Energy Dependent Calculations of Fission Product, Prompt, and Delayed Neutron Yields for Neutron Induced Fission on ^{235}U, ^{238}U, and ^{239}Pu
AARTICLE TEMPLATE
Energy Dependent Calculations of Fission Product, Prompt, andDelayed Neutron Yields for Neutron Induced Fission on U, U,and Pu S. Okumura a , T. Kawano b , A.E. Lovell b , and T. Yoshida c a NAPC–Nuclear Data Section, International Atomic Energy Agency, Vienna A-1400, Austria; b Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA; c Laboratory for Advanced Nuclear Energy, Tokyo Institute of Technology, Tokyo 152-8550,Japan
ARTICLE HISTORY
Compiled February 2, 2021
ABSTRACT
We perform energy dependent calculations of independent and cumulative fissionproduct yields for U, U, and
Pu in the first chance fission region. Startingwith the primary fission fragment distributions taken from available experimentaldata and analytical functions based on assumptions for the excitation energy andspin-parity distributions, the Hauser-Feshbach statistical decay treatment for fissionfragment de-excitation is applied to more than 1,000 fission fragments for the in-cident neutron energies up to 5 MeV. The calculated independent fission productyields are then used as an input of β -decay to produce the cumulative yield, andsummation calculations are performed. Model parameters in these procedures areadjusted by applying the Bayesian technique at the thermal energy for U and
Pu and in the fast energy range for
U. The calculated fission observable quan-tities, such as the energy-dependent fission yields, and prompt and delayed neutronyields, are compared with available experimental data. We also study a possible im-pact of the second chance fission opening on the energy dependence of the delayedneutron yield by extrapolating the calculation.
CONTACT S. Okumura. Email: [email protected] a r X i v : . [ nu c l - t h ] F e b EYWORDS
Fission Product Yield, Hauser-Feshbach Stastical Decay, Prompt NeutronMultiplicity, β Decay, Delayed Neutron Yield
1. Introduction
Nuclear systems that involve the nuclear fission process often require very high accu-racy of both prompt and delayed neutron multiplicity data, ν p and ν d , albeit modelpredictions for these quantities are not yet at the satisfactory level. For major fission-ing systems, such as the neutron-induced reaction on U, more than 99% neutronsare the prompt fission neutrons, which are emitted from highly excited two fissionfragments formed just after fission. Typically there are more than 1000 fission frag-ments, and 2–3 prompt neutrons per fission are emitted. Whereas a small fraction( ∼ β -decay chain of fissionproducts, and approximately 270 nuclides have been identified as precursors for thedelayed neutron emission [1,2]. Ideally we can calculate ν p and ν d by summing up allthe decaying compound nuclei weighted by the fission yields, which is the so-calledsummation calculation ( e.g. Ref. [3]). This method, however, requires a lot of well-tuned model inputs. This was partly done in our previous study [4] for the promptneutron emission.Since the discovery of delayed neutron by Roberts et al. shortly after the dis-covery of nuclear fission in 1939 [5], despite its tiny fraction, the delayed neutronhas attracted people in various scientific communities, as its quite important role inkeeping the thermal reactors critical, as well as the reactor systems containing highburn-up fuel, and transmutation of minor actinides. The delayed neutron yield hasbeen measured [6–9] and repeatedly evaluated [1,10–12] for various fissioning systemsat several incident neutron energies. Some models for predicting the time-dependentdelayed neutron yield have been proposed [13–15], and these studies pointed out theimportance of fission yield data to perform these model calculations.When an incident neutron energy goes higher, it is natural that ν p also increasesmonotonously, since the formed compound nucleus has larger available total energy.However, in contrast to ν p , ν d shows totally different behavior, depending on how the2elayed neutron precursors are produced. There still exists challenges to understandpeculiar energy-dependence of ν d , i.e., a slight increase in the yield from thermalto 3 MeV and a steep decrease above 4 MeV as seen in , U. To account forthe abrupt changes, the evaluated ν d data in nuclear data libraries, JENDL-4.0 [16]and ENDF/B-VIII [17], include a very crude piecewise linear function to representexperimental data.Such the energy-dependent behavior has not yet been explained theoretically.Alexander et al. [18] first interpreted the energy-dependence in ν d by taking into ac-count the odd-even effect of fission products. Ohsawa et al. [19,20] introduced themultimodal random neck-rupture model [21] and fission mode fluctuations [22] to ex-plain the energy-dependence. Minato [23] proposed a model to reproduce the energy-dependence of ν d based on the fission yield using Katakura’s systematics [24]. Althoughan explicit statistical decay was not performed in Minato’s model —– hence the cal-culated ν p and ν d are independent of one another —– it also supports Alexander’sobservation: the odd-even effect in the the charge distribution is important. Recentlythe odd-even effect was explained by applying the microscopic number projectionmethod [25].By extending