FACTORS DETERMINING THE ENERGY DEPENDENCE OF DELAYED NEUTRON YIELDS IN NEUTRON INDUCED FISSION

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1 ÓÄÊ FACTORS DETERMINING THE ENERGY DEPENDENCE OF DELAYED NEUTRON YIELDS IN NEUTRON INDUCED FISSION V. G. Pronyaev, V. M. Piksaikin Institute of Physics and Power Engineering Obninsk, Russia The scheme of parametrization of the energy dependence of delayed neutron yields (DNY) in neutron induced fission is proposed. It is based on "microscopic" approach, where the DNY for each precursor is obtained as a product of cumulative yield (CY) of this precursor on neutron emission probability P n. The dependence of cumulative yields from incident neutron energy can be obtained from independent yields calculated with account of the following factors having a physical sense: i) contributions from different chances of fission; ii) variation of mass independent yields with energy due to variations of contributions of different modes of fission (Symmetric, Standard I and Standard II); iii) washing out of even-odd effects in charge distribution of fission fragment yields with increase of the neutron energy; iv) shift of distribution of fragment mass yields for a given isotope formed after emission of prompt neutrons because of variation of number of prompt neutrons with energy.. The analysis was done for the energy region of the first fission chance of 235 U and 237 Np nuclei (neutron energy below 5.5 MeV). The different experimental data were used for a choice of parameters. It is shown, that DNY for fissioning nuclei with even and odd Z have rather different energy dependence and general decrease of DNY is explained mainly by variation of number of prompt neutrons with energy. 1. INTRODUCTION There are only a few direct experimental data on the energy dependence of the delayed neutron yields (DNY) for neutron induced fission [1] (see for example Fig. 1 for 235 U). These data show that integral DNY for even Z fissioning system probably has some small increase below 3 MeV and then rather sharp decrease which definitely can not be connected with a threshold of second chance fission. Some semi-empirical formulas for introduction of energy dependence in integral DNY were given in paper [2]. The formulas introduce the energy dependence through the dependence of prompt neutron yield (PNY) from energy and some additional corrective function. But this connection between DNY and PNY is not so evident in "macroscopic" parametrization. The most of modern files of evaluated data contain (see dashed curves in Fig. 10) the DNY values which are constant up to 4 MeV. The analysis made in paper [3] had demonstrated a high sensitivity of the fast reactor kinetic parameters to the uncertainty in the energy dependence of DNY. The aim of this work is to find the way of parametrization of the DNY energy dependence basing on simple physical considerations and physical model parameters. It means that we want to avoid the discussion of complicated problems of physics of fission and to use direct experimental data for introducing of energy dependence of the parameters where it is possible. 2. FACTORIZATION OF THE ENERGY DEPENDENCE OF THE DELAYED NEUTRON YIELDS 2.1. General scheme of parametrization The so-called "microscopic" model of the DNY will be used for introducing of the energy dependence. Let CY(Z,A,E n ) is cumulative yield of fragment (precursor) with charge Z and mass number A after fission induced by neutron with energy E n and P n (Z,A) is a probability of the neutron emission after beta-decay of (Z,A) nucleus. Because the slow process of beta-decay occurs always from the same fixed states (ground or metastable), P n (Z,A) is turned out not depending from incident energy. Then the integral DNY can be calculated as: DNY( E ) = CY( Z, A, E ) P ( Z, A ) (1) n n n ZA, Thus the energy dependence of DNY is completely explained by energy dependence of precursors cumulative yields. Cumulative yields can be obtained from independent yields IY(Z,A,E n ) if we know the branching coefficients of different transitions. There is well known evaluation [4] of P n (Z,A) for 272 precursors determining the main contribution in integral DNY for most practical applications. Independent yields IY(Z,A,E n ) are parametrized only for fission induced by thermal neutrons (THRM), neutrons of fission spectrum (FAST) and 14 MeV neutrons (HIGH) [5,6]. It is difficult to expect that linear

