Pre-neutron-emission mass distributions for reaction 238 U(n, f) up to 60 MeV
aa r X i v : . [ nu c l - t h ] O c t EPJ manuscript No. (will be inserted by the editor)
Pre-neutron-emission mass distributions for reaction
U(n, f )up to 60 MeV
Xiaojun Sun , Chenggang Yu , Ning Wang , and Yongxu Yang College of Physical Science and Technology, Guangxi Normal University, Guilin 541004, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, ChinaReceived: date / Revised version: date
Abstract.
The pre-neutron-emission mass distributions for reaction
U(n, f) up to 60 MeV are system-atically studied with an empirical fission potential model. The energy dependence of the peaks and valleysof the pre-neutron-emission mass distributions is described by an exponential form based on the newlymeasured data. The energy dependence of evaporation neutrons before scission is also considered, whichplays a crucial role for the reasonable description of the mass distributions. The measured data for thepre-neutron-emission mass distributions for reaction
U(n, f) are reasonably well reproduced up to 60MeV. The mass distributions at unmeasured energies are also predicted with this approach.
PACS.
Since the discovery of neutron-induced fission of Uraniumin 1938 [1], the neutron-induced fission is the subject ofboth theoretical and experimental studies. In the past,tremendous efforts have been focused on the low-energyactinide fission because of the particular importance fornuclear energy applications. Nowadays, there is an in-creasing interest in studying neutron-induced fission of ac-tinides at intermediate energies. It is motivated by nucleardata needs for new applications such as accelerator-drivensystem, thorium-based fuel cycle, and the next generationof exotic beam facilities. The pre-neutron-emission massdistribution is one of the most important quantities forneutron-induced fission. Its precise description is of greatimportance for both understanding the fission mechanismand the practical application. In addition,
U is one ofthe most important actinides, and its disposal in spentfuel (
U is up to 95%) is an important feature of theutilization of nuclear power.Although one can qualitatively describe the nuclear fis-sion process as a deformation of a single nucleus, the ex-actly understanding the fission process or quantitativelypredicting the pre-neutron-emission fragment mass distri-butions or product yields are still very elusive for the ex-isting theories and models [2]. An international workinggroup has studied the overall problem and recommendedthe assembly of the required nuclear data (including fis-sion products) at intermediate incident neutron energiesup to 150 MeV [3]. Compared with low-energy fission, the a E-mail: [email protected] modeling of neutron-induced fission at intermediate ener-gies is severely complicated by the fact that fission followspre-equilibrium particle emission and competes with neu-tron evaporation [4].Several important theories and models [2,5,6,7,8,9,10,11,12,13,14,15,16,17] have been developed for under-standing the fission mechanism or quantitatively calcu-lating the fragment mass distributions or fission productyields. These models are mainly focused on the dynami-cal processes. The systematic approaches which consist ofthree to seven Gaussian functions have been developed forquantitatively predicting the fragment mass distributionsor product yields [18,19,20,21].A combination method based on the driving potentialfrom the Skyrme energy-density functional [22,23] and thephenomenological fission potential is proposed in our pre-vious work [24], and the experimental pre-neutron-emissionmass distributions of neutron-induced actinide fission atlow energies have been reasonably well reproduced. Thepresent study is an extension of this combination methodfor reaction
U(n, f) at incident energies up to 60 MeV.This paper is organized as follows. In section 1, the com-bination method and the potential parameters are intro-duced in detail. In section 2, the comparisons of the calcu-lated results and the measured data for reaction
U(n,f) are presented and analyzed. A simple summary is alsogiven in the this section.
Xiaojun Sun et al.: Pre-neutron-emission mass distributions for reaction
U(n, f) up to 60 MeV
10 20 30 40 50 600.00.40.81.21.62.0 (n,3nf)
O. Shcherbakov, 2002 P. W. Lisowski, 1992 P. W. Lisowski, 1991 ENDF/B-VII C r o ss S e c t i on ( ba r n ) U(n,f) E n (MeV) (n,f) (n,nf) (n,2nf) Fig. 1.