the Hauser-Feshbach Fission Fragment Decay (HF D) model [4]to the β -decay process, consistency among the independent and cumulative fissionyields Y I ( Z, A ) and Y C ( Z, A ), and neutron multiplicities ν p and ν d , is automat-ically guaranteed. In this model, we start with the fission fragment distribution Y ( Z, A, E ex , J, Π) characterized by the distributions of mass and charge, excitationenergy, and spin/parity. We perform the Hauser-Feshbach statistical decay for the ex-cited fission fragments to calculate the independent fission yields Y I ( Z, A ) and ν p . Asuccessive β -decay calculation gives the cumulative yields Y C ( Z, A ) and ν d . The modelparameters are adjusted to reproduce experimental data at thermal by applying theBayesian technique, and we extrapolate the calculation to the second chance fissionthreshold. In this paper, we limit ourselves mainly to first-chance fission, because moreuncertain parameters will be involved in the multi-chance fission case. Although westudy the multi-chance fission case elsewhere [26], here, we briefly explore a possibleimpact of the second-chance opening with a particular focus on ν d .3 . Methods β -decay calculations The energy dependence of the independent and cumulative fission product yields(FPY) arises from properties of some model parameters. The primary fission frag-ment distribution Y P ( Z, A ), often approximated by a few Gaussian forms, graduallychanges the shape as the incident neutron energy increases. When the excitation en-ergy of the fissioning compound system increases, the fission path after the secondbarrier spreads along the most probable path, hence the asymmetric terms will havewider width, and the peaks of distributions will be lower to satisfy the normalizationcondition.The energy dependence of total kinetic energy (TKE) is also the one of therelated physical observables of predicting energy-dependent FPY. We often see that theexperimental data of TKE decrease monotonously for some major fissioning nuclidessuch as , U and
Pu [27,28,49], except at very low energies [29,30].The anisothermal parameter R T , which changes the number of prompt neutronsremoved from the fission fragments, often needs to be larger than unity to reproducethe neutron multiplicity distribution as a function of fragment mass number, ν ( A ). Thereason of this is still unclear. It might be natural to assume R T = 1 by the phase-spaceargument, where the total excitation energy would be shared by the two fragmentsaccording to the number of available states. The odd-even effect in the charge distri-bution of Wahl’s Z p model [31,32] might decrease at higher excitation energies, wherea particular nuclear structure effect no longer persists. Since the original Whal system-atics does not consider any energy dependence of the odd-even effect, we incorporatethe energy dependencies of these parameters, yet phenomenological parameterizationis applied. 4 .1.2. Generation of the fission fragment distribution The primary fission fragment distributions are the key ingredient in the prompt neu-tron emission calculation. While this is a complicated multi-dimensional distribution,including energy, spin, parity, etc., we demonstrated that the numerical integrationover all these distributions is feasible by the Hauser-Feshbach Fission Fragment Decay(HF D) model. The model produces various fission observables simultaneously, e.g. ,the prompt neutron multiplicity ν p , independent FPY Y I ( Z, A ), and isomeric ratio(IR) [4]. Since the method and relevant equations are explained elsewhere [4], a briefdescription as well as newly developed components will be given here.The primary fission fragment yield Y P ( Z, A ) is constructed by five (or seven ifneeded) Gaussians fitted to experimental primary fission fragment mass distributionsof neutron induced reaction on U, U, and
Pu. A charge distribution for a givenmass number is generated by the Z p model [32] of Wahl’s systematics [31] implementedin the HF D model.TKE as a function of primary fission fragment mass TKE( A ) is also generatedbased on the experimental data, which yields the average excitation energy of eachfragment. An A -average of TKE( A ) gives a TKE value at a given neutron incidentenergy TKE( E ), and the variance of TKE( A ) gives the excitation energy distribution.By combining with the distributions of excitation energy E ex , spin J , and parity Πdescribed in the previous work [4], an initial configuration of fission fragment com-pound nucleus Y P ( Z, A, E ex , J, Π) is fully characterized. The Hauser-Feshbach theoryis applied to the statistical decay of generated Y P ( Z, A, E ex , J, Π). The experimentaldata sets used in this study are listed in Tables 1, 2, and 3.The functional forms for TKE( A ) and TKE( E ) are given in our former work [4],and the parameters of these functions for U are the same as before. Those for
Puwere taken from the CGMF code [33]. Because there is no primary fission fragmentdata for
U at thermal, the parameters in Y P ( Z, A ) and TKE( A ) are determined inthe 1.1 – 1.3 MeV region. The obtained Y P ( Z, A ) is given later, and TKE( A ) isTKE( A h ) = (348 . − . A h ) (cid:26) − . (cid:18) − ( A h − A m ) . (cid:19)(cid:27) MeV , (1)5nd TKE( E ) is TKE( E ) = 171 . − . E n MeV , (2)where the incident energy E n is in MeV. Table 1.
