to reproduce exactly the semi-empirical result

A MeV / 12   (4.59)
The pairing correction for either neutrons or protons is given then by

PC PC PC E E E    (4.60)
The pairing correction can be calculated in this manner with a numerical accuracy of about 0.1 MeV. This approach employs a saddle point approximation which is not adequate at excitation energies below 1 MeV in doubly even or odd-A nuclei where the specific character of the states involved in fission or neutron deexcitation becomes important. For this reason the effective number of transition states N A , N B , and N n were calculated as a sum of two separate contributions when the decay involved an even-even nucleus (i.e., neutron decay from even Z, odd N) or an odd-A nucleus. The first contribution came from states above 1 MeV where a continuous level density was used, whereas the second contribution from states below 2 MeV used a discrete spectrum of levels. The levels used are shown in Figs. 4.6-4.8 for the decay of even Z -odd N, odd Z -even N, and odd Z -odd N fissioning nuclei, respectively.

The continuous level densities were determined with the above presented method.

They were determined from single particle spectra calculated for 240 Pu at the ground state deformation (including the hexadecapole deformation), at the first saddle (not including the stable γ deformation), and at the second asymmetric saddle. At the second saddle point the level densities are multiplied by 2 to take into account the two degenerate solutions at the mass asymmetric saddle point.

In principle the spectrum of discrete levels is different for each nucleus, but in most cases it is not possible to obtain a very reliable estimate of the detailed discrete spectra. Therefore, Back et al. use a single discrete spectrum for the neutron and fission decay to each type of nucleus (even Z -odd N, odd Z -even N, odd Z -odd N).

For neutron decay to an even -even residual nucleus Back (Back et al, 1974) assumes a spectrum of vibrational excitations (Fig. 4.6), which are obtained from an average of the excitations experimentally observed in the uranium-curium region. Each vibration is assumed to contain a rotational band with spacings given by a rotational constant (Back has used the value of 7 keV).

For neutron decay to odd Z -even N nuclei (Fig. 4.7) the discrete level spectru, is taken as the spectrum of one-quasiparticle proton states generated from the calculated single proton states for 240 Pu using the Bolsterli method (Bolsterli et al, 1972). The energies of the one-quasiparticle states are obtained from the relation

0 2 2      i sp i q E E (4.61)
where 0

 and  are the pairing gaps at the ground states and at i q E respectively. The pairing gap Δ(E) is obtained from above described the level density calculations of Bolsterli, which use the same set of single proton states. Each one-quasiparticle state is assumed to have a rotational band built on it with a rotational constant of 7 keV.

Fig. 4.6.

Level spectra for an even Z -odd N fissioning nucleus (Back et al, 1974).

Fig. 4.7.

Level spectra for an odd Z -even N fissioning nucleus (Back et al, 1974).

Fig. 4.8.

Level spectra for an odd Z -odd N fissioning nucleus (Back et al, 1974).

For the fission decay of even Z-odd N nuclei (Fig. 4.6.) and odd Z -even N nuclei (Fig. 4.7) the single neutron or single proton spectra, calculated with the Bolsterli et al.

method, at the appropriate saddle points are used to generate one-quasiparticle states. These one-quasiparticle states are then used to calculate the contribution to N A and N B from the discrete levels for each case. At the second saddle the levels correspond to an asymmetric shape and are, therefore, taken as doubly degenerate. Such one-quasiparticle states are assumed to head a rotational band with a rotational constant of 5 keV.

Figures 4.7 and 4.8 also show the discrete level spectra at the first saddle point taken from (Back et al, 1974). The details of the two sets of discrete spectra are different but the total number of levels available between 0 and 1 MeV are similar.

If a 0 Π 0 J band is considered, on this state some vibration state can be built as follows (Vladuca et al, 1994):

-a mass-asymmetry vibration with J = J 0 and Π = -Π 0 -a beta vibration with J = J 0 and Π = Π 0 -two gamma vibrations with J = J 0 + 2 and Π = Π 0 J = |J 0 -2| and Π = Π 0 -two bending vibrations with J = |J 0 + 1| and Π = -Π 0 J = |J 0 -1| and Π = -Π 0

On each state, with the energy 0  , the six vibration states can be built using the 0 Π 0 J rules presented above; their energies will be:

    i A, ε i A, ε 0 f  beta vibration     ) ( i A, ε i A, ε 1 0 f A k v   mass-asymmetry vibration     ) ( i A, ε i A, ε 2 0 f A k v   gamma vibrations     ) ( i A, ε i A, ε 3 0 f A k v   bending vibrations     i B, ε i B, ε 0 f  beta vibration     ) ( i B, ε i B, ε 1 0 f B k v   mass-asymmetry vibration     ) ( i B, ε i B, ε 2 0 f B k v   gamma vibrations     ) ( i B, ε i B, ε 3 0 f B k v   bending vibrations
where k v1 (A), k v2 (A), k v3 (A), k v1 (B), k v2 (B) and k v3 (B) are constants.

The discrete transitional state spectrum is obtained by building the rotational levels on the band heads, according to the equation

          1 K K 1 J J 2 Π K ε Π J K, E f 2 f f                (4.62)
where

  Π K ε f
are the rotational band heads energies (measured from the top of the fundamental barrier), K is the spin projection along the symmetry axis and the index f means the inner (A) or outer (B) barrier. The quantities   f 2 2  are the inertial parameters.

In order to build the rotational bands, usually the following values are used

MeV 005 . 0 2 A 2            MeV 003 . 0 2 I 2            MeV 002 . 0 2 B 2          
 taking into account that on the path forwards fission, the elongation of the nuclei grows, so also the moment of inertia is increasing, leading to a decrease of the inertial parameters.

Level density models for the fission path

The collective enhancement in the codes STATIS and GNASH

Many authors calculate the level density function on the fission path as

sph fis K     ,
in other words K being an absolute enhancement factor. It is more physically correct to

consider . . . deform s g fis K    
, where the level density function for a deformed nucleus is obtained by multiplying the density corresponding to the ground state deformation with an enhancement factor K which is taken relative to the ground state. This enhancement factor is defined according to the nuclear shape symmetries at the respective saddle point (Vladuca et al, 2006):

             
2 2 0 3 2 0 2 2 0 2           f f f f f K (4.63)
where σ 2 are are the spin cut-off.

The values of a, Δ and σ 0 can be used as parameters. The nuclear temperatures T A and T B may be considered also free parameters, which can be deduced from fitting the cumulative number of the discrete transitional spectrum.

For the spin cut-off the folowing expression is considered

     E a A 3 2 2 0888 . 0  (4.64)
In order to determine the level density parameters for the fission path, the matching problem has to be solved. The same method, as presented in chapter 3.3.3 will be used; the values of i low N (the cumulative number of levels at which the fit begins) and T i will be used as input parameters.