2 interpolation of parameters chosen in this semi-empirical parametrization can give right prediction of IY(Z,A,E n ) and therefore the DNY energy dependence. The IY(Z,A,E n ) for thermal or fast neutron induced fission were used as initial (benchmark) distributions for the introducing of the energy dependence in this work. The following factors which may influence on this dependence have been selected: contribution of different fission chances, contribution of different fission modes for the same chance, change of IY due to change of PNY with energy and washing out of even-odd effects in charge distribution of fission fragment yield Separation of fission chances The influence of fission chances on the energy dependence of DNY is demonstrated in Fig. 1 for 235 U neutron induced fission. For this, the DNY for each fission chance was taken from file of evaluated data of ENDF/B-VI library for corresponding fissioning nucleus at FAST neutron energy. These partial DNY values were fixed and total energy dependent DNY was obtained by summation of contribution from each fission chance weighted on the contribution of this chance in the total fission cross section: DNY( U, E n ) = ( DNY( U, FAST ) σ( n, 0nf ) + DNY( U, FAST ) σ( n, 1nf ) (2) DNY( U, FAST ) σ( n, 2nf ) + DNY( U, FAST ) σ( n, 3nf ))/ σ( n, f ) where σ( n, f ) = σ( n, 0nf ) + σ( n, 1nf ) + σ( n, 2nf ) + σ( n, 3 nf ) and all cross sections and DNY were taken from ENDF/B-VI library. It is seen from Fig. 1 that at least part of the energy dependence of DNY may be connected with a variation of contribution of different fission chances. Relative contribution of each fission chance in the total delayed ( i ) neutron yield will be R En n i nf n f 1 ( ) = σ(,( ) )/ σ (, ), where i is a number of fission chance. In this work we will limit our consideration only by the energy region where the contribution of first fission chance is dominating Variations caused by contributions of different fission modes The variation of mass independent yields due to variation of contribution of different modes of fission with energy can be parametrized by means of five Gaussians which describe the contribution of symmetrical fission, and two asymmetrical modes (Standard I and Standard II) [7,8,9]. But because we consider here a narrow energy range En < 7.0 MeV, where mass independent yields for main precursor nuclei (85 < A < 105 and 130 < A < 150) have no big changes we prefer to use existing experimental data for parametrization of these changes. Ratio of primary fragment mass yields measured for 235 U [10] at neutron energy 5.5 MeV to the yields measured at 0.5 MeV is shown in Fig. 2 by thin solid line. The intermediate structure in ratio is partially connected as it will be shown below (see III.I) with odd-even effects in charge distribution. Because the uncertainties of the experimental data are not given, some narrow sharp peaks in ratio may be caused by data uncertainties. To exclude the influence of these uncertainties and odd-even effects which will be parametrized separately, we made an averaging of yields on primary fragment masses within an interval at 5 atomic mass numbers. Ratio for these averaged yields is shown in Fig. 2 by thick solid curve. The linear interpolation and extrapolation of the ratio R 2 (A',E n ) for each primary fragment mass A' in dependence from neutron incident energy in a range between thermal and 7.0 MeV and normalized to 1 at thermal neutron energy was used by us in this work for description of variation of mass distributions due to variation of contribution of different fission modes Parametrization of odd-even differences Discussion of the odd-even effects in charge distributions of fission fragments can be found in papers [11,12]. This effect is rather strong for 235 U neutron induced fission and absent for 237 Np as for nuclei with even Z. The value of D p (in per cent) can be considered as amplitude of the effect [11]: IYe( En ) IYo( En ) Dp( En ) = 100 (3) IY ( E ) + IY ( E ) e n o n where IY e (E n ) and IY o (E n ) are sums of IY(Z,A,E n ) on A with odd or even Z for IY e (E n ) or IY o (E n ) correspondingly. The existing experimental data for D p (E n ) in 235 U are shown in Fig. 3 in dependence from excitation energy at the saddle point E s =B n -B f +E n. Because in rough statistical approximation IY e (E n )/IY o (E n )=exp(c/sqrt(e s )) is not able to fit the variation of even-odd differences with excitation energy observed in experiment [11], we have used the pure empirical dependence IY e (E n )/IY o (E n )=exp(c/e s ) with