Experimental data of mass distributions included in the parameter fitting of Y P ( A, E ). Nuclide Energy (MeV) Author & Reference
U 2 . × − Baba et al. [34]2 . × − Hambsch [35]2 . × − Pleasonton et al. [36]2 . × − Simon et al. [37]2 . × − Straede et al. [38]2 . × − Zeynalov et al. [39]2 . × − – 7 D’yachenko et al. [40] U 1.11, 1.25 Goverdovskiy et al. [41]1.2 – 5.8 Vives et al. [42]
Pu 2 . × − – 4.48 Akimov et al. [43]2 . × − Surin et al. [44]2 . × − Wagemans et al. [45]2 . × − Schillebeeckx et al. [46]2 . × − Nishio et al. [47]2 . × − Tsuchiya et al. [48]
Table 2.
Experimental data included in the parameter fitting of TKE( A ). Nuclide Energy (MeV) Author & Reference
U 2 . × − Baba et al. [34]2 . × − Hambsch [35]2 . × − Simon et al. [37]2 . × − Zeynalov et al. [39]2 . × − D’yachenko et al. [40]
U 1.2 Vives et al. [42]
Pu 2 . × − Surin et al. [44]2 . × − Wagemans et al. [45]2 . × − Nishio et al. [47]2 . × − Tsuchiya et al. [48]
The Gaussian terms for Y P ( A ) are parameterized as Y P ( A ) = (cid:88) i =1 F i √ πσ i exp (cid:26) − ( A − A m + ∆ i ) σ i (cid:27) , (3)6 able 3. Experimental data included in the parameter fitting of TKE( E ). Nuclide Energy (MeV) Author & Reference
U 0.18 – 8.83 Meadows and Budtz-Jørgensen [27]2 . × − – 35.5 Duke [29] U 1.5 – 400.0 Z¨oller et al. [49]1.4 – 28.3 Duke et al. [50]
Pu 0.05 – 5.3 Akimov et al. [43]2 . × − – 3.55 Vorobeva et al. [51]0.5 – 50 Meierbachtol et al. [52]where σ i and ∆ i are the Gaussian parameters, the index i runs from the low mass side,and the component of i = 3 is for the symmetric distribution (∆ = 0). A m = A CN / A CN is the mass number of fissioning compoundnucleus, and F i is the fraction of each Gaussian component. The symmetric shape of Y P ( A ) ensures implict relations of F = F , F = F , etc.We assume that the energy sharing between the complementary light and heavyfragments is followed by the anisothermal model [53,54], which is defined by the ratioof effective temperature T L and T H in the light and heavy fission fragments, R T = T L T H = (cid:114) a H U L a L U H , (4)where U is the excitation energy corrected by the pairing energy [55], and a is thelevel density parameter including the shell correction energy.There are several estimates of R T for different fissioning systems. In the case ofthermal neutron induced fission on U, a constant R T reasonably reproduces theexperimental ν ( A ) data [4], and Talou et al. [56,57] showed the cases of Pu(n th ,f),and Cf spontaneous fission. However, it has been reported that better reproductionof experimental data is achieved by mass-dependent R T parameters [58–61]. In thepresent work, we do not explore all possible functional forms of R T . Instead, a simpleenergy-dependence is introduced as R T = R T + E n R T , R T + E n R T ≥ , otherwise , (5)where R T and R T are model parameters. As we expect R T decreases as the incident7nergy, R T < Z p model the even-odd effect in the Z -distribution is given as f = F Z F N Z even N even F Z /F N Z even N odd F N /F Z Z odd N even1 / ( F Z F N ) Z odd N odd , (6)where F Z ≥ F N ≥ Z and/or N are the even number. We expect such even-oddstaggering will be mitigated when a fissioning system has higher excitation energy. Wemodel the reduction in the even-odd effect by F Z = 1 . F W Z − . f Z , (7) F N = 1 . F W N − . f N , (8)where F W Z and F W N are the parameters in Wahl’s systematics, and f Z and f N arethe scaling factor as inputs. These scaling factors are also linear functions of incidentneutron energy, f i = f i + E n f i , i = Z, N . β -decay calculation The HF D model produces the independent fission product yields Y I ( Z, A ), as wellas the meta-stable state production when the nuclear structure data indicate thatthe level half-life is long enough (typically more than 1 ms.) Here we add a meta-state index M to specify the isomers explicitly, Y I ( Z, A, M ) and Y C ( Z, A, M ). Thecumulative yields are calculated in a time-independent manner, hence Y I ( Z, A, M )and Y C ( Z, A, M ) are simply connected by the decay branching ratios [62]. The decaydata included are the half-lives T / , the decay mode ( α -decay, β − -decay, delayedneutron emission, etc. ), and the branching ratios to each decay mode. They are takenfrom ENDF/B-VIII decay data library. We also considered JENDL-4.0 decay datalibrary, however the result is not so different.8hen a decay branch includes a neutron emission mode, this nuclide is identifiedas a β -delayed neutron precursor. The delayed neutron yield from this i -th precursor iscalculated as ν d ( i ) = Y C ( i ) b i N d , where b i is the branching ratio to the neutron-decaymode, and N d is usually one unless multiple neutron