Fission cross section of reaction
U(n, f) for incidentneutron energies from threshold energy to 60 MeV. The experi-mental data are obtained from Refs. [26](squares), [27](circles)and [28](triangles), respectively. The solid line denotes the eval-uated results of ENDF/B-VII, and the dash lines label the inci-dent energy regions corresponding to the different multi-chancefission channels such as (n, f), (n, nf), (n, 2nf) and (n, 3nf),respectively.
The sequential products of neutron-induced binary fissionare elaborated on Refs. [24,25]. A combination method forcalculating the pre-neutron-emission mass distributions ofneutron-induced actinide fission at low energies has beenproposed in our previous work [24]. In this model, thepre-neutron-emission mass distributions are described as P ( A ) = C exp[ − U ( A )] . (1)Where C is the normalization constant, and the variable A denotes the mass number of the primary fragment. Thephenomenological fission potential U ( A ) is described bythree harmonic-oscillator functions, i.e., U ( A ) = u ( A − A ) A ≤ a − u ( A − A ) + R a ≤ A ≤ bu ( A − A ) A ≥ b. (2)Where, A and A are the positions of the light and heavyfragment peaks of the pre-neutron-emission mass distri-butions, respectively. A denotes the corresponding posi-tion for symmetric fission. The fission potential parame-ters u , u , u , a, b and R , which are the functions of A , A and A , have been uniquely derived as Eq. (6) - Eq. (9)in our previous paper [24].A particular attention should be payed that these pa-rameters are closely relative to the evaporation neutronsbefore scission at different incident energies. For reaction U(n, f) at low incident energies ( E n ≤ . A and A are obtained from the nucleus-nucleusdriving potential of the fissile nucleus U [22,23]. With P ( A ) U(n,f)
C.M. Zoller et al. 1995 F.Vives et al. 2000 V. Simutkin et al. 2011 This work P ( A ) En (MeV)
Fig. 2.
Peak P ( A ) and valley P ( A ) of the pre-neutron-emission mass distributions for reaction U(n, f) as a func-tion of incident neutron energy. The experimental data arederived from the white neutron beam (circles) [29], monoen-ergetic neutron (triangles) [30] and the quasi-monoenergeticneutron (squares) [4,31]. The solid lines denote the results ofthis work. the incident neutron energy increasing, the excitation en-ergy of the compound nucleus will become higher, and afew neutrons will be evaporated before scission. The num-bers of the evaporation neutron can be derived from thecorresponding multi-chance fission cross sections. There-fore, the fission cross sections of reaction
U(n, f) havebeen investigated as shown in Fig. 1. The scattering dotsdenote the experimental data derived from Refs. [26,27,28], and the solid line does the evaluated results of ENDF/B-VII, which is recommended as the standard cross sections.The dash lines denote the incident energy regions corre-sponding to the different multi-chance fission channels aslabeled (n, f), (n, nf), (n, 2nf) and (n, 3nf), respectively.From Fig. 1, one established that the number ˜ n ( E n ) ofevaporation neutrons before scission can be roughly ex-pressed as follow˜ n ( E n ) = , E th ≤ E n ≤ . , . < E n ≤ . , . < E n ≤ . . < E n ≤
60 MeV . (3)Where, E th is the threshold energy for U(n, f) reac-tion. Eq. (3) is consistent with the result at low incidentenergies as shown in Ref. [24].It is assumed that a compound nucleus A CN after evap-orating neutrons ˜ n ( E n ) separates into a pair of fragmentsin the fission process, so the mass number of the fissilenucleus is A F N = A CN − ˜ n ( E n ) at different incident en-ergy regions. For reaction U(n, f), the fissile nuclei are U, U, U and
U, respectively, at different in-cident energy regions as listed in Eq. (3). Based on thenucleus-nucleus potential with the Skyrme energy-densityfunctional [22,23], the driving potentials of these fissile iaojun Sun et al.: Pre-neutron-emission mass distributions for reaction
U(n, f) up to 60 MeV 3
Table 1.
