PROMPT EMISSION MODELING IN THE FISSION PROCESS

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1 PROMPT EMISSION MODELING IN THE FISSION PROCESS I.VISAN 1,, G.GIUBEGA 1, A. TUDORA 1 1 University of Bucharest, Faculty of Physics, Bucharest-Magurele, POB MG-11, R-769, Romania iuliana.visan@yahoo.com; giubega_georgiana@yahoo.com; anabellatudora@hotmail.com Institute for Nuclear Research, POB 7, 115-Mioveni, AG, Romania, iuliana.visan@nuclear.ro Received August 3, 13 The present paper gives a short description of the deterministic Point-by-Point model of prompt emission in fission. The model can provide almost all quantities characterizing the fission fragments and the prompt neutron and gamma rays emission. During the time, the Point-by-Point model was successfully applied to many spontaneous and neutron induced fissioning systems as 5 Cf(SF), 36- Pu(SF),, Cm(SF), 33,3,35,36,3 U(n,f), 39 Pu(n,f), 37 Np(n,f), Pa(n,f), 3 Th(n,f)), being validated by the excellent description of all existing experimental data. Key words: Point-by-Point model, prompt emission data. 1. INTRODUCTION The high accuracy of the evaluated nuclear data concerning the prompt fission process is very important for nuclear reactors calculations and other applications based on fission. The deterministic Point-by-Point (PbP) model is considered today one of the most powerful approaches for the prompt neutron and gamma-rays emission taking into account the full fragmentation range of the fissioning nucleus. It provides almost all quantities characterizing both the fission fragments (FF) and the prompt emission. The primary results of the model consist in the so-called multi-parametric matrices, meaning different quantities as a function of fragment (Z, A) and as a function of total kinetic energy (TKE). The model is based on the neutron evaporation from fully accelerated fission fragments, but the neutron emission from other sources (such as the scission neutrons and the neutrons evaporated during the fragment acceleration) can be also considered. The sequential neutron emission is taken into account by the fragment residual temperature distribution. The compound nucleus cross-sections of the inverse process of neutron evaporation from fragments are provided by optical model calculations using phenomenological potential parameterizations adequate Rom. Journ. Phys., Vol. 59, Nos. 3, P. 7, Bucharest, 1

2 Prompt emission modeling in the fission process 73 for nuclei appearing as fission fragments. The level density parameters of fragments are calculated in the frame of the generalized super-fluid model. The total excitation energy (TXE) partition between fully accelerated FF is made by modeling at scission or by using different parameterizations (deduced frim the results of modeling at scission). By averaging the multi-parametric matrices of different quantities over the fragment mass, charge and TKE distributions, it is possible to obtain average quantities characterizing both the fragments and the prompt emission as following: average quantities as a function of fragment mass number (e.g. prompt neutron multiplicity ν(a), prompt gamma-ray energy Eγ(A)), as a function of TKE (e.g. < ν >(TKE), < ε >(TKE)) or total average ones (e.g. < ν > tot, spectra, < Eγ >, prompt gamma multiplicity < n γ >) as a function of the incident energy.. FEATURES OF THE POINT-BY-POINT MODEL.1. FRAGMENTATION RANGE The fragmentation range plays a very important role. To calculate the multiparametric matrices of different quantities, generically labeled q(z,a,tke), it is necessary to generate firstly the fragmentation range [1]. This is built by taking into account all mass pairs covering a convenient range (from symmetric fission up to a far asymmetric split), for each mass pair usually two up to four fragments being considered with the charge numbers Z as the nearest integers above and below the most probable charge (taken as unchanged charge distribution (Z UCD ) corrected with a possible charge polarization ( Z)). For each fragment of the fragmentation range all quantities are calculated at TKE values covering a convenient range... BASIC FEATURES OF Q(Z,A,TKE) CALCULATIONS The multi-parametric matrices do not depend on experimental data. To provide q(z,a,tke), the PbP model only needs data taken from reference parameter libraries such as mass excesses and shell corrections from RIPL3 [] and optical model potential parameterizations necessary for the calculation of compound nucleus cross-sections σ c (ε) of the inverse process (also taken from RIPL3 [3]). Such that, the total excitation energy at full acceleration of each pair of fragments (of the fragmentation range) at a given TKE value is: TXE = E r + E n + B n TKE (1) where E r is the energy release in the fission process, E n and B n are the incident neutron energy and the compound nucleus binding energy, respectively and TKE is the total kinetic energy of the respective pair of fragments.