emission is allowed. The totaldelayed neutron yield ν d is (cid:80) i ν d ( i ). An optimization procedure of the HF D model parameters is a non-linear multi-dimensional least-squares problem. Albeit such complex problem might be solved bythe modern technology, this will be a hefty computation and beyond our scope. Instead,we perform a relatively small-scale adjustment of the model parameters to reproducesome of the fission product yield data by applying the Bayesian technique with theKALMAN code [63]. The model parameters are first estimated by comparing withthe most sensitive quantities. They are our prior. Then the prior parameters are ad-justed simultaneously by fitting to the experimental data. Although it is always idealto use raw experimental data, we use the evaluated values that should be representa-tive of available experimental data. However, it should be noted that we are not tyingto reproduce the evaluation, but to find a consistent solution among different fissionobservable.The model parameters to be included in the KALMAN calculation are the firstand second Gaussian parameters (fraction F i , width σ i , and mass shift ∆ i for i = 1 and2.) We fix the symmetric Gaussian, because it does not have any sensitivities to theexperimental data included in this study, and its fraction is too small anyway. We alsoinclude the anisothermal R T parameter, the spin factor f J , and the scaling factor inEqs. (7) and (8). The adjustment is performed at the thermal energy (or at relativelylow energy for U), and the energy-dependent parts in these model parameters arefixed. 9he sensitivity matrix C is defined as c ij = ∂d i ∂p j , ≤ i ≤ N , ≤ j ≤ M , (9)where P = ( p , p , . . . ) is the model parameter vector, and D = ( d , d , . . . ) is thedata vector containing the calculated values. The partial derivatives are calculatednumerically. The KALMAN code linearizes the model calculation as D = F ( P ) (cid:39) F ( P ) + C ( P − P ) , (10)where F ( P ) stands for a model calculation with a given parameter P , and P is theprior parameter vector.It is not so easy to impose a constraint 2 F + 2 F + F = 2 on the Gaussianfractions during the adjustment process, e.g. , when the F parameter is perturbed as F + δ , the sum exceeds 2; 2( F + δ ) + 2 F + F = 2 + 2 δ . However, we renormalizethe fractions internally F (cid:48) j = F j − δ δ F j = F j (cid:18) − δ δ (cid:19) (11)to assure the sum to be 2. F (cid:48) j is the actual fraction inside the calculations, and F j isnot necessarily normalized but represents a model input.
3. Results , U and Pu The prior Gaussian parameters, R T , f J , and TKE for U at the thermal energy aretaken from our previous study [4]. When we modify TKE, TKE( A ) is automaticallyshifted to make sure the A -average coincides with the given TKE value. The originalWalh’s Z p model is also employed as the prior parameter, which means f Z = f N = 1.10hey are shown in the second column of Table 4. These parameters are adjusted toreproduce the cumulative fission product yields of Zr, Zr, Mo,
Te,
Ba, and
Nd at thermal, as well as ν p and ν d . Now we have 11 parameters ( M = 11) and 8data ( N = 8.)With the prior parameters, the calculated ν p of 2.38 is slightly lower than theevaluated values of 2.41 (ENDF/B-VIII) and 2.42 (JENDL-4.0), while the prior ν d of0.0195 is 23% larger than the value found in both libraries, 0.0159. The adjustment rec-onciles these discrepancies with the better known values, and the posterior parametersyield ν p = 2 .
415 and ν d = 0 . Y P ( A ) arevery modest, and the posterior parameters equally reproduce the experimental dataof mass distribution, we do not include the comparison plot here. Figure 1 (a) is themass chain yield with the prior and posterior parameters. The ENDF evaluated val-ues are also compared. This figure also shows some mass-chains that contain major β -delayed neutron emitters. The reduction in ν d is, in part, caused by the smallerposterior yields of A = 137 and 94, which include I and Rb. While these masseswere not included in the adjustment, the sensitivity of ν d to these masses implicitlydemands the reduction of these mass-chains.When the prior R T and f J parameters are determined, we compare the neutronmultiplicity distribution P ( ν ) with the experimental data. The posterior parametersmodify the calculated P ( ν ) but not so significantly. The calculated P ( ν ) still agreesfairly well with the data. 11 U C u m u l a t i v e M a ss Y i e l d Mass NumberPriorPosteriorincludeddn emitter Pu C u m u l a t i v e M a ss Y i e l d Mass NumberPriorPosteriorincludeddn emitter U C u m u l a t i v e M a ss Y i e l d Mass NumberPriorPosteriorincludeddn emitter
Figure 1.
