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Correlated Neutron Emission in Fission

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Correlated Neutron Emission in Fission Correlated neutron emission in fission S. Lemaire , P. Talou , T. Kawano , D. G. Madland and M. B. Chadwick ¡ Nuclear Physics group, Los Alamos National Laboratory, Los Alamos, NM, 87545 Abstract. We have implemented a Monte-Carlo simulation of the fission fragments statistical decay by sequential neutron emission. Within this approach, we calculate both the center-of-mass and laboratory prompt neutron energy spectra, the ¢ prompt neutron multiplicity distribution P ν £ , and the average total number of emitted neutrons as a function of the mass of ¢ the fission fragment ν¯ A £ . Two assumptions for partitioning the total available excitation energy among the light and heavy fragments are considered. Preliminary results are reported for the neutron-induced fission of 235U (at 0.53 MeV neutron energy) and for the spontaneous fission of 252Cf. INTRODUCTION Methodology In this work, we extend the Los Alamos model [1] by A Monte Carlo approach allows to follow in detail any implementing a Monte-Carlo simulation of the statistical reaction chain and to record the result in a history-type decay (Weisskopf-Ewing) of the fission fragments (FF) file, which basically mimics the results of an experiment. by sequential neutron emission. This approach leads to a We first sample the FF mass and charge distributions, much more detailed picture of the decay process and var- and pick a pair of light and heavy nuclei that will then de- ious physical quantities can then be assessed: the center- cay by emitting zero, one or several neutrons. This decay of-mass and laboratory prompt neutron energy spectrum sequence is governed by neutrons emission probabilities ¤ ¤ ¥ N en ¥ , the prompt neutron multiplicity distribution P n , at different temperatures of the compound nucleus and the average number of emitted neutrons as a function of by the energies of the emitted neutrons. ¤ ¦ the FF n¯ A Z ¥ , and all possible neutron-neutron correla- The FF mass and charge distributions is given by ¤ ¦ ¤ ¤ ¤ © ¥ ¥ ¥ tions. Y A Z ¥¨§ Y A exp P Z , where Y A exp represents The layout of this paper is as follows: first, our theo- an experimental pre-neutron FF mass distribution. The ¤ retical and modeling approach are introduced. Numeri- charge distribution P Z ¥ is assumed Gaussian in shape. cal results on both the spontaneous fission of 252Cf and Of course, the particular decay path followed by this 235 neutron-induced (en=0.53 MeV) fission of U are then pair of nuclei depends on the available excitation en- presented. Finally, a discussion of these preliminary re- ergies, which can be deduced in the following manner. ¤ ¦ sults and open questions conclude the paper. The total excitation energy available for the pair A Z ¥ l ¤ ¦ (light), A Z ¥ h (heavy) reads ¤ ¦ ¦ ¦ ¤ ¦ ¦ ¦ ¥ THEORETICAL APPROACH ET Al Ah Zl Zh ¥ § Er Al Ah Zl Zh ¤ ¦ ¤ ¦ ¦ ¥ Bn Ac Zc ¥ en TKE Al Ah (1) Unlike in the Los Alamos model where many quantities ¤ ¦ ¦ ¦ are lumped together, our approach tries to follow in detail where Er Al Ah Zl Zh ¥ is the energy release in the fis- the statistical decay of the FF by sequential emission sion process, which is given, in the case of binary fission, of individual neutrons. Many quantities and parameters, by the difference between the compound nucleus and the ¤ ¦ more or less known, enter as input in the calculation. We FF masses. Bn Ac Zc ¥ and en are the separation and ki- will now briefly describe the methodology used in the netic energies of the neutron inducing fission (in the case ¤ ¦ present work, and then go into some detail over the list of spontaneous fission, both Bn Ac Zc ¥ and en terms in ¤ ¦ of input quantities that enter in our calculations. Eq. (1) disappear). T KE Al Ah ¥ is the total FF kinetic energy. In fact, TKE is not a single value but rather a distribution, assumed to be Gaussian, whose mean value and width are taken from experiment. One of the long-standing questions about the nuclear fission process is how does the available total excitation energy get partitioned among the light and heavy frag- frame of a FF. So, sampling over the angle of emission ¦ ! ments. In the present study, we have considered two hy- of the neutron qn 0 p for each nucleus (A, Z), we potheses for partitioning this energy: infer the neutron energy in the laboratory frame, taking into account the recoil energy of the residual nucleus. • Partitioning so that both light and heavy fragments share the same temperature (hypothesis identical to the one made in the Los Alamos model [1]) at the instant of scission. From this condition, it follows Input Parameters that the initial excitation energy