3 constant C=0.511 MeV for 235 U which through expression (3) fits it rather well (see solid curve in Fig. 3). Then using a simple expressions IY e (E n )=1+D p (E n ) and IY o (E n )=1-D p (E n ) we may calculate the ratios R 3 e =IY e (E n )/IY e (THRM) and R 3 o =IY o (E n )/IY o (THRM) which give us the qualitative evaluation of washing out of even-odd effects with energy. The energy dependence in this approximation is taken the same for all odd or even Z fragments Energy variation of number of prompt neutrons and independent yields Influence of variation of average number of prompt neutrons with energy on independent mass yields was used in empirical parametrization considered by A.C. Wahl [13]. But this influence on the mass distributions for a fragment with a fixed Z can be much stronger. The emission of prompt neutrons transforms the distribution from primary fission fragment yields (PY) for a given isotope to the independent yields by shifting PY(A',Z,E n ) for each primary fragment (A',Z) to IY(A,Z,E n ) (where A=A'-(n p (A',Z,E n ))). It may lead to substantial decrease of yield for neutron redundant nuclei contributing in formation of DNY precursors. Unfortunately there are no available experimental data on n p (A',Z,E n ). In this work we have used the experimental data for νp( A, En ) = νp( A, Z, En ) measured at Karlsruhe [10] at two neutron incident Z energies: 0.5 MeV and 5.5 MeV. The data are shown in Fig. 4. Because in this work we are going to take into account the influence of the change of prompt neutron number with energy in average we have found the average number of prompt neutrons emitted by light (85<A'<105) and heavy (130<A'<150) fragments for these two energies. As appeared, the average number of prompt neutrons is practically not changed for light fragment and dn p =n p (E n =5.5 MeV)-n p (E n =0.5 MeV)=0.65 for heavy fragment in considered energy range. The last value is consistent with the value dn p =0.651 which can be obtained from the evaluated data files as growth of total n p between these two energies. We can introduce the factor R 4 (Z,A,E n ) which gives just increase (decrease) of IY(Z,A,E n ) with energy due to shift of distribution caused by variation of n p with energy. In this work we have used the following approximations for change of mass distributions of independent yields of fragments with fixed Z in energy range of first fission chance: no shift for any light fragment (lf) (R 4 (Z,A lf,e n )=1) and da=(n p (E n )-n p (E th )) for any heavy fragment (hf). The da values are the following: 0.1 (E n =1 MeV), 0.21 (2.0), 0.33 (3.0), 0.46 (4.0), 0.70 (5.5), 0.98 (7.0). The simple relation A=A'-n p av (A',E n ), where n p av (A',E n ) is n p (A',E n ) averaged on A', was applied for reduction of all data to the same independent mass variable. 3. TEST OF THE MODEL AND PREDICTION OF THE DNY ENERGY DEPENDENCE 3.1. Independent and cumulative yields Results of calculations of mass distributions of light and heavy fragment independent yields with a fixed Z for a chosen scheme of parametrization are shown in Fig. 5 and 6 in comparison with experimental data. Dashed lines and open symbols show the parametrization [6] and experimental yields for 235 U thermal neutron induced fission [14]. Solid lines show yields obtained in parametrization scheme of this work (but the dashed line distributions were used as initial) in comparison with experimental data (filled symbols) [15], all for 3.0 MeV neutron induced fission. It is seen, that the results of empirical parametrization [6] of charge-mass distributions for thermal neutron induced fission can substantially deviate from experimental data for fragments with low yields. To increase the accuracy of the DNY for thermal neutron induced fission evaluated in "microscopic" approach we can to correct empirical model yields, giveen in [6] by existing experimental data. Fig. 7 demonstrates the ratio of primary mass yields at 5.5 MeV to the yields at 0.5 MeV neutron energy. Thin solid line presents the experimental data (the same as shown in Fig.2) and filled circles connected by dashed line - ratios calculated in parametrization scheme of this work with account of all factors (R(E n )=R 1 (E n )R 2 (E n )R 3 (E n )R 4 (E n ), but R 1 (E n )=1 and R 4 (E n )=1 for light fragment as discussed in chapters II.1. and II.4.). Intermediate structure in ratio can be explained by disappearance of odd-even effects with increase of neutron energy (see II.3.). Charge numbers near local minima and maxima in ratio show which isotopic yield determines this structure (see also Fig. 5 which clearly explains the origin of these structures). Experimental data demonstrate more sharp structure which probably can be explained partly by low statistical uncertainty of these data. The results of calculations of cumulative yields for a few nuclides presenting light or heavy fragment are shown in Fig. 8 and 9 in comparison with existing experimental data. There is the contribution of second chance fission for neutron energy higher than 5.5 MeV which was not accounted here. Because the accuracy of the empirical parametrization of independent charge-mass yields for thermal neutron induced fission (as seen for example from Fig. 5) is not higher than %, we have the same order discrepancies for cumulative yields of fragments. But the general behavior of cumulative yields energy dependence is described rather well.