Calculated mass chain yields with the prior and posterior model parameters for U, Pu, and
U. They are compared with some selected cumulative fission product yield data by the circles. The squaresare the mass chains that include major delayed neutron emitters. able 4. Prior and posterior model parameters defined in Eqs. (3), (5), (7), and (8), as well as the spin salingfactor f J . These parameters are dimensionless quantities, except TKE is in MeV. pri post uncertainty[%] and correlation [%] F σ − −
56 100 F −
40 36 100 σ −
33 1 28 34 100∆ − f Z − f N − R T − − −
64 3 − − −
51 0 100 f J −
30 16 − − −
23 7 0 0 100TKE 170.5 170.1 0.1 − −
27 6 11 1 − − −
82 100 Pu The Gaussian parameters obtained by fitting to the experimental Y P ( A ) for Pu are∆ = − ∆ = 14 .
09 + 0 . E n , (12)∆ = − ∆ = 20 .
08 + 0 . E n , (13) σ = σ = 3 .
26 + 0 . E n , (14) σ = σ = 6 .
58 + 0 . E n , (15) σ = 10 . , (16)where E n in MeV. The fractions of each Gaussian are given by F = F = 0 . . E n , (17) F = F = 0 . − . E n , (18) F = 0 .
003 + 0 . E n . (19)The adjustment procedure for Pu at thermal includes the same parameters asthose in the
U case. These parameters are fitted to ν p , ν d , and cumulative FPY of Kr, Rb, Kr, Sr,
Xe,
Xe,
Xe,
Xe,
Ba,
Ce,
Pr,
Nd,
Nd,
Nd,
Nd,
Nd, and
Nd. They were chosen from the mass chain evaluation by13 able 5.
Prior and posterior model parameters for
Pu. See Table 4 for parameter descriptions. pri post uncertainty[%] and correlation [%] F σ F − − σ −
20 21 17 20 100∆ − − − −
66 100 f Z − −
21 30 100 f N − − R T − − − − f J −
24 10 −
24 24 10 −
33 14 1 28 100TKE 178.2 179.4 0.1 26 −
21 1 − − −
28 0 − −
92 100England and Rider [64], where relatively small uncertainties are assigned. The priorand posterior model parameters are given in Table 5, and the comparison of massyields are in Fig. 1 (b). Similar to the
U case, the prior parameter set produces ν d = 0 . ν d is achieved by the adjustment, and the posterior set gives 0.00665. U Because fission observable data for
U are only available in the fast energy range andabove, the procedure is slightly different from the
U and
Pu cases. The adjustedGaussian parameters were obtained at 1.1 and 1.25 MeV by Goverdovskiy [41] and1.2 MeV [42] by Vives. The adjusted Gaussian parameters are∆ = − ∆ = 15 . − . E n , (20)∆ = − ∆ = 23 . − . E n , (21) σ = σ = 3 .
31 + 0 . E n , (22) σ = σ = 5 .
43 + 0 . E n , (23) σ = 4 .
50 + 0 . E n . (24)14 able 6. Prior and posterior model parameters for
U at ≈ pri post uncertainty[%] and correlation [%] F σ − −
18 100 F −
10 55 100 σ −
40 8 42 31 100∆ −
11 27 43 30 13 100 f Z −
16 25 20 −
14 48 100 f N − − − R T − − − − f J − − − − − − − − −
30 28 0 − −
18 100The fractions of each Gaussian are given by F = F = 0 . . E n , (25) F = F = 0 . − . E n , (26) F = 0 . . E n , (27)and the covariance matrix is given in Table 6. These parameters are fitted to ν p , ν d ,and cumulative FPY of Zr, I, Xe,
Cs,
Ba,
Ce, Ce,
Pr,
Nd, and
Nd. ν p and ν d Some of the Gaussian parameters are weakly energy-dependent, and often expressedby a linear function of the incident energy as in Eqs. (17) – (19). The energy-dependentterms are obtained by fitting to the experimental Y P ( A ) data, and we do not attemptto tune these parameters. We consider other parameters, R T , f Z , f N , and TKE, to beenergy-dependent, and simple linear functions are assumed as in Eqs.(5), (7) and (8).Since the energy-dependence of TKE is rather well know experimentally, we studythe energy-dependence of the FPYs, ν p , and ν d by assuming a simple form for the15ode inputs for U and
Pu first. We exclude
U for now, as it is a thresholdfissioner. The R T parameter in Eq. (5) is roughly − ( R T (0) − / . − to make R T = 1 at the opening of second chance fission, hence R T = − . − . U and
Pu. Similarly, f Z and f N are estimated to be f Z = − .