If it is considered that K f multiplies both branches of the composite Gilbert-Cameron formula (the constant temperature part, and the Fermi gas part) then at the matching point

) ( ) ( ) ( ) ( ) ( ) ( i M FG i i i M T i i E K E K    (4.65)
where i stands for the index A or B. If

) (i L E is the excitation energy corresponding to the cumulative number of levels i low N , then    i i L E E f low i dE E K N N ) ( ) (  (4.66)
By applying a logarithm to this equation it is obtained that

                             i i L i i i f i L i T E T E T E K E N E N ) ( ) ( 0 ) ( exp exp ln ln ) ( ) ( ln (4.67)
Solving this equation and the one resulting from the continuity condition

) ( d d ) ( d d ) ( ) ( ) ( ) ( i M FG i M T E E E E    (4.68)
the level density parameters

) (i M E and ) ( 0 i E are       ) ) 6 ( 3 ( 2 ) ( i i i i i M aT aT aT T E (4.69) )) ( ln( ) ( ) ( ) ( ) ( 0     i M i FG i i i M i E T T E E  (4.70)
In STATIS there is also a method implemented, that allows to multiply only the Fermi-gas part of the Gilbert-Cameron composite formula (if collectivity is assumed to be already implicitly included in the constant temperature part)

) ( ) ( ) ( ) ( ) ( ) ( i M FG i i i M T i E K E    (4.71)
The same input parameters will be used: i low N and T i . Solving the two resulting equations for the different values of the collective enhancement factor, the following results are obtained (Tudora, 2011):

1. Ground state 1  i K          M M FG E E a dU d T 2 3 ln 1             6 3 2 gs gs gs gs M aT aT aT T E 2. Axial and mass symmetry       gs M i M f i E E K ) ( 2 0 2          ) ( ) ( 1 ln 1 i M i M FG E E a dU d T             4 2 2 ) ( i i i i i M aT aT aT T E

Axial symmetry and mass asymmetry

      gs M i M f i E E K ) ( 2 0 2 2 2          ) ( ) ( 1 ln 1 i M i M FG E E a dU d T             4 2 2 ) ( i i i i i M aT aT aT T E

Axial asymmetry and mass symmetry

        gs M i M f i E E K 4 / 3 ) ( 2 0 3 2 2              ) ( ) ( 4 3 ln 1 i M i M FG E E a dU d T                 3 2 3 2 ) ( i i i i i M aT aT aT T E 5. No symmetry       gs M i M f i E E K ) ( 2 0 2 8 8            ) ( ) ( 1 ln 1 i M i M FG E E a dU d T             4 2 2 ) ( i i i i i M aT aT aT T E

The collective effects in the level density, as present in TALYS

In order to account for the collective effects, TALYS [START_REF] Koning | TALYS: Comprehensive nuclear reaction modeling[END_REF] uses also a phenomenological way by explicitly introducing collective enhancement factors for an intrinsic Fermi gas level density ) , , ( int ,

 J E x F 
. This method is applied for the level density at the fundamental ground state deformation, as well as for the level density on the fission path. Then the deformed Fermi gas level density ) , , ( ,

 J E x def F  is ) , , ( ) ( ) ( ) , , ( int , , 
  J E E K E K J E x F x vib x rot x def F   (4.72)
where K rot and K vib are called the rotational and vibrational enhancement factors, respectively.

The vibrational enhancement of the level density is approximated in the RIPL-2 handbook by

        T U S E K x vib   exp ) ( (4.73)
In this equation T is the nuclear temperature given by a U T  (4.74)

and δS and δU are changes in the entropy and excitation energy, respectively, resulting from the vibrational modes, and are described by the Bose gas relationships

        i i i i i i n n n n S ln ) 1 ln( ) 1 ( ) 1 2 (   (4.75)    i i i i n U    ) 1 2 ( (4.76)
where ω i are the energies, λ i the multipolarities, and n i the occupation numbers for vibrational excitations at a given temperature. The disappearance of vibrational enhancement of the level density at high temperatures can be taken into account by defining the occupation numbers in terms of the equation

1 exp 2 exp                T n i i i i    (4.77)
where γ i are the spreading widths of the vibrational excitations, and can be written as

) 4 ( 2 2 2 T c i i      (4.78)
In TALYS the following values are used A more important contribution to the collective enhancement of the level density originates from rotational excitations. Its effect is not only much stronger (K rot ≈ 10 -100 whereas K vib ≈ 3), but the the expression for the rotational enhancement factor depends also on the deformation. For purely deformed cases, K rot is equal to the perpendicular spin cut-off

parameter 2   T I E K x rot     2 ) (  (4.82)
where the rigid-body moment of inertia perpendicular to the symmetry axis given by

                 3 1 01389 . 0 3 1 2 3 5 2 0   A I I (4.83) It results a U A k rot         3 1 01389 . 0 2 3 5  (4.84)
For high excitation energies, it is known that the rotational behavior vanishes. To take this into account, in TALYS a phenomenological damping function f(E x ) is introduced, which is equal to 1 for the purely deformed case and 0 for the spherical case. The expression for the level density is then

  ) , , ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( 1 ) , , ( int , int , int ,           J E E K E K J E E K E k E f J E E K E f J E x F x vib x rot x F x vib x rot x x F x vib x x     (4.85) where     1 , 1 ) ( 1 ) ( max ) (    x x rot x rot E f E k E K (4.86) The function ) ( x E f
is taken as a combination of Fermi functions The constant temperature model formalism can be extended with explicit collective enhancement. In this case the total level density is

           gs col gs col x x d E E E f exp 1 1 ) (
        M x x tot F x vib x rot M x x tot T x tot E E E E K E K E E E E ), ( ) ( ) ( ), ( ) ( int ,    (4.88)
The collective enhancement is not applied to the constant temperature region, since collectivity is assumed to be already implicitly included in the discrete levels.

Equation (4.87) for the rotational enhancement also holds for inner barriers with neutron number N≤144, which are all assumed to be axially symmetric. Inner barriers with N > 144 (for example the Am isotopes) are taken to be axially asymmetric. In this case the rotational enhancement is

                          1 , 1 ) ( 1 3 2 1 2 max ) , ( ) ( || 2 2 2 x x asym rot x rot E f E K E K      (4.89) 2 ||
 designates the parallel spin cut-off parameter, obtained from the projection of the angular momentum of the single-particle states on the symmetry axis and is defined as

aU A parl ) 3 2 1 ( 24 . 0 6 2 3 / 2 2 2 .      (4.90)
For outer barriers, due to the mass asymmetry, an extra factor of 2 to is applied to K rot . The temperature plays an important role in the determination of the level densities. In these calculations they have been considered to be fit parameters. In the process of determining their values it has been taken into account, that the temperature increases as the nucleus evolves from the equilibrium deformation to the deformations on the fission path. In addition, also the odd or even character of the studied isotopes has been considered. It is likely that, for the same charge number Z, the level density of a nucleus with an odd number of neutrons is higher that the level density of an isopote with an even number of neutrons. If this judgement is applied to the constant temperature part of the Gilbert-Cameron composite formula, it follows that, for nuclei with the same Z, the temperature corresponding to nuclei with an odd number of neutrons must be lower than the temperature corresponding to nuclei with an even number of neutrons.

Fission level density results

In order to be able to use the code TALYS with the same level density parameters as STATIS and GNASH, some subroutines have been added. A soubroutine has been added which makes it possible to introduce as input parameters the cumulative number of levels at which the fit begins (N low ) and the temperature T in order to calculate the parameters of the Gilbert-Cameron composite formula (as presented in chapter 3.3.3), as well for the ground state, as for the fission path. Another addition to TALYS is the possibility to calculate the collective enhancement factor according to the possible saddle point symmetries, as presented in chapter 4.4.1.

All americium nuclei ( 242 -239 Am) are taken as being axially asymmetric and mass symmetric (AS) (Vladuca et al., 2006). An important set of parameters for the statistical model calculations is represented by the transition states on the fission path. As for the fundamental deformation, the spectra of the transition states corresponding to the two fission barriers have two components: the discret and the continuum, the second one being described by the Gilbert-Cameron composite formula.