4 3.2. Energy dependence of DNY for 235 U and 237 Np in the fission induced by neutrons with energy MeV. The results of calculations of total DNY for neutron induced fission of 235 U and 237 Np nuclei in parametrization scheme of this work are compared with existing experimental data in Fig. 10. The parameters of the IY description for 237 Np have been chosen from analysis of experimental data [10] similar to those for 235 U. Stepped dashed lines in this figure show the evaluated data from ENDF/B-VI library, dotted line - result of calculations of DNY in the frame-work of the present parametrization with IY for thermal neutron taken from [6] and thick solid lines - with IY for thermal ( 235 U) and fast ( 237 Np) neutron induced fission taken from [6] and corrected with account of available experimental data [14,15,16,17] for IY and CY. It is seen, that account of experimental data for IY and CY improves the consistency with the existing experimental data for DNY. The energy dependence of DNY for odd and even Z nuclei is appeared rather different. The initial increase of DNY for odd Z 235 U is explained by odd-even effects in yields of nuclei contributing in precursors formation. These effects are absent for even Z 237 Np. If the parametrization of energy dependence of washing out of odd-even effects will differ from that used in this work (see Fig. 3), the other shape of DNY energy dependence can be obtained. For parametrization, we have used in this work an approximation that light fragment independent yields have no variations in considered energy range caused by change of number of prompt neutrons emitted by primary light fragment. But this approximation was based on consideration of average values of n p for the all region of light fragment nuclei. Analysis of data in Fig. 4 shows that it is not so at least for some particular nuclei with mass numbers below 90. The average number for prompt neutron emitted by primary fragment with mass A can be presented as ν ( AE, ) = ν ( Z, AE, ) PY( Z, AE, ). Because the primary yields of many light fragments, contributing p n p n n Z in formation of precursor nuclei are rather low, it means that constancy of n p (A,E n ) with energy does not obligatory means the constancy of n p (Z,A,E n ) To demonstrate the influence of possible change of number of prompt neutrons with energy for light fragment on total delayed neutron yield we have made the calculations with the following values of da (=dn) for light fragment: 0.0 below 4 MeV, +0.3 for 5.5 MeV and +0.6 for 7.0 MeV. The total DNY is shown in this case in Fig. 10 for 235 U by dotted line. Thin solid line shows the DNY energy dependence, if we account also the contribution of second chance fission with relative contribution of different chances and n p =1.29 for second chance nuclei (all taken from 235 U file of ENDF/B-VI library). General tendency of the decreasing of DNY with energy is clearly connected with the shift of isotopic mass distributions due to the growth of number of emitted prompt neutrons with energy. 4. CONCLUSION Chosen factorization scheme connects the energy dependence of DNY with variation of parameters determining the charge-mass distribution of fission fragments in dependence from energy of neutron inducing a fission. It is still far from parametrization which can be based on introducing of approximations in theoretical model approaches. The selected factors are considered as main factors with help of which it was possible to describe the experimentally observed energy dependence of independent, cumulative yields of fragments and DNY. The origin of sharp decrease of the DNY with energy below the threshold of second chance fission is clearly shown. The difference in energy dependence of DNY for neutron induced fission of odd and even Z nuclei is also demonstrated. More accurate quantitative description of the DNY energy dependence requires the knowledge of variation of number of prompt neutrons with energy emitted by primary fission fragments with fixed Z and A and new experimental data on charge-mass distributions of fission fragment primary yields with energy. As an alternative to this, the new experimental data on charge mass-distributions of independent yields of fission fragments with energy may improve the accuracy of microscopic parametrization of DNY energy dependence. Authors are grateful to Dr. A. Ignatyuk for support and help in realization of this work. The work was done in the frame work of ISTC contract N 304. References 1. Evans E.A., Thorpe M.M., Krick M.S.//Nucl. Sci. Eng., 50 (1973) p Lendel A. et al.//atomnaja Energija, 61 (1986) p. 215 (in Russian). 3. D'Angelo A., Filip A.//Nucl. Sci. Eng., 114 (1993) p England T.R. et al. In: Proceedings of the Specialist's Meeting on Delayed Neutron Properties, University of Birmingham Report (1987) p Wahl A.C.//Atomic Data and Nucl. Data Tables 39 (1988) p Ihara H.(ed.), Report JAERI-M (1989)

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