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ARTICLE Monte Carlo Simulation of Prompt Neutron Evaporation by Fragments in Low Energy Nuclear Fission 1,* 23 Djelloul BENZAID, Mohamed DJEBARAand Abdeslam SEGHOUR, 1 Centre Universitaire de Khemis Miliana, Route de Theniat El Had BP 44225, Ain Defla, Algeria 2 Université des Sciences et de la Technologie Houari Boumediene, Alger, Algeria 3 Centre de Recherche Nucléaire d'Alger, Algeria In this work, we performed a Monte Carlo simulation of the evaporation of prompt neutron in the fission at low 249 ergy. We are interested in the rhave a detailed experimental study of isotonic distribu en eaction(,) we Cf nf th tions at different kinetic energies. The statistical Fong model is used to calculate the intrinsic excitation energy shared between the two fragments of fission. The simulation reproduces correctly mass and isotonic distributions. However, only the peak located at N=59 of the local evenodd effect has been found. KEYWORDS: MonteCarlo, Neutron, evenodd effect, nuclear fission a I. Introductionvestigating the scission point. The charge evenodd effect is defined as: In this work we have simulated the process of evaporation Y−Y of prompt neutrons using the MonteCarlo method, in ordere oδ= ×100(%) p(1) Y+Y to provide possible explanations to some experimental factse o by simulating the phase between the excited state of the representsr where:YandYe espectively,the yields of all compound nucleus and the end of the evaporation of neu o charges in the odd and even charge distribution of fragments. trons by the fragments, a phase that is practically impossible We also define similarly the neutron pairing effectδ. to detect experimentally. We go back to the primary massn distributions, i.e., before evaporation of neutrons, from the Unlike the charge distributions, the pairing effect of neu measured charge distributions of the reaction, representing trons represents the final isotonic distribution since it is 1) the fragments at the scission phase. measured after evaporation of neutrons. The potential energy of deformationPis calculated in the 2) framework of the liquid drop model,and the sharing of III. Simulationof Neutron Evaporation Process 3) intrinsic excitation energy is done using the Fong mode, Using MonteCarlo Method assuming that the two fragments are at the same temperature, The process of evaporation of prompt neutrons in fission while minimizingPrespect to deformation parameters with 249 at low energy of( )simulated by the is α andα. In our simulation the evaporation of neutronsCf nth,f 31 32 is supposed certain once the total excitation energy of fragMonteCarlo method. This method is essentially based on ment isgreater than the sum of the evaporated neutronthe random drawing of quantities, such as the number of E ex initial mass, evaporated neutron energy and the excitation separation energyBits kinetic energy andε whichis nn energy of fission fragments, following distributions assumed to follow a maxwellian distribution. preselected on the basis of experimental results and if ne In this work we discuss the effect of evaporation of neu cessary assumptions. Our simulated results are compared 1) trons on different observables, in particular isotonic initial with those obtained by Djebaraet al. distribution. We also try to see to what extent the observed The path of fission starts from by absorption of the target structures in the local evenodd effect can be or not related to nucleus of a thermal neutron and ends with the fragmenta the process of evaporation. tion of the formed compound nucleus into two lighter fragments in excited states. II. EvenOdd Effect Our simulation program is based on the following key elements: The charge distribution represents the charge measured at 1. Determination of yields of primary masses from expe the scission point since there is no proton evaporation. The rimental charge yields by drawing random mass of the measurement of this observable is an effective means of in fragment following a gaussian distribution. 2.Potential energy is calculated from the liquid drop *Corresponding author, Email:benzaidhd@yahoo.com © Atomic Energy Society of Japan
6.0 : Contribution of each charge 5.5 : Primary simulated 5.0 4.5 44 42 4.0 3.5 46 3.0 2.540 2.0 38 1.5 Z=36 Z=48 1.0 0.5 0.0 80 90100 110 120 130 Mass number
Fig. 1 Experimentalisotopic distribution
16 14 12 10 8 6 4 2 0 32 34 36 38 40 42 44 46 48 50 Atomic number Fig. 2 Independentand total simulated mass distributions