The energetic positioning of the intrinsic levels in the two saddle points are estimated using microscopic models. An important point to consider are the saddle point symmetries of the nucleus. For example, if the nucleus is in the second saddle point symmetric to rotation and asymmetric to reflection, the mass-asymmetric levels (which are asymmetric to reflection) will have a lower energy than the β and γ levels (which are symmetric to reflection).

The energies, spins and parities of the head band states for the isotope 242 Am are given in Table 4.2. For the other isotopes 241 -239 Am are the headband states given in Annex 2. For the neptunium nuclei, the first two fissioning nuclei ( 238, 237 Np) are taken with axially asymmetric and mass symmetric (AS) shapes while the next ones ( 236 -235 Np) are considered to have axially and mass symmetry (SS) (Vladuca et al, 2006).

The energies, spins and parities of the head band states for the isotope 238 Np are given in Table 4.4. For the other isotopes 237 -235 Np are the headband states given in Annex 2.

Np-238

Barrier A Barrier B Fig. 4.11.: Cumulative number of levels calculated with fission level densities and the cumulative number of discrete transitional spectra of fissioning nuclei 238-237 Am. The transmission coefficients provided by the ECIS code as results of the calculations presented in chapter 2, have been used in the calculation of the transmission coefficients for the neutronic channels.

The calculations regarding the compound nucleus mechanism, which is treated by statistical model, for the reactions n+ 241 Am, n+ 237 Np, n+ 235 U and n+ 233 U were obtained using the codes STATIS, GNASH-FKK and TALYS. STATIS, which contains a refined fission model, can be used only for the first fission chance, and the calculations for the secondary fission chances have been performed with the code GNASH-FKK. The version of GNASH which has been used contains some modifications to the original code (Tudora et al, 2005;Vladuca et al, 2006): the possibility to use for the two saddle points different values for the moment of inertia has been implemented; the parameters of the level density can be given to be different in the two saddle points, and these parameters can be calculated automatically, to find the best coupling between the discrete spectrum and the level density function; the enhancement factors in the two saddle points can be computed automatically, by just mentioning the symmetries the nuclei has in these two points.

A comparison with the results given by TALYS has been done.

Cross section results for n+ 241 Am

The parameters for the fundamental fission barriers for the reaction n+ 241 Am are given, in comparison to the parameter retrieved from the database RIPL3, in Table 4.9. The results are similar, although for the (n,2n) cross section TALYS fits better the experimental data, while GNASH slightly underestimates a part of the experimental data in the energy range 8-13MeV, and overestimates for the energies 13-20 MeV. The differences are very small, the highest one being of 0.1b. 1MeV, is the cross section obtained using TALYS a better fit for the experimental data; also for the energy region 1.5 -5 MeV is a difference visible between the results obtained with STATIS and those obtained using TALYS. For the second fission chance are the results obtained with TALYS and GNASH almost identical, while for the third and fourth fission chance are the results obtained with GNASH closer to the experimental data of Dabbs. Symmetry in B SA(MA) MA SA(MA) MA SA(MA) MA SA(MA) n/a

The GDR parameters that have been used are given in comparison to the RIPL3 values, in Table 4.12. the TALYS code gives higher values for the capture cross section. At these energies the values of the capture cross section are already very small (lower than 0.1 b), so that this difference will not have a significant influence on the result of the other cross sections. The good agreement of the calculations with the experimental data in the low energy region proves again the correct normalization of gamma-ray transmission coefficients. data. The energy range is presented once in linear scale, so the whole domain is visible, and once in logarithmic scale, to show the fission cross section at small energies. For the first fission chance the results of the calculations with STATIS have been used, and for the secondary fission chances the results of GNASH are plotted. It can be seen that the results are in good agreement with the experimental data. The fission cross section obtained using the TALYS code is very similar to the one obtained with GNASH; a difference can be seen in the energy range 2 -6MeV, where the result obtained with TALYS is higher than the one obtained with GNASH. The cross section of the first fission chance is higher in the TALYS calculations than in the GNASH ones, while the secondary fission chances have an opposite behavior (being lower for the calculations performed with GNASH than those obtained with TALYS); these two effects compensate each other, so that the total fission cross section has approximately the same value for both calculations. For the reaction n+ 235 U are the parameters for the fundamental fission barriers given, in comparison to the parameter retrieved from the database RIPL3, in Tables 4.13.-4.14. In the case of the uranium nuclei, both systematic, the one from (Vladuca et al, 2006) and the RIPL3 database, assume for the isotopes 236 U-231 U SS symmetry, for which V A < V B . For reasons which will be shown later in chapter 4.6., only the results of the calculations with the codes STATIS and GNASH will be given in comparison with the experimental data and the nuclear evaluated files. The energy range is presented once in linear scale, so that the whole domain is visible, and once in logarithmic scale, in order to focus on the fission cross section at small energies.

It can be seen that the results are in very good agreement with the experimental data. All parameters that have been used in GNASH for the reaction n+ 235 U have been also used to do calculations with the code TALYS. The same optical potential, deformation parameters and number of coupled levels as in ECIS calculation have been used for the direct interaction calculations with the code TALYS. The total cross section is in good agreement with the experimental data, and with the result obtained by using the ECIS code (as shown in chapter 2). For the level density at the ground state the parameters from Table 3.3 have been introduced as input parameters, and for the level density on the fission path the parameters shown in Table 4.8. were used. As input parameters for the fission channel the ones from table 4.13 have been used. The results show a significant difference to the ones obtained with the GNASH code and are in visible disagreement with the experimental data, too. shows this big discrepancy, for the first fission chance. data. The energy range is presented once in linear scale, so the whole domain is visible, and once in logarithmic scale, to show the fission cross section at small energies. It can be seen that the results are in good agreement with the experimental data. The fission cross section obtained using the TALYS code is very similar to the one obtained with GNASH for the energy range 0.001 -0.8 MeV; above 0.8 MeV the cross section obtained with TALYS decreases significantly for the first fission chance, as it can be well seen in Fig. 4.44.

As it has been shown in Section 4.6.1 and 4.6.2., in the case of fissile nuclei 233 U and 235 U the TALYS calculations using the same parameters as in GNASH lead to results that are in a large discrepancy with the GNASH results and the experimental data. This fact suggests that a mistake is possible in the programming which is followed by the fissile nuclei, maybe in the manner to take the symmetry or the gamma channel treatment (normalization). In the neutron induced cross section calculations of two fertile ( 237 Np and 241 Am) and two fissile ( 235 U and 233 U) actinides the computer codes STATIS, GNASH and TAYLS were used. With the parameters determined by performing calculations with the STATIS and GNASH codes, the TALYS code has been tested in order to improve the treatment of the fission channel. To achieve that, some developments have been added to this code as following:

a new subroutine has been implemented which gives a better solution of the matching problem between the two parts of the Gilbert-Cameron composite formula, the constant temperature and the Fermi gas functions;

the possibility to calculate the level density parameter a from the average s-wave level spacing at the binding energy, by solving a transcendental equation; -the possibility of using enhancement factors relative to the ground state deformation, which take into account the symmetries of the nuclei on the fission path was introduced, as it is used in the codes STATIS, GNASH and EMPIRE.

The results obtained with the codes ECIS, STATIS and GNASH are in good agreement with all existing experimental data for all studied reactions. In the case of the TALYS code the calculations performed for the two fertile nuclei give good results, which describe well the experimental data and are very close to the results provided by the STATIS and GNASH codes.