296 MeV − and f N = − .
161 MeV − for U, and f Z = − .
430 MeV − and f N = − .
156 MeV − for Pu, which ensures that the even-odd effect disappears at E n = 6 MeV.First we consider four cases; (1) both R T and f Z,N are constant, (2) constant R T and energy-dependent f Z,N , (3) energy-dependent R T and constant f Z,N , and (4)both energy-dependent. By comparing the calculated ν p and ν d with experimentaldata, we found that the energy-dependence of R T modestly impacts on the results,and probably the modeling uncertainty conceals the importance of R T . Whereas wealso noticed that the energy-dependence of f Z,N is crucial for ν d . Hereafter we assume R T is constant, while f Z,N is energy-dependent.When an independent or cumulative FPY is almost energy-independent, dY I,C ( Z, A, E ) dE (cid:39) , (28)it is easier to see the mass region where this condition happens by calculating thederivative of mass yields, dY I,C ( A, E ) dE = (cid:88) Z dY I,C ( Z, A, E ) dE (cid:39) . (29)We approximate the derivative by coarse numerical derivative ( Y I,C ( A, − Y I,C ( A, /
2, which is shown in Fig. 2. The general shape of dY I /dE does notchange too much in the energy range below the second chance fission. This implies thecumulative FPYs vary monotonously with the incident neutron energy.The derivative plot for U indicates FPYs near A = 85, 100, and 135 varyslowly with the energy, while FPYs near A = 90, 104, 129, and 143 should havesteeper energy-dependence. These energy-independent regions, or the pivots, appeardue to complicated interplay among the energy-dependent model parameters. In thecase of Pu, the pivots locate near A = 92, 109, 129, and 142, and the FPYs in the16eak regions ( A = 103 and 133) may show the largest reduction rate.In Fig. 3 we compare some of our calculated Y C ( Z, A, E ) with the experimentaldata of Gooden et al. [65], measurements at LANL in the critical assemblies [66],as well as other published data. From the derivative plot in Fig. 2, we expect Y C of U decreases in the A (cid:39)
90 region, while Y C of Pu increases. For the bothisotopes, the pivot will be seen in A = 95 – 100. The comparisons of Sr, Zr, and Mo clearly show these behavior, and
Ru now shows an opposite tendency as theincident neutron energy.On the heavier mass side, the slope of Y C ( Z, A, E ) changes the sign from positiveto negative around A = 134 for U, with one exception of the A = 137 case that hasthe positive slope. For Pu, the sign change happens twice, near A = 130 and 145.This is shown in Fig. 4; Te,
Cs,
Ba, and
Nd. Our model calculation alsoreproduces other isotopes with the similar quality.Although we didn’t include the
U case in Fig. 2 as FPY at thermal is onlygiven by extrapolation, Figs. 3 and 4 include
U too. ν p and ν d The calculated ν p and ν d for , U and
Pu are compared with experimentaldata in Figs. 5 and 6. The evaluated ν p and ν d in ENDF/B-VIII and JENDL-4.0,which are evaluated by least-squares fitting to the available experimental data, are alsocompared. In general ν p increases as the incident neutron energy goes higher, simplybecause of the energy conservation. However, its slope dν p /dE strongly depends onthe behavior of TKE. Although the mechanism for the incident-energy dependence ofTKE is still unclear, we take the energy dependence of TKE from experimental data,and it enables us to reproduce ν p by our model. Other parameters, the Gaussian shapeand f Z,N , also change the slope of ν p , but they have a much more modest impact onthe calculated result.The energy-dependence of ν d is caused mainly by changing the yields of thedelayed neutron precursors. Interestingly the calculated and experimental ν d ’s revealvery weak energy-dependency for these isotopes. As we noted large fractions of delayedneutron emission are from the mass regions of A = 137 and 94, and according to Fig. 2,17 d Y ( A , E ) / d E [/ M e V ] Mass Number U235Pu239
Figure 2.
Energy-dependence of the cumulative mass yields, ∂Y ( A, E ) /∂E , approximated by ( Y ( A, − Y ( A, / U Sr C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenpresent Zr C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenpresent Mo C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenpresent Ru C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenpresent Pu Sr C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent Zr C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent Mo C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent Ru C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent U Sr C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent Zr C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent Mo C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent Ru C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent
Figure 3.