of a given FF is: The fission mass yields have been measured exten- 1 sively and precisely for several nuclei and energies. In ¦ § E E (2) l h T ah l the present calculation, we sample over the pre-neutron 1 al h ¤ fragments yields Y A ¥ , i.e., before neutron evaporation, where l and h refer to the light and heavy system. as reconstructed from the experimentally measured fis- ¤ sion products mass distribution. In particular, we use the • Partitioning using the experimental n¯ A ¥ to infer the 252 initial excitation of each fragment. This condition data by Hambsch [3] in the case of Cf(sf), and the data writes as follow: by Schmitt [4] in the case of the neutron-induced fission (at 0.53 MeV) on 235U. ¤ ¤ ¦ ¥ n¯ Al h e l h 255 fragments were used to represent the Y A Z ¥ for ¦ § El h ET (3) 235 ¤ the neutron induced n(0.53 MeV) U reaction. In par- ∑ n¯ Ai ¥ e i i l h ticular, we considered 85 equispaced fragment masses " (between 76 " A 160) with 3 isobars per fragment where e l h is equal to the average energy removed mass, around the most probable charge Zp. In the case per emitted neutron (e § 1 265 and 1 511 MeV for 252 the n (0.53 MeV) + 235U and 252Cf(sf) respectively). of spontaneous fission of Cf, we used 315 FF between " It is the sum of the average center-of-mass energy of 74 " A 178 among which 105 fragment masses. the emitted neutrons and of the average FF neutron Nuclear masses are used to calculate the energy re- separation energy. lease for a given pair of FF. It is a function of both mass and charge number of complementary fragments. The Within the Fermi-gas model, the initial FF excitation data tables by Audi, Wapstra, Thibault [5] were used in energy E is simply related to the nuclear temperature l h the present calculation. Tl h. The probability for the FF to emit a neutron at a We use in our calculation the level density parameter given kinetic energy is obtained by sampling over the to be: Weisskopf spectrum at this particular temperature [2] : ¤ ¦ ¥ %& dW A Z gU # ¤ ¦ ¦ a A Z U ¥§ a 1 1 e (7) e n $ en T U A Z ¤ ¦ ¦ ¦ ¦ f A Z en T ¥§ 2 e (4) ¤ ¦ T § ¥ § A Z where U E D A Z , g 0 05, a is the asymptotic level density parameter [6]. The pairing D and shell where TA Z is the nuclear temperature of the residual correction dW energies for the FF were taken from the nucleus given by nuclear mass formula of Koura et al.[7]. The level density ' parameters a approximate to A 7 25. ¤ ¦ ¤ ¦ ¥ E A Z ¥ Bn A Z The total kinetic energy is used to assess the total FF ¦ § T (5) A Z excitation energy distribution. It is assumed to be approx- aA 1 Z imately Gaussian in shape with an average value and with aA Z the level density parameter of the nucleus. width taken from the experiment (Ref. [3] for sponta- 252 The emission of a neutron of energy en from the FF neous fission of Cf and Ref. [4] for the neutron in- 235 at the excitation energy E produces a residual nucleus duced n(0.53 MeV) U reaction). with the excitation energy For sake of simplicity, we have assumed no mass, charge or energy dependence of the cross section for ¤ ¤ ¦ ¦ ¤ ¦ ¥§ ¥ ¥ E A 1 Z E A Z en Bn A Z (6) the inverse process of compound nucleus formation. This approximation will be reviewed later on. The sequential neutron emission ends when the exci- We have used the average number of emitted neutrons tation energy of the residual nucleus is less than the sum ¤ n¯ A ¥ as a way of partitionning the total excitation en- of its neutron separation energy and pairing energy. ergy distribution between the light and heavy fragment. The transformation of the center-of-mass spectrum For the spontaneous fission of 252Cf we used data from to the laboratory spectrum is done by assuming that 235 Refs. [8, 9]. For the neutron induced n(0.53 MeV) U neutrons are emitted isotropically in the center-of-mass reaction, we used data from Ref. [10, 11]. 1 0.4 Diven et al. (1956) Calculation with Eq. (2) 0.1 Calculation with Eq. (3) 0.3 0.01 n) 0.2 P( 0.001 Johansson et al. (1977) 0.1 0.0001 Calculation with Eq. (2) Energy spectrum (1/MeV) Calculation with Eq. (3) 1e-05 0 0 5 10 15 0 1 2 3 4 5 6 7 Laboratory neutron energy (MeV) Neutron multiplicity FIGURE 1. Neutron energy spectrum for n(0.53 FIGURE 2. Neutron multiplicity distribution for n(0.53 235 ( 235 + MeV) U reaction. The thick line is our Monte-Carlo MeV) ( U reaction . Open square symbols are from our calculation assuming partitioning of FF total excitation ¢ Monte-Carlo calculation assuming partitioning of FF total ex- ¢ £ , energy as a function of ν¯ A and the thin line is the result citation energy as a function of ν¯ A £ , triangles are the re- obtained under the assumption of an equal temperature sult obtained under the assumption of an equal temperature of of complementary FF.
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