The direct interaction results provided by TALYS are identical with the results provided by ECIS for all four studied cases. The disagreement of the results obtained with TALYS in the case of the two studied fissile nuclei suggest that there is an inconsistency in the parameter transmission inside the TALYS code; also some corrections have to be made in the programming path in order to obtain good results in the case of fissile nuclei, too.

Annex 1. Used parameterisations of optical model potentials

The optical model potential of Capote et al.

The deformed optical model parameterisation proposed by R. Capote (Capote et al., 2005a(Capote et al., , 2005b(Capote et al., , 2008) ) is given by the following equations:

                                     ) a , R (r, V l σ â , R r, f dr d r 1 E iW E ΔV E V c m a , R r, f dr d E iW E ΔV E ΔV a , R r, f E iW E ΔV E ΔV E ΔV E V E a, , R r, V c c Coul so so S W so so so 2 π s s S W s Coul s s V V S W v Coul v Coul HF v HF                                                                             ) , θ R (r, V l σ , θ R r, f dr d r 1 E iW E ΔV E V c m , θ R r, g E iW E ΔV E ΔV , θ R r, f E iW E ΔV E ΔV , θ R r, f E ΔV E V E , , θ R r, V c Coul so S W so so so 2 π s S W s Coul s s v S W v Coul v v HF S W Coul HF HF                                                 F HF HF HF E E λ exp A E V                  A Z N V C 1 1 V A 0 viso 1 Z 0 HF         2 v 2 F 2 F v v B E E E E A E W               F s 2 s 2 F 2 F s s E E C exp B E E E E A E W                     A Z N W C 1 1 W A 0 wiso 1 Z 0 s       F so SO so E E λ exp V E V            2 so 2 F 2 F SO so B E E E E W E W    
The following parameters are used: Efrem Sh Soukhovitskiĩ (Soukhovitskiĩ et al., 2004) proposed an optical potential for which:

                                                                 , θ R r, V L σ , θ R r, f dr d r 1 iW V c μ , θ R r, f W ) , θ R (r, f dr d a 4W i , θ R r, f V , θ R r, V Coul so so so 2 π V V D D D R R                                           E A Z Z C V 232 A V V A 2Z A C 1 1 e V E V E V 232) (A V V V Coul 1/3 Coul DISP R A R 0 R viso 1 Z E λ DISP R 2 2 R 1 R A R 0 R R R    2 D 2 2 E λ wiso 1 Z A D DISP D D WID E E e A 2Z A C 1 232) (A W W W D                    2 V 2 2 DISP V V WID E E W W        E so λ e 0 so V so V 2 so 2 2 DISP so so WID E E W W                     2 R 2 2 R R R WID E E R C 1 r E r .
It is considered: The optical model potential of Bruyères-le-Châtel

Another deformed optical model parameterisation was developed by the Bruyères-le-Châtel group, labeled BRC in the following (Morillon and Romain, 2004), and is defined as:

                            2 HF 4 2 2 HF 2 2 0 HF V E hc γ 4μ exp V E hc 2 β μ exp V E V     2 v 2 F F v v v B E E E E B A E ΔV          2 v 2 F F v v B E E E E A E W               F s 2 s 2 F 2 F s s E E C exp B E E E E A E W               E ΔV E E 0.004 exp 6 E V so F so         2 2 F F so 160 ) E (E E E 160 3 E ΔV              2 2 F 2 F so 60 1 E E E E -3 E W     
The following parameters are used: Energy KΠ Energy KΠ 1 0 2.5 1 0 2.5 2 0.025 -2.5 2 0.05 -2.5 3 0.1 4.5 3 0.1 4.5 4 0.1 0.5 4 0.1 0.5 5 0.05 -3.5 5 0.05 -3.5 6 0.05 -1.5 6 0.05 -1.5

  Fig. 1.1. The fission cross section for the fissile nuclei 235 U and 239 Pu

  with T aL the kinetic energy of the projectile in the laboratory system. normalize to the unity the incident flux.

  the unitarity of the matrix S. It follows:

  Fig.2.8. 235 U(n,tot) coupled channel calculation using the deformed optical model parameterizations BRC, Capote and Soukhovitskiĩ in comparison with the existing experimental data taken from EXFOR, in the energy range 0.001 -1 MeV.

Fig. 2 .

 2 Fig.2.13. 233 U(n,tot) coupled channel calculation using the deformed optical model parameterizations BRC, Capote and Soukhovitskiĩ in comparison with the existing experimental data taken from EXFOR, in the energy range 1 -30 MeV.

  redistribute the energy on one or more nucleons.

  the compound nucleus mechanism.

From

  In order to evaluate the contribution of the integrated cross section given by the compound nucleus mechanism, it is required to evaluate the

  Applying a similar procedure to that used to obtain the average energy of the S

  taking into account (3.26) and (3.29), equation (3.34) becomes

  26) a relation between the elements of the matrix P and a

  where a  are real phases. Neglecting ab M in (3.41) leads to For equation (3.44) not to violate the unitarity expressed by equation (3.43), it is required that relation (3.45) takes place; this is achieved only if a T « 1. In this case, equation (3.43) is identical with (3.44)random real values, not correlated for b a  , and as such 2 a g  has a 2 distribution with one degree of freedom, in agreement with the distribution obtained by Porter and Thomas for the neutron widths for the isolated resonances )

  relation which, up to the correction of the level fluctuations ab W , or "Moldauer correction", is identical with expression (3.22) obtained by Hauser and Feshbach. Knowing that

  the value of 2 (if the absorption is strong and  a a T has a greater value). In actual calculations the variation of the a  coefficients with the channel a is neglected and all channels are considered to have one or two degrees of freedom -according to a strong or weak absorption (a threshold value for  a a T is set, above which two degrees of freedom are considered).

  the number of open channels. Unlike equation (3.60), in (3.67) the dependence of the correction a G of all transmission coefficients has been taken into account. Additionally, with this new correction applied to a G , the first iteration ) 1 ( a

  as before, that the direct reactions are negligible. In this case, using for the S-matrix the equation (3.58), the average

  The matching problem gives two conditions, given by (3.103) and (3.105), with three unknowns: T, E 0 and E M .Another constraint is obtained by demanding that in the discrete level region the constant temperature law reproduces the experimental discrete levels 108) is solved by an iterative procedure. The levels N high and N low are chosen such that T

  Fig. 3.1. Possible solutions if N low and N high are used as input parameters

  the calculated values of the parameters which characterize the level density for the compound nuclei involved in the studied reactions n+ 241 Am, n+ 237 Np, n+ 235 U and n+ 233 U . The good agreement of the calculated cumulative number of levels with the experimental ones, taken from the EXFOR database, for the nuclei239-242 Am,235-238 Np and 231-236 U are shown in the Figures 3.2-3.4.

  Fig. 3.2. Level density at fundamental deformation, cumulative number of levels of 242-239 Am

Fig. 3 . 4 .

 34 Fig. 3.4. Level density at fundamental deformation, cumulative number of levels of 236-233 U

Fig. 4 . 1 .

 41 Fig. 4.1. Experimental near-barrier cross-section slopes of n + 237 Np and n + 241 Am. The average value of the fission cross-section in the plateau of the first chance fission is normalized to 1. For comparison the normalised cross-section of n + 232 Th is plotted.