Energy dependence of cumulative fission product yields of Sr, Zr, Mo, and
Ru for theneutron-induced fission on
U (left),
Pu (middle), and
U (right) of calculated data (solid line) comparedwith with the experimental data of Gooden et al. [65], as well as other published data. U Te C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenpresent Cs C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]LANL C-NRpresent Ba C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenLANL C-NRpresent Nd C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenLANL C-NRpresent Pu Te C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent Cs C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]LANL C-NRExperimentsPresent Ba C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]LANL C-NRExperimentsGoodenPresent Nd C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]LANL C-NRExperimentsGoodenPresent U Te C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]ExperimentsGoodenPresent Cs C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]LANL C-NRExperimentsPresent Ba C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]LANL C-NRExperimentsGoodenPresent Nd C u m u l a t i v e F i ss i on P r odu c t Y i e l d [ % ] Incident Neutron Energy [MeV]LANL C-NRExperimentsGoodenPresent
Figure 4.
Incident neutron energy dependence of cumulative fission product yields of
Te,
Cs,
Ba,and
Nd for the neutron-induced fission on
U (left),
Pu (middle), and
U (right) of calculated data(solid line) compared with with the experimental data of Gooden et al. [65] and measurements at LANL in thecritical assemblies (LANL CN-R) [66], as well as other published data.
20e expect ν d to decrease.As it is not so convenient to survey the delayed neutron precursors individually,we lump the precursors into the well-known six groups according to their half-lives T / ,and calculate the energy-dependence of the six-group yields. The group structure isusually defined by the isotopes included in each group. This is convenient for the longer T / groups, but it is ambiguous for the shorter groups. For the sake of convenience, wedefine the six-group structure as (1) T / >
40 s, (2) 8 < T / ≤
40 s, (3) 3 < T / ≤ < T / ≤ . < T / ≤ T / ≤ . U, the largest contribution is from theGroup 4, which slightly decreases as the incident neutron energy. This is compensatedby the increasing Group 2, resulting in the flat behavior of ν d . The energy variationof each group is more visible for the Pu case. Obviously the energy-dependenceof ν d does not originate from specific fission products, but a consequence of theircompetition.We studied sensitivities of the model parameters to dν d /dE , and found that the f Z and f N terms change the slope. When f Z = f N = 0, or a constant odd-eveneffect, ν d decreases for both U and
Pu cases. We briefly estimated the energy-dependence of the odd-even term so that this effect fades away toward the secondchance fission. Nonetheless, this ansatz was not so unrealistic. Better reproduction ofthe experimental data can be achieved by adjusting the f Z and f N parameters, yetthe currently available data have rather large uncertainties to estimate these parame-ters precisely. The experimental data of ν d for U drop sharply near 5 MeV [8,10], and the evaluateddata often include a curious kink to reproduce this behavior. As we demonstratedthat ν d is weakly energy-dependent up to the second chance fission, the kink couldbe hypothetically the evidence of the second-chance contribution, namely transitionof major fissioning system from U to
U. The full-extension of our FPY modelby including the multi-chance fission is underway [26], and here we extrapolate ourcalculations beyond the second-chance fission threshold. We do not intend to perform21 U A v e r age N u m be r o f P r o m p t N eu t r on s Incident Neutron Energy [MeV]ENDF/B-VIIIJENDL-4Present Pu A v e r age N u m be r o f P r o m p t N eu t r on s Incident Neutron Energy [MeV]ENDF/B-VIIIJENDL-4Present U A v e r age N u m be r o f P r o m p t N eu t r on s Incident Neutron Energy [MeV]ENDF/B-VIIIJENDL-4Present
Figure 5.
Calculated incident neutron energy dependence of ν p for U, Pu, and
U (solid line) com-pared with the evaluated ν p in ENDF/B-VIII (dotted line) and JENDL-4.0 (dot-dashed line) and availableexperimental data. U A v e r age N u m be r o f D e l a y ed N eu t r on s Incident Neutron Energy [MeV]KrickENDF/B-VIIIJENDL-4Present Pu A v e r age N u m be r o f D e l a y ed N eu t r on s Incident Neutron Energy [MeV]KrickENDF/B-VIIIJENDL-4Present U A v e r age N u m be r o f D e l a y ed N eu t r on s Incident Neutron Energy [MeV]KrickENDF/B-VIIIJENDL-4Present
Figure 6.
Calculated incident neutron energy dependence of ν d for U, Pu, and
U (solid line) com-pared with the evaluated ν d in ENDF/B-VIII (dotted line) and JENDL-4.0 (dot-dashed line), and experimentalmeasurements by Krick et.al [8] (filled circle) and the other available experimental data (open circle). U F r a c t i on o f D e l a y ed N eu t r on G r oup Incident Neutron Energy [MeV]Group 1Group 2Group 3Group 4Group 5Group 6 Pu F r a c t i on o f D e l a y ed N eu t r on G r oup Incident Neutron Energy [MeV]Group 1Group 2Group 3Group 4Group 5Group 6 U F r a c t i on o f D e l a y ed N eu t r on G r oup Incident Neutron Energy [MeV]Group 1Group 2Group 3Group 4Group 5Group 6
Figure 7.