The positions ( A , A ) for the mass number of the light and heavy fragments mass distributions for reaction U(n,f) at different incident energy regions. E n (MeV) 9-11 16-18 24-26 33 45 60 Ref.Experiment (99, 138) (99, 138) (98, 138) (99, 137) (99, 137) (99, 136) [31]TALYS (99, 139) (99, 138) (99, 138) (98, 137) (98, 137) (98, 136) [17]This work (99, 139) (99, 138) (99, 137) (99, 137) (99, 137) (99, 137) -2 -1 -2 -1
80 120 16010 -3 -2 -1
80 120 160 80 120 160 A (a) P ( A ) ( % ) U(n,f)
En=1.3 MeV (d) (c)(b)
En=1.6 MeV
En=1.7 MeV P ( A ) ( % ) En=2.5 MeV
En=3.0MeV (e) (f)
En=3.5 MeV (i)(h)(g) P ( A ) ( % ) En=4.5 MeV A En=5.0 MeV A En=5.5 MeV
Fig. 3.
Pre-neutron-emission mass distributions at incident energies from 1.3 to 5.5 MeV for reaction
U(n, f). The scatteredsymbols denote the experimental data, which are taken from Ref. [30] (squares, measured by the monoenergetic neutron) andfrom Ref. [29] (circles, measured by the white neutron beam), respectively. Xiaojun Sun et al.: Pre-neutron-emission mass distributions for reaction
U(n, f) up to 60 MeV
60 80 100 120 140 16010 -2 -1
60 80 100 120 140 160 60 80 100 120 140 16010 -2 -1 (c)(b)
33 MeV 33.0-45.0 MeV (d) P ( A ) ( % ) A
45 MeV 45.0-55.0 MeV (e) A
60 MeV 55.0-71.0 MeV (f) A TALYS This work 9-11 MeV 8.5-11.5 MeV P ( A ) ( % ) U(n,f) (a)
Fig. 4.
Pre-neutron-emission mass distributions at incident energies from 10 to 60 MeV for reaction
U(n, f). The scatteredsymbols denote the experimental data, which are taken from Ref. [4,31] (squares, measured by the quasi-monoenergetic neutron)and from Ref. [29] (circles, measured by the white neutron beam), respectively. The dash and solid curves denote the calculatedresults of TALYS code [17] and in this work, respectively. systems are studied considering the deformations of frag-ments. One sees that these driving potentials generallyshow a valley at A ∼
140 for the mass distributions ofheavy fragments, as elaborated Fig. 1 in Ref. [24]. It isshould be noted that the driving potentials are only de-rived from the ground state or low excited energies of thefragments. However, the fissile nuclei still hold highly ex-cited energies after evaporating neutrons at different in-cident energy regions. So the position A of the heavyfragment peaks, as well as A of the light fragment peaksand A of the symmetrical fission, should be modified as A = A g.s. − ˜ n ( E n ) ,A = A F N − A ,A = A F N / . (4)Where A g.s. ≃
140 denotes the lowest position of the driv-ing potential derived from the ground state of the frag-ments. These results of Eqs. (3) and (4) agree exactlywith the positions of the maximal mass distributions ofthe heavy fragments measured by the quasi-monoenergeticneutrons beam from 10 MeV up 60 MeV [4,31] as listed inTable 1. For comparison, the results of the famous TALYScode [17] are also listed in this Table. Based on monoenergetic experimental data [30] andthe quasi-monoenergetic experimental data [4,31], the heights P ( A ) and P ( A ) of the valleys and peaks of the pre-neutron-emission mass distributions have been fitted asthe functions of incident neutron energy. For reaction U(n,f) up to 60 MeV, the energy dependence of P ( A ) and P ( A ) is written as P ( A ) = 3 .
850 + 2 . e − . E n ,P ( A ) = 0 .