  Fig. 4.2. Qualitative representation of a double-humped barrier

  Fig.4.3. Systematic of nucleus shape symmetry at the inner barrier saddle point.(Vladuca et. al, 2006)

Fig. 4

 4 Fig. 4.5. The dependence of single-neutron energies upon single-particle number for a spherical 208 PbNucleus(Bolsterli et al, 1972)

  possible for K rot to go to 1 for high excitation energy. The values for fission used in TALYS for gs col

  for the compound nuclei involved in the reactions n + 241 Am, 237 Np, 235,233 U considering enhancement factors relative to the ground state deformation The level densities at nuclear deformations through the fission path are important ingredients in the fission transmission coefficient calculations. The collective enhancement factors of Eq. (4.71) are taken according to the systematic of nuclear shape symmetries at inner barrier deformation Fig. 4.3. At the outer barrier deformation shapes with axial symmetry and mass asymmetry (SA) are taken for all fissioning nuclei.

  Fig.4.9.: Cumulative number of levels calculated with fission level densities and the cumulative number of discrete transitional spectra of fissioning nuclei 242-241 Am.

  Fig.4.12.: Cumulative number of levels calculated with fission level densities and the cumulative number of discrete transitional spectra of fissioning nuclei236-235 Am.

Fig. 4 .Fig. 4 .

 44 Fig. 4.13.: Cumulative number of levels calculated with fission level densities and the cumulative number of discrete transitional spectra of fissioning nuclei 236-235 U.

4. 5 .

 5 Neutron induced cross section calculations for the reactions n+ 241 Am, n+ 237 Np, n+ 235 U and n+ 233 U One important step in evaluating the cross sections for the different reactions, is to determine the parameters that describe the fundamental fission barriers. In order to do this, both systematics and microscopic calculations are used. They provide some approximate values of the parameters. Starting from these initial values, the parameters are varied for a few percents (between 1-10%) to obtain a better agreement of the calculated fission cross section with the experimental data.

Fig. 4 .

 4 Fig.4.16. shows the calculated capture cross section for the reaction n+ 241 Am; the results of the calculations with STATIS, GNASH and TALYS are shown in comparison to the experimental data. These results are very close to each other, except in the energy range 3 -20MeV, where the GNASH code gives higher values. As in this energy range the values of the capture cross section being already very small (lower than 0.03 b), they will not influence the result of the other cross sections. The good agreement of the calculations with the

Fig. 4 .

 4 Fig. 4.19. The (n,xn) cross sections for the reaction n+ 241 Am

  Fig. 4.20. Fission cross section of n+ 241 Am compared to EXFOR data

Fig. 4 .

 4 Fig. 4.22. shows the calculated capture cross section for the reaction n+ 237 Am; the results of the calculations with STATIS, GNASH and TALYS are shown in comparison to the experimental data. Their results are very close to each other. In the energy range 0.8 -20MeV

  Fig. 4.22. The capture cross section for the reaction n+ 237 Np

Fig. 4 .

 4 Fig. 4.23. The elastic cross section for the reaction n+ 237 Np

Fission

  Fig. 4.26. Fission cross section of n+ 237 Np compared to EXFOR data

FissionFig. 4 .

 4 Fig. 4.27. Fission cross section of n+ 237 Np compared to EXFOR data

Fig. 4 .

 4 Fig.4.28. shows the calculated capture cross section for the reaction n+ 235 U; the results of the calculations with the STATIS code are given in comparison to the experimental data. The results are in very good agreement with the experimental data. In Figs.4.29.-4.30. the elastic and inelastic cross sections are given in comparison as well with the experimental data, as with the evaluation files ENDF/B-VII, JEFF-3.1.2 and JENDL 4.0. The results are in good agreement with both, the experimental data and the nuclear evaluated files.

Fig. 4 .Fig. 4 .

 44 Fig. 4.28. The capture cross section for the reaction n+ 235 U

Fig. 4 .

 4 Fig. 4.32. The fission cross section for the reaction n+ 235 U, in comparison with the experimental data

Fig. 4 .

 4 Fig. 4.35. The elastic cross section for the reaction n+ 233 U

FissionFig. 4 .

 4 Fig. 4.37. The fission cross section for the reaction n+ 233 U, in comparison with the experimental data

Fig

Fig. 4 .

 4 Fig.4.38. The fission cross section for the reaction n+ 235 U, obtained with the code TALYS, in comparison with the experimental data and the calculation using the same parameters performed with GNASH

  Fig.4.39. The capture cross section for the reaction n+ 235 U, obtained with the code TALYS, in comparison with the experimental data and the calculation using the same parameters performed with GNASH

Fig. 4 .

 4 Fig. 4.42. The elastic cross section for the reaction n+ 233 U

Fission

  Fig. 4.43. The fission cross section for the reaction n+ 233 U, in comparison with the experimental data

Fission

  Fig. 4.44. The fission cross section for the reaction n+ 233 U, in comparison with the experimental data

  2516 + 0.001367 (238 -A) a HF = 0.636 -0.002 (238 -A) r v = 1.253 a v = 0.680 -0.00033 ( 238 -A) r s = 1.1808 a s = 0.603 -0.0005 ( 238 -

  Z,A) denotes the separation energy of nucleon i from a nucleus (Z,A)) and μ stands for the reduced mass.

Fig.2.1. Real and imaginary depths of volume, surface and spin-orbit for 237 Np. The depths for the BRC potential are plotted with solid lines, the ones for the Capote potential with dotted lines and for the Soukhovitskiĩ potential with dashed lines.

		50
		40
	Potential depth (MeV)	10 20 30	BRC	Vr Wv Vs Ws Vso Wso	Capote Vr Wv Vs Ws Vso Wso	Soukhovitskii Vr Wv Vs=0 Ws Vso Wso
		0
		0		5	10	15	20	25	30
						En (MeV)

in Table 2.1. were used. Table 2.1.Deformation parameters for 237 Np

  237 Np the first five levels of the ground state rotational band were

		237 Np(n,tot)						Kornilov 2000 Lychagin 1997
											Auchampaugh 1984
	Total cross-section (MeV)										Capote Soukhovitskii BRC
	-3				10 -2					10 -1					10 0
	coupled						En (MeV)
	2.5 + Fig.2.4. 237 Np(n,tot) coupled channel calculation using the deformed optical model parameterizations 0.0 MeV
	3.5 + BRC, Capote and Soukhovitskiĩ in comparison with the existing experimental data taken from 0.03319 MeV
	4.5 +	0.07592 MeV EXFOR, in the energy range 0.001 -1 MeV.
	5.5 +		0.13000 MeV
	6.5 +		0.19146 MeV
	8
	6 The deformation parameters presented Capote Soukhovitskiĩ BRC 7
			β 2 0.207 0.213			0.207
	5		β 4 0.070 0.08				0.070
			β 6 -0.01	0.0015			-0.01
	4
	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30
	The total cross section calculations are given in Figs. 2.4 and 2.5 for the three studied
	potentials. In both energy ranges (0.001 -1 MeV and 1 -30 MeV) the best agreement
	between the experimental data and the coupled channel calculations have been obtained by
	using the Capote potential, with the deformation parameters given in Table 2.1.

237 Np(n,tot)

	2.4
		Kornilov 2000
		Lychagin 1997
		Auchampaugh 1984
	Total cross-section (MeV)	Capote Soukhovitskii BRC
	En (MeV)
	Fig.2.5. 237 Np(n,tot) coupled channel calculation using the deformed optical model parameterizations
	BRC, Capote and Soukhovitskiĩ in comparison with the existing experimental data taken from
	EXFOR, in the energy range 1 -30 MeV.