Calculated relative contribution to ν d from each of the delayed neutron groups for U, Pu,and
U. Evaluated ν d fractions were taken from JENDL-4.0 and are shown by the symbols (+:Group 1, × :Group 2, ∗ : Group 3, (cid:3) : Group 4, (cid:4) : Group 5, and (cid:13) : Group 6) at thermal and 1 MeV for U, and atthermal and 2 MeV for
Pu, and U.
24 detailed model parameter adjustment as done for the first chance case, but similarparameters obtained by the first-chance calculation were plugged into the second-chance fission to see if we will be able to reproduce the kink. This exercise is done forthe
U case only.The fission probabilities P f ( E ) for the first and second chances are calculatedwith the CoH code [67]. The fission parameters, such as the fission barrier, curvature,and level density, are adjusted to reproduce the evaluated fission cross section of U.We use the same Y P ( A ) for the second chance, but shifted the mid-point by 1/2 massunit to the lower mass side. f J , TKE, and f Z,N for both
U and
U are the same.The calculated ν d is shown in Fig. 8. Albeit the calculated ν d drops at the energythat is about 1.5 MeV higher than the experimental data, the shape is well reproduced.This supports our hypothesis of the transition of fissioning systems from the firstcompound nucleus to the second one. At 8 MeV the probability of second chancefission reaches 80%, and a new set of delayed neutron emitters again forms a newplateau above that energy. The step-function-like behavior of ν d is thus understood.The calculated transition energy, which is basically the second-chance fissionthreshold, is higher than the experimental data, and this is still an open question.Despite the fact that our fission barrier parameters could have some uncertainties, the1.5-MeV change in the fission barriers makes a significant suppression of the fissioncross section above 5 MeV. At this moment we don’t have a simple solution of matchingthe kink point in the experimental data and theoretical calculation.
4. Conclusion
The Hauser-Feshbach Fission Fragment Decay (HF D) model was extended to cal-culate β -delayed quantities such as the cumulative yield calculation and the delayedneutron yield ν d , where consistency of prompt products retained. The model param-eters for U, Pu, and
U — the Gaussian functions to characterize the primaryfission yields, the anisothermal parameter R T , the spin parameter f J , TKE, and theodd-even term of Wahl’s Z p model — were estimated by employing the Bayesian tech-nique with the KALMAN code at the thermal energy for U, Pu and 1.2 MeV for25 A v e r age N u m be r o f D e l a y ed N eu t r on s Incident Neutron Energy [MeV]ENDF/B-VIIIJENDL-4Present
Figure 8.
Energy dependence of ν d for U when the second-chance fission is involved.238
U. The result implies that a stronger odd-even effect is required to reproduce theexperimental ν d , which is also reported by Minato [23].Anchoring the statistical decay calculations to experimental data available at thethermal energy for U, Pu and 1.2 MeV for
U, we extrapolated the HF Dmodel to the second chance fission threshold energy, and demonstrated that the calcu-lated cumulative FPYs fairly reproduced the experimental data, as well as ν p and ν d simultaneously. The flat behavior of ν d along the neutron-incident energy seen in theexperimental data of U and
Pu was attributed to a coincidental compensationof increasing and decreasing delayed neutron precursors.To examine the sudden change in ν d near 5 MeV, we extrapolated our calcula-tions beyond the second-chance fission by assuming the same parameters as the firstchance. Indeed this is a crude assumption, nevertheless we were able to reproducethe step-function-like variation of ν d . This is promising, and our HF D model for theindependent and cumulative FPY should be the most advanced tool for evaluatingthe FPY data, because it produces many fission observable quantities in a consistentmanner. Unfortunately our calculation drops at around 5.5 MeV, despite the kink in26he experimental data is seen near 4 MeV. This discrepancy should be explained byfurther investigation in both the theory and experimental data. Having said that, theHF D model qualitatively explains that the variation seen in ν d is a result of differentprecursors produced by fission at each fission-chance. Acknowledgements
We thank Dr. Minato for valuable discussions on the delayed neutron emission calcu-lation. TK thanks P. Talou, M.B. Chadwick, T. Bredeweg, and M. Gooden of LANLand A. Tonchev of LLNL for encouraging and continuous support of this work. TKand AL performed this work under the auspice of the U.S. Department of Energy byLos Alamos National Laboratory under Contract 89233218CNA000001.
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