model minimizing it with respect to deformation para2. Potential Ener metersα andαmotion of a. Theoint on the nuclear surface can be de 31 32 3. Totalexcitation energy is shared between the twoscribed by a series expansion of legendary polynomials: fragments using the model of Fong. The energy re R(θ)=R[1+αP(cosθ)+αP(cosθ)+] 0 22 33leasedQis calculated from the mass table of Audi and 4) Wapstra. Assuminthat the deformation is most likeloctu ole 4. Simulationof evaporation according the model ofdeformation, we can content ourselves with the termP in 3 3) Fong. the reviousdevelo ment.In this casethe mutual Coulomb ener oftwo framents isiven as a function of the para IV. Definition of Different Important Quantities meters of deformation, by the expression: 1. Primary Mass Yields 2 Z Ze L H C(α,α)= The primary mass distributions are constructed from the31 32 R(1+0.9314α)+R(1+0.9314α)(6) 0L3L0H3H measured charge distributions (Fig. 1) by drawing random mass of light fragment following a gaussian distribution : representthe radius of the light and whereR0L andR 2 0H Y(Z)(A'−A)Y(A'Z)=exp− heavy fragments respectively. These are given by: (2) 2 σ π2σ( ) A2AZ 1 / 3 R0=r0A(7) where:σrepresent, respectively, experimental Y(Z) andA withr=1.5Fermi. charge distribution and the standard deviation of the distri0 bution and is given by:As for the deformation eneries, if we limit ourselves to terms ,they can be calculated from the expression: P σ( )= +AZ0.04Z0.20(3) 3 2 2 D(α)=0.7143αE(0)−0.2041αE(0) A isthe mean of mass number before evaporation of neui3i3i Si3i Ci(8) a trons. It is given by UCDhypothesis as: withi≡( )an L,H, and whereESi0dECi(0) represent,re AFA=(Z−0.5)the surface enerand Coulomb eners ectivelof the (4) Z fra mentand are given according to the model of the liquid F drop model by: d are,respectively, the mass and charge of the AF anZF 2 / 3 nucleus undergoing fission.( )ESi0=0.014A amu(9) The primary yields are obtained simply by adding all iso2 Z i = E(0)0.000627amu(10) topic contributions:Ci1/ 2 A i 2 Y(Z)(A'−A)Y(A')=Y(A'Z)=exp− The potential energy is, then, given by: ∑i∑2(5) i(Z) iσA2π2σA P(α,α)=C(α,α)+D(α)+D(α)31 323L3H3L3H(11) Figure 2simulated total and independent mass represents yields.of Excitation Energy between the Heavy and3. Sharin Li htFra ments LetGof the comthe total intrinsic excitation ener be ound nucleus formed bthe thermal incident neutron and a UCDhypothesis (Unchanged Charge Density) supposes that the the fissile nucleus, it is defined as the difference between the ratio (Z/A) remains unchanged for both of nuclei undergoing fission energy releaseQthe potential energy of the compound and and the two fragments of fission before evaporation.
6 100000: Experiment : Primary simulated 5 : Final simulated 80000 4 60000 3 40000 2 20000 1 0 0 0 2 4 6 8 80 90100 110 120 130 Neutron energy [MeV] Mass number Fig. 3neutron energy distribution evaporatedFig. 4yield Mass nucleus: amaxwelienne. The neutrons can be evaporated if the fol lowing condition is satisfied: = −G QP(12) E>B+ε exL nn(17) The internal excitation enercorres ondsmost likelto the minimum of potential ener. Theroblem is therefore whereε andB represent,respectively, the kinetic ener nn to m(α) inimizePα31,32 withrespect to the parameters of gy of evaporated neutron and the separation energy. deformationα andα. 31 32our siies obtained bThe distribution of neutron ener Substitutin Es. 6and 8 bE . 11one can obtainmulationFi .3 canbe represented by the following the otentialener exressed as a function of thearame maxwelienneformula: tersα andα. On the other hand, it is assumed that the 31 32 ε 5n N( )−= × 8) εn2.63 10εnexp (1 confi urationat the scissionoint is that which corresonds T to the minimumotential energy and thus to the maximum of the excitation en: ergy,EexLwithT≈1 MeV.Note that the most probable value of energy of neutrons is ∂P(α,α)∂P(α,α) andaround 1MeV and is identical to the experimental value 31 3231 32 =0=0 (13) ∂α∂α 31 32showing a satisfactory drawing of these quantities. EnergyGis the sum of the energies of the two fragments: V. Results and Discussions G=G+G(14) L H 1. Final Mass Distribution such thatand theintrinsic excitation energy GLGHof The arefinal mass distribution inte rated over the kinetic energy is obtained after evaporation of neutrons.Figure 4 light and heavfra ments,res ectivel . represents the result of our simulation. Yields are normalized The two framents are formed in contact; their temera to 100%. tures should be the sameTAccording to the1 MeV. ~ 5) We have introduced therimar massdistribution for statistical model of the nucleus,we have: com arison. Note that the distribution after eva oration of G A neutronsis shifted to lihter masses. This shift is more im L L = (15) G A H Hortion of the hiortant