.2. Results for the reaction n+ 241 Am

	In the case of n+ 241 Am the first six levels of the ground state rotational band were
	coupled
	2.5 -	0.0	MeV
	3.5 -	0.04118 MeV
	4.5 -	0.09370 MeV
	5.5 -	0.15750 MeV
	6.5 -	0.23370 MeV
	7.5 -	0.33400 MeV
	For the Capote potential, the parameters suggested by the JENDL-4.0 evaluation have
	been used:
	 Volume parameters:
	V 0 = 48	λ HF = 0.004	C viso = 15.9
	A v = 12.04	B v = 81.36	E a = 385
	 Surface parameters:
	W 0 = 17.2	B s = 11.19	C s = 0.01361	C wiso = 23.5

Table 2 .2. Deformation parameters for 241 Am

 2

	Capote Soukhovitskiĩ BRC
	β 2 0.213 0.213	0.207
	β 4 0.070 0.08	0.070
	β 6 0.015 0.0015	-0.01
	The total cross section calculations are given in Figs. 2.6 and 2.7 for the three studied
	potentials. At small energies, between 0.001 and 1 MeV, the Soukhovitskiĩ potential is in the
	best agreement with the experimental data, but for the whole energy range the coupled
	channel calculations with the Capote potential, modified with the JENDL-4.0 parameters, and
	using the deformation parameters β 2 =0.213, β 4 =0.070 and β 6 =0.015, give the best agreement
	with the experimental data.

.3. Results for the reaction n+ 235 U

	The total cross section calculations are given in Figs. 2.8 and 2.9 for the three studied
	potentials. In the energy range 0.001 -1 MeV the best agreement of the calculations with the
	experimental date are obtained using the BRC parameterization, with the deformation
	parameters β 2 =0.215, β 4 =0.110 and β 6 =-0.010, while for the energy range 1 -30 MeV the
	Soukhovitskiĩ parameterization with β 2 =0.198, β 4 =0.099 and β 6 =-0.0097 is in the best
	agreement with the experimental data.
	For the reaction n+ 235 U exist also experimental data for the differential elastic cross-
	section. The results of the coupled-channel calculations are compared to this experimental
	data in Figs. 2.10 and 2.11. The results for the differential elastic cross sections obtained with
	all three potentials are in good agreement with the experimental data, with no significant
	differences between them.
	In the case of n+ 235 U the first five levels of the ground state rotational band were
	coupled
	3.5 -	0.0	MeV
	4.5 -	0.04621 MeV
	5.5 -	0.10304 MeV
	6.5 -	0.17071 MeV
	7.5 -	0.24913 MeV
	The deformation parameters given in Table 2.3 were used.
	Table 2.3. Deformation parameters for 235 U
		Capote Soukhovitskiĩ BRC
		β 2 0.211 0.198	0.215
		β 4 0.107 0.099	0.110
		β 6 -0.021 -0.0097	-0.01

Table 2

 2

	.4. Deformation parameters for 233 U
	Capote Soukhovitskiĩ BRC
	β 2 0.220 0.183	0.207
	β 4 0.100 0.120	0.117
	β 6 0.0015 0.003	0.008
	The total cross section calculations are given in Figs. 2.12 and 2.13 for the three
	studied potentials. In the energy range 0.001 -1 MeV, as well as in the energy range 1 -30
	MeV the best agreement of the coupled channel calculations with the experimental date is
	obtained using the optical model parameterization of Capote, with the deformation parameters
	β 2 =0.220, β 4 =0.100 and β 6 =0.0015.

Table 3

 3

		.1. Am isotopes: parameters at the fundamental deformation
			Am-242 Am-241	Am-240 Am-239
	B n (MeV) 5.537567 6.641373	5.957443 7.101967
	Δ (MeV)	0	0.43	0	0.49
	a (MeV)	27.79175 26.06914	25.67486 25.58980
	N low	8	10	10	6
	T (MeV)	0.400	0.417	0.405	0.420
	E m (MeV) 3.1317	3.5883	3.2876	3.6171
	E 0 (MeV) -1.2351	-0.7771	-1.0230	-0.6862
	σ r	2	32.1716 31.2041	29.4263 30.5927

Table 3

 3

	.2. Np isotopes: parameters at the fundamental deformation
		Np-238 Np-237 Np-236 Np-235
	B n (MeV) 5.4881 6.5739 5.7390 6.9840
	Δ (MeV) 0	0.49	0	0.57
	a (MeV)	27.908 27.3751 27.3462 27.2518
	N low	48	20	3	9
	T (MeV) 0.412	0.417	0.4174 0.4185
	E m (MeV) 3.3885 3.8838 3.3874 3.9711
	E 0 (MeV) -1.3999 -0.8998 -1.3918 -0.8222 σ r 2 33.1479 33.6037 32.6838 32.5584

Table 3

 3

		.3. U isotopes: parameters at the fundamental deformation
		U-236	U-235	U-234	U-233	U-232	U-231
	B n (MeV)	6.544758	5.297840	6.844166	5.759434	7.2729	5.8714
	Δ (MeV)	1.18	0.69	1.26	0.69	1.29	0.69
	A (MeV)	28.7852	28.9970	28.1180	27.9814	27.96765	27.99537
	N low	26	22	26	26	20	3
	T (MeV)	0.425	0.420	0.423	0.420	0.423	0.420
	E m (MeV)	4.8913	4.4392	4.9119	4.2546	4.8727	4.2778
	E 0 (MeV) σ r 2	-0.4340 34.7000	-0.9701 35.2591	-0.3147 34.1697	-0.8260 33.5806	-0.2377 33.5613	-0.8423 33.5050

Table 4

 4 It results that the slopes for n + 237 Np and n + 241 Am are significantly higher than the slope obtained for n + 232 Th, which has a triple-humped fission barrier, this being an indication for the fact that 238 Np and 242 Am have only two fission barriers.

	.1. Slopes of the near-barrier cross-sections
	Reaction	Slope
	N + 237 Np	0.6869
	N + 241 Am	0.7902
	N + 232 Th	0.2713

Table 4 .

 4 All level density parameters for the fission path for the nuclei242-239 Am, which are involved in the reaction n+ 241 Am, are summarized in Table 4.3.; the cumulative number of levels calculated with fission level densities and the cumulative number of discrete transitional spectra for the americium nuclei are given in Fig 4.9.-4.10. With the red, respectively blue line, is the cumulative number of levels for barrier A, respectively barrier B, represented. With the magenta, respectively cyan colored line is the level density, with the parameters from Table 4.3., shown. The very good agreement between the cumulative number of the discrete spectrum and the level density is visible.

		2. Head band states for 242 Am
		A			B
		Energy KΠ		Energy KΠ
	1	0	-2	1	0	-2
	2	0.125	2	2	0.15	2
	3	0.3	-4	3	0.1	-4
	4	0.3	-0	4	0.1	-0
	5	0.05	3	5	0.15	3
	6	0.05	1	6	0.15	1
	7	0.008	3	7	0.008	3
	8	0.133	-3	8	0.158	-3
	9	0.308	5	9	0.108	5
	10	0.308	1 10	0.108	1
	11	0.058	-4 11	0.158	-4
	12	0.058	-2 12	0.158	-2
	13	0.006	0 13	0.006	0
	14	0.131	-0 14	0.156	-0
	15	0.306	2 15	0.106	2
	16	0.306	2 16	0.106	2
	17	0.056	-1 17	0.156	-1
	18	0.056	-1 18	0.156	-1
	19	0.01	-2 19	0.01	-2
	20	0.135	2 20	0.16	2
	21	0.31	-4 21	0.11	-4
	22	0.31	-0 22	0.11	-0
	23	0.06	3 23	0.16	3
	24	0.06	1 24	0.16	1

Table 4

 4 All level density parameters for the fission path for the isotopes238-235 Np are summarized in Table 4.5.; the cumulative number of levels calculated with fission level densities and the cumulative number of discrete transitional spectra for the neptunium nuclei are given in Figs 4.11.-12. With the red, respectively blue line, is the cumulative number of levels for barrier A, respectively barrier B, represented. With the magenta, respectively cyan colored line is the level density, with the parameters from Table 4.5., shown. The very good agreement between the cumulative number of the discrete spectrum and the level density is visible.