044 + 4 . e − . /E n . (5)The results of the measurement, including from the whiteneutron beam [29], monoenergetic neutron [30] and thequasi-monoenergetic neutron [4,31], and the calculationare shown in Fig. 2. So the parameter R in Eq. (2) can bederived easily through Eq. (6) as shown R = ln P ( A ) P ( A ) . (6)Furthermore, Eq. (5) approximatively equals the resultsof Ref. [24] at low incident energy ( E n ≤ . P ( A ) and P ( A ) exponentiallychange with the incident energies in general, which couldprovide some useful information at unmeasured energies. iaojun Sun et al.: Pre-neutron-emission mass distributions for reaction U(n, f) up to 60 MeV 5
80 100 120 140 16010100 . . . . . . E n ( M e V ) A U(n,f)
Fig. 5. (Color online) The calculated pre-neutron-emissionmass distributions (%) at incident energies from threshold en-ergy up to 100 MeV for reaction
U(n, f) involved the frag-ment mass number A and the incident energy E n . In this work, the evaporation neutrons ˜ n ( E n ) at differentincident energy regions are derived from the fission crosssections in multi-chance fission channels as shown in Fig.1. In terms of the evaporation neutrons ˜ n ( E n ), the posi-tions A of the heavy fragment peaks of the pre-neutron-emission mass distributions are determined. So the posi-tions A of the light fragment peaks can be also obtainedeasily. Combined the heights P ( A ) and P ( A ) of the val-leys and peaks of the pre-neutron-emission mass distribu-tions as shown in Fig. 2, the parameter R is also obtainedin terms of Eq. (6). So the pre-neutron-emission mass dis-tributions can be calculated using Eq. (1) and (2) up to60 MeV, as shown in Fig. 3 and 4. Fig. 3 shows the pre-neutron-emission mass distributions of reaction U(n, f)at low incident energies from 1.3 MeV to 5.5 MeV, andone can see that this results agree with the previous resultsof Ref. [24]. Fig. 4 shows the calculated results up to 60MeV. In this figure, the scattered symbols denote the ex-perimental data, which are taken from Ref. [4,31] (squares,measured by the quasi-monoenergetic neutron) and fromRef. [29] (circles, measured by the white neutron beam),respectively. The solid curves denote the calculated resultsin this work. The dash curves denote the results calculatedby TALYS code [17]. One can see that the experimentaldata of reaction
U(n, f) can be reproduced well at dif-ferent incident neutron energies from threshold energy upto 60 MeV. It indicates that the method combined thedriving potential with phenomenological fission potentialis reasonable to describe the pre-neutron-emission massdistributions of reaction
U(n, f) up to 60 MeV.Fig. 5 gives the contours of the predicted pre-neutron-emission mass distributions of reaction
U(n, f) fromthreshold energy to 100 MeV. One can see that severaldistinct characters of the pre-neutron-emission mass dis- tributions can be reasonably reproduced: 1) the doublebump shape; 2) the increase of the valley heights, as wellas the decrease of the peak heights, with the incident en-ergy increasing; 3) the position A of the heavy fragmentpeak locates roughly 140 at low energies, and gradually de-creases because of the evaporation neutrons before scissionat E n > A of the lightfragment peak always locates roughly 99 from thresholdenergy up to 100 MeV. This implies that the combinationmethod in this paper can provide some additionally usefulinformation for the intermediate energies neutron inducedactinides fission. Acknowledgements
We thank our colleagues L. Ou and M. Liu for somevaluable suggestions. This work was supported by GuangxiUniversity Science and Technology Research Projects (GrantNo. 2013ZD007), Guangxi Natural Science Foundation (GrantNo. 2012GXNSFAA053008), the Th-based Molten SaltReactor Power System of the Strategic Pioneer Scienceand Technology Projects from the Chinese Academy ofSciences, and National Natural Science Foundation of China(Grants No. 11265004).
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Fragment Mass Distributions in Neutron-Induced Fission of
Th and