for theh masses than low ones. A fact which is due to the increased number of neutrons eva We finally obtain the total excitation energyof the E orated with the mass numberA. This imlies both that the exL light fragment by adding the intrinsic excitation energyGL width of the final distribution is smaller than the initial dis nergy andfinally: and the deformation eDLmaximum final distributionsecondl thetribution and increases sli htlfrom the initial distribution. The surface E=G+DexL L L(16) distributionis always maintained. This energy will allow to simulate the evaporation of neu 2. Isotonic Distribution trons from light fragments, and thus determines the mass Isotonic distributions inte rated over kinetic eneris distribution and the final isotonic distributions and other obtained after addinall inde endent yields of fragments important quantities. having the same number of neutronsN: 4. Neutron EmissionY(N)=Y(A,N) ∑(19) A The distribution of kinetic energies of evaporated neu trons is assumed to follow a maxwelienne distribution with We have shown in5Fi .simulated and measured neutron nuclear temperatureThave also introducedfor com arisonthe ofaround 1 MeV. The kinetic energydistributions we of a neutron evaporated is drawn, then, randomly followingprimary distribution. Note that the experimental distribution
12 : Experiment 10: Primary simulated : Final simulated 8 6 4 2 0 50 55 60 65 70 75 neutron number Fig. 5 Isotonicyield
is well reproduced by our simulation. We also note the great similarity between the experimen tal curve and simulated one that are very structured, fragments with an even number (N) of neutrons are more advantaged than those with neighbors odd number of neu trons (N± 1). The favored production of even isotones compared to odd ones is taken into account by the average evenodd effect given by: Y−Y e o δ= ×100(%) n(20) Y+Y e o where and representthe sum of all even and odd YY eo isotones ieldin the isotonic distribution of framents. The neutron evenoddcalculated accordinto E .20 is e ual to11% avalue to be compared to the experimental value (9.5 ± 0.7)% . 3. Averae Number of Evaorated Neutrons Fi ure6resents a com rearison between the evolution of the number of neutrons evaorated bfission framents to that obtained bour simulation. The a reement is satisfac tor ;the eneralsha e of the ex erimental s ectrum is re roducedexce tthat the simulated values are sli htl lower than the ex erimental values. The disagreement is more important around the massA= 120. 4. Local EvenOdd EffectFor more detailed informationwe can studthe so called 6) local evenodd effect usinTrac method. Thismethod consist of estimating the local evenodd effect by the fol lowing formula: 3N+1 dN+ =exp{(−1)[(L−L)−3(L−L)]} 3 30 21(21) 2 aret L,L,L2andL3he natural logarithms of isotonic 01 ieldsNN +1N +2NOn this intervalthe evenodd +3. effect can be estimated bthe reviousformula. We re resent in7Fi .simulated and the measured the local evenodd effect. The exerimental eaklocated atN~ 60 is re roduced in an acce table manner. A fact which su eststhat the peak may be caused to evaporation of neu trons.
4.0 : Experiment 3.5 : Simulation 3.0 2.5 2.0 1.5 1.0 0.5 0.0 80 90100 110 120 130 mass number Fig. 6 Averageevaporated neutron number 20 : Experiment 18 : Simulation 16 14 12 10 8 6 4 54 56 58 60 62 64 66 68 70 72 Neutron number Fig. 7evaporated neutron number Average
VI. Conclusion In our work we have simulated the process of evaporation of neutrons by light fragments of fission. We used the model of Fong to calculate the excitation energy and its partition between the two fragments. The yields of primary masses are constructed assuming that the yield of for each partial charge Z follow a Gaussian distribution. Mass and neutron yields are reproduced in a very satisfactory way. Neutron evenodd simulated is about 11%. The peak located at N ~ 60 in the local evenodd effect is reproduced. Thus, we con clude that it is due to evaporation of neutrons. The second peak is not reproduced. It can not be combined with a simple neutron evaporation. References 1)M. Djebara,Mass, nuclear charge and kinetic energy distribu 249 tions of fragments of fission ofand Cf(n,f) 98th 229 , Doctoratedissertation, USTHB (1994), [In Th(n,f) 98th French]. 2)N. Bohr, J. A. Wheeler,Phys. Rev.,56, 426450 (1939). 3)P. Fong,Phys. Rev.,102, 434448 (1956). 4)G. Audi ,A. H. Wapstra,Nucl. Phys.,A595, 409480 (1995). 5)R. R. Roy, B. P. Nigam,Nuclear Physics, theory and experi ment, John Wiley and sons (1993). 6)B. L. Tracy, J. Chaumont, R. Klapisch, J. M. Nitschke, A. M. Poskanzer, E. Roeckl, C. Thibault,Phys. Rev.,C5, 222234 (1972).