		.4. Head band states for 238 Np
		A	B
		Energy KΠ	Energy KΠ
	1	0 0	1	0 0
	2	0.004 2	2	0 2
	3	0.008 3	3	0	3
	4	0.01 -1	4	0	-1
	5	0.025 -0	5	0	-0
	6	0.029 -2	6	0	-1
	7	0.033 -3	7	0	-1
	8	0.035 1	8	0	-2
	9	0.05 -1	9	0	-3
	10	0.05 -1	10	0	-1
	11	0.054 -3	11	0	-3
	12	0.054 -1	12	0	-4
	13	0.058 -4	13	0	-2
	14	0.058 -2	14	0	1
	15	0.1 2	15	0	2

Table 4 .

 4 8. Level density parameters at the saddle points for 236-231 U

		U-236	U-235	U-234	U-233	U-232	U-231
	T A (MeV)	0.430	0.435	0.430	0.435	0.430	0.435
	E m A (MeV)	5.1064	4.7675	5.052	4.570	5.061	4.573
	E 0 A (MeV)	-0.5889	-1.1843	-0.404	-1.028	-0.358	-1.032
	K A	1.0286	1.0429	1.029	1.040	1.026	1.040
	N low A	20	100	20	90	20	100
	E low A (MeV)	0.38	0.57	0.98	0.47	0.980	0.56
	T B (MeV)	0.435	0.440	0.435	0.440	0.435	0.440
	E m B (MeV)	5.2164	4.8799	5.159	4.678	5.167	4.681
	E 0 B (MeV)	-0.6608	-1.2584	-0.473	-1.098	-0.427	-1.102
	K B	2.0858	2.1143	2.087	2.138	2.081	2.109
	N low B	25	100	20	100	20	100
	E low B (MeV)	0.24	0.52	0.872	0.51	0.872	0.51

Table 4

 4

									.9. Fission barrier parameters for 242-239 Am
									Am-242	Am-241	Am-240	Am-239
									Used	RIPL3 Used	RIPL3 used	RIPL3 Used	RIPL3
									parameters	parameters	parameters	Parameters
	V A						6.38	6.32 5.90	6.06 6.10	6.10 5.80	6.00
			A					0.65	0.60 1.00	0.80 0.60	0.60 1.00	0.80
			2	2		A	(keV) 5	5	5	5
	Symmetry in A AS(GA) GA	AS(GA) GA	AS(GA) GA	AS(GA) GA
	V B						5.54	5.78 4.90	5.35 6.00	6.00 4.80	5.40
			B					0.41	0.40 0.50	0.50 0.40	0.40 0.50	0.56
			2	2		B	(keV) 2	2	2	2
	Symmetry in B SA(MA) MA SA(MA) MA SA(MA) MA SA(MA) MA
									Table 4.10. GDR parameters for 242 Am
	Nucleus	Am-242	Am-242 RIPL3
	β 2							0.213	0.213
	E 01 (MeV)	11.4397	10.80
	Γ 01 (MeV)	2.5966	2.42
	E 02 (MeV)	13.9934	13.95
	Γ 02 (MeV)	4.4804	3.87
	σ 02 / σ 01	0.221
	Γ 0 exp (meV)	46	46
	D 0 (eV)		0.58	0.58

The GDR parameters entering the transmission coefficients for gamma in the frame of the Kopecky-Uhl formalism with 2 resonances that have been used are given in comparison to the RIPL3 values, in Table

4

.10.

2. Cross section results for n+ 237 Np For

  the reaction n+237 Np are the parameters for the fundamental fission barriers given, in comparison to the parameter retrieved from the database RIPL3, in Table4.11. The difference which appears in the value of the barrier height for236 Np is related to the fact that in this work the systematics from(Vladuca et al, 2006) has been used, where for 236 Np in the first saddle point a SS symmetry is assumed (symmetry for which V A < V B ), while in the RIPL3 database a AS symmetry is considered (symmetry for which V A > V B ).

									Table 4.11. Fission barrier parameters for 238-235 Np
									Np-238	Np-237	Np-236	Np-235
									Used	RIPL3 used	RIPL3 Used	RIPL3 Used	RIPL3
									Parameters	parameters	Parameters	Parameters
	V A						6.09	6.50 6.12	6.00 5.22	5.90 5.25	n/a
			A					0.46	0.60 1.00	1.00 0.46	0.60 1.00	n/a
			2	2		A	(keV) 5	5	5	5
	Symmetry in A AS(GA) GA	AS(GA) GA	SS	GA	SS	n/a
	V B						5.21	5.75 5.23	5.40 5.27	5.40 5.29	n/a
			B					0.40	0.40 0.50	0.50 0.40	0.40 0.50	n/a
			2	2		B	(keV) 2	2	2	2
				Fission cross section (b)				,	Vorotnikov 1986 Alexandrov 1983 Dabbs 1983 Cance 1981 Prindle 1979 Fomushin 1969 Knitter 1979 Hage 1981 -ratio to 235
										E(MeV)

U Fig. 4.21. Fission cross section of n+ 241 Am compared to EXFOR data 4.5.

Table 4 .12. GDR parameters for 238 Np

 4

	Nucleus	Np-238	Np-238 RIPL3
	β 2	0.211	0.215
	E 01 (MeV)	11.462	10.80
	Γ 01 (MeV)	2.596	2.43
	E 02 (MeV)	14.048	14.03
	Γ 02 (MeV)	4.494	3.91
	σ 02 / σ 01	0.243
	Γ 0 exp (meV)	40.8	40.8
	D 0 (eV)	0.57	0.57

Table 4 .

 4

									13. Fission barrier parameters for 236-234 U
									U-236	U-235	U-234
									Used	RIPL3 Used	RIPL3 Used	RIPL3
									Parameters	parameters	Parameters
	V A							5.25	5.00 4.95	5.25 5.25	4.80
	 	A						1.00	0.90 1.00	0.70 0.90	0.90
		2	2		A	(keV)	5	5	5
	Symmetry in A	SS	S	SS	S	SS	S
	V B							5.90	5.67 5.90	6.00 5. 90	5.50
			B						0.60	0.60 0.60	0.50 0.60	0.60
		2	2		B	(keV)	2	2	2
	Symmetry in B	SA(MA) MA	SA(MA) MA	SA(MA) MA

Table 4

 4

								.14. Fission barrier parameters for 233-231 U
									U-233	U-232	U-231
									Used	RIPL3 Used	RIPL3 Used	RIPL3
									Parameters	Parameters	Parameters
	V A							5.10	4.35 5.10	4.90 5.00	4.40
	 	A						0.80	0.80 0.90	0.90 0.80	0.70
		2	2		A	(keV)	5	5	5
	Symmetry in A SS	S	SS	S	SS	S
	V B							5.70	5.55 5.65	5.40 5.65	5.50
	 	B						0.50	0.50 0.60	0.60 0.50	0.50
		2	2		B	(keV)	2	2	2
	Symmetry in B	SA(MA) MA	SA(MA) MA	SA(MA) MA
	The GDR parameters for 236 U which have been used in the calculations are given in
	comparison to the RIPL3 values, in Table 4.15.
	.
								Table 4.15. GDR parameters for 236 U
								Nucleus		U-236	U-236 RIPL3
								β 2		0.215	0.211
								E 01 (MeV)	10.84	10.92
								Γ 01 (MeV)	2.45	2.55

  The fission cross section for the reaction n+ 233 U, in comparison with the experimental data

			233 U(n,f)
			Kanda JPNTOH 86		Tovesson 2004
			Kanda JPNTOH 85		Hambsch 2004
			Shpak CCPFEI 80
			Carlson USALAS 78	Shpak RUSFEI 98
			Fursov CCPFEI 78	Poenitz USAANL 78
			Shpak CCPFEI 75	Li Jing-Wen CPRAEP 77
			Meadows USALAS 74	Shcherbakov RUSLIN 2003
	Fission cross-section (b)		Pfletschinger GERKFK 70 Smirenkin CCPFEI 67 Albert USALAS 66 Lamphere USAORL 56		GNASH Lisovski USALAS 91 Shcherbakov 2003 Carlson, Behrenz 78
			En (MeV)
	Fig. 4.36. 0	2	4 Kanda JPNTOH 86 6 Kanda JPNTOH 85 Shpak CCPFEI 80 Carlson USALAS 78 Fursov CCPFEI 78 8 233 U(n,f) 10 Lisovski USALAS 91 12 Shcherbakov 2003 Carlson, Behrenz 78 Tovesson 2004 Hambsch 2004 Shpak RUSFEI 98 Poenitz USAANL 78 Li Jing-Wen CPRAEP 77 14 Shcherbakov RUSLIN 2003 Shpak CCPFEI 75 Meadows USALAS 74 Pfletschinger GERKFK 70 Smirenkin CCPFEI 67 Albert USALAS 66 Lamphere USAORL 56	16	18 GNASH	20

Table A2 .

 A2 1. Head band states for 241,239 Am A B

	Table A2.3. Head band states for 237 Np
		A			B
		Energy KΠ		Energy KΠ
	1	0 2.5	1	0 2.5
	2	0.025 -2.5	2	0.05 -2.5
	3	0.1 4.5	3	0.1 4.5
	4	0.1 0.5	4	0.1 0.5
	5	0.05 -3.5	5	0.05 -3.5
	6	0.05 -1.5	6	0.05 -1.5
	7	0.001 -0.5	7	0.001 -0.5
	8	0.026 0.5	8	0.51 0.5
	9	0.101 -2.5	9	0.101 -2.5
	10	0.101 -1.5 10	0.101 -1.5
	11	0.051 1.5 11	0.051 1.5
	12	0.051 0.5 12	0.051 0.5
	Table A2.2. Head band states for 240 Am
		A			B
		Energy KΠ		Energy KΠ
		0	0	1	0	0
		0.025	-0	2	0.05	-0
		0.1	2	3	0.1	2
		0.1	2	4	0.1	2
		0.05	-1	5	0.05	-1
		0.05	-1	6	0.05	-1
		0.008	3	7	0.008	3
		0.033	-3	8	0.058	-3
		0.108	5	9	0.108	5
		0.108	1 10	0.108	1
		0.058	-4 11	0.058	-4
		0.058	-2 12	0.058	-2
		0.004	2 13	0.004	2
		0.029	-2 14	0.054	-2
		0.104	4 15	0.104	4
		0.104	0 16	0.104	0
		0.054	-3 17	0.054	-3
		0.054	-1 18	0.054	-1
		0.01	-1 19	0.01	-1
		0.035	1 20	0.16	1
		0.11	-3 21	0.11	-3
		0.11	-1 22	0.11	-1

Table A2

 A2

		.4. Head band states for 236 Np
		A		B
		Energy KΠ	Energy KΠ
	1	0	0 1	0	0
	2	0.008	3 2	0.008	3
	3	0.004	2 3	0.004	2
	4	0.01	-1 4	0.01	-1
	5	0.009	-0 5	0.001	-0
	6	0	-2 6	0.006	-2
	Table A2.5. Head band states for 235 Np
		A		B
		Energy KΠ	Energy KΠ
	1	0 2.5 1	0 2.5
	2	0.125 1.5 2	0.05 1.5

* * * * * * *In the hypothesis of a total K-mixing effect, it must be taken into account that once reached the isomeric well by the penetration of an interior barrier, characterized by a projection K on the symmetry axis, the compound nucleus can penetrate any interior or exterior barrier preserving the momentum J, no matter their K' projections. Expression (4.18) becomes(Vladuca et al, 1997 a,b,c)

In the case of the U nuclei, all nuclei ( 236-231 U) are considered to have axial and mass symmetric shape (SS).

The energies, spins and parities of the head band states for the isotopes 236 U and 234 U are given in Table 4.6. and 4.7. For the other isotopes 235,233-231 U are the headband states given in Annex 2. The same parameters for the level density, for the ground state and for the fission saddles, have been used for the isotopes 234 U and 233 U in both cases: as well in the calculations for the reaction n+ 235 U, where they are the third and fourth secondary chances, as well as in the calculations for n+ 233 U, where they are the primary and secondary fission chance. This proves the consistency of the calculations.

Cross section results for n+ 233 U

For the reaction n+ 233 U are the parameters for the fundamental fission barriers given in Tables 4.13-4.14., in comparison to the parameters retrieved from the database RIPL3.

The GDR parameters for 234 U which have been used in the calculations are very close to those given in RIPL3 as it can be seen in Table 4.16. Chapter 5

Conclusions

This work is devoted to the study of the fast neutron induced cross section of actinides, focusing on the fission cross section. The nuclear models for the theoretical prediction of nuclear reaction cross sections have been described. The computer codes used in this work, including the models as well as methods and procedures for determining their input parameters have been described. On this basis, neutron cross sections were calculated for the fertile actinides 241 Am, 237 Np and the fissile nuclei 235 U and 233 U.

The subject of this work fits well in the international effort of revision and improvement of nuclear data evaluation, being a topic of interest for both fundamental and practical points of view.

The direct interaction mechanism has been treated in the frame of the coupled channel formalism using recent deformed optical model parameterizations, which take the dispersion into account, and the computer code ECIS. Studies regarding the influence of coupling levels and deformation parameters were performed, too. The calculated total cross section of the four studied reactions is obtained in good agreement with the experimental data for all studied deformed optical parameterizations. In the case of 235 U the existing experimental differential elastic cross-section data are also well described by the coupled channel results with all studied potentials. In order to chose the deformed optical parameterization to be used in the neutron induced cross section calculations a χ 2 test has been performed to determine the best agreement of the coupled channel total and differential elastic cross-sections with the experimental data. Both used codes, GNASH and TALYS, include the other two mechanisms, compound nucleus and pre-equilibrium, involved in the studied fast neutron induced reactions.

In the incident energy range of interest in this work the compound nucleus mechanism has been treated by statistical model using the correction of the width fluctuations given by the extended statistical HRTW and/or the Moldauer model. At the ground state deformation the most important roles are played by the discrete spectrum, the level density (with level density parameters obtained from average resonance data and described by the Gilbert-Cameron composite formula), the gamma-ray strength function and its normalization to the swave experimental data. For the fission path accurate parameters regarding the discrete