3 7 I. Visan, G. Giubega, A. Tudora 3.3. THE TOTAL EXCITATION ENERGY PARTITION AT FULL ACCELERATION Two methods were proposed and used for the TXE partition between the fully accelerated fission fragments in order to obtain the fission fragment excitation energy at full acceleration needed in PbP model calculations. One of these methods is based on modeling at scission []. This method is consisting in two steps: 1) the calculation of the extra-deformation energy at scission (with deformability provided by liquid drop model with shell corrections taken into account and deformation parameters from Hartree-Fock-Bogoliubov calculations [5] and ) the partition of the available excitation energy at scission (obtained by subtracting the extra-deformation energies from TXE) by assuming thermodynamically equilibrium at scission and fragment level density description by Fermi-Gas (details can be found in Refs. [1, ]). The level density parameters of fragments at scission are calculated using an iterative procedure in the frame of the generalized super-fluid model. Finally, the excitation energy at full acceleration of each fragment of the fragmentation range is obtained as a sum of the extra-deformation and excitation energies at scission. The method based on modeling at scission was firstly verified by comparing the E*(A) results with the indirect experimental E*(A) obtained from ν(a) data (for details see Ref.[]) as in the example given in Fig..1 [6] (red circles and blue diamonds): E * (MeV) modelling at scission: En=.5 MeV En=5.5 MeV parameterization En=.5 MeV En=5.5 MeV 35 U(n,f) 1 5 'Índirect' exp.e*(a) from ν H /ν pair of: Mueller and Naqvi GERKFK (En=.5 MeV) Mueller and Naqvi GERKFK (En=5.55 MeV) exp. Z (Wahl) FF range: Z/A A Fig U(n,f): E*(A) resulted from the modeling at scission (red circles and blue diamonds) and from the parameterization (magenta and cyan squares connected with lines) in comparison with the indirect E*(A) experimental data (plotted with different open symbols).

4 Prompt emission modeling in the fission process 75 Another method of TXE partition is based on the using of different parameterizations, as following: The TXE partition according to the ratio of prompt neutrons number emitted by the complementary fission fragments. This is based on the systematic behaviour of the experimental ratio ν H/ ν pair (A H ) (e.g. Fig..) (details in Ref. [7]). The parameterization of the excitation energy obtained by modeling at scission (e.g. Fig..3) [6]. This method is based on the same behaviour of the ratio of heavy fragment excitation energy to the total excitation energy as a function of heavy fragment mass (E * H/TXE(A H )) as the experimental ratio ν H /ν pair (A H ) [6,7] consisting in: the ratio is less than.5 for fragments pairs with A H < 1, with a minimum around A H = (driven by the magic numbers Z = 5 and/or N = ); it is approximately.5 for fragmentations with A H around 1 and it exhibits an almost linear increase for A H above U(n th,f) ν H /ν pair parameterization ν H /ν pair ν H /ν pair U(n th,f) ν H /ν pair parameterization A of HF Nishio 199 JPNKTO Nishio JPNKTO 199 Maslin UKALD 1967 Fig.. ν H /ν pair parameterization (red line) in comparison with the experimental data (different symbols) for 33 U(n th,f) (upper part) and for 35 U(n th,f) (lower part).

5 76 I. Visan, G. Giubega, A. Tudora 5 E* H /TXE E* H /TXE Pu(n th,f) modelling at scission parameterization 35 U(n.5MeV,f) modelling at scission parameterization Z/A, Z=.5 ν H /ν pair from ν exp (A) of: Nishio 1995 JPNKTO Alpalin 1965 RUSKUR Tsuchiya JPNKTO AH Z/A, exp. Z(Wahl) ν H /ν pair from ν exp (A) of Naqvi GERKFK Fig..3 E* H /TXE parameterization (red line) in comparison with the results of modeling at scission (blue circles) and with the experimental data (different open symbols) for 35 U (n.5mev,f) (upper part) and 39 Pu(n th,f) (lower part)... THE PROMPT FISSION NEUTRON MULTIPLICITY AND SPECTRUM OF AN INDIVIDUAL FRAGMENT The prompt fission neutron spectrum (PFNS) in the center-of-mass system (CMS) corresponding to an individual fission fragment is given by []: T m 1+ b cos θ CMS Φε ( ) = ( σ c ( εε ) KTPT ( ) ( ) exp( ε/ T)d T) () 1 + b /3 with the normalization constant given by: KT ( ) = σ c ( ε) εexp( ε/ T)dε (3) The maxim value of the fragment residual temperature distribution P(T) is Tm = E*/ a with E* obtained from the TXE partition and the fragment level density at full acceleration calculated in the frame of superfluid model. 1

6 6 Prompt emission modeling in the fission process 77 The expression of prompt neutron spectrum associated to an individual fragment in the laboratory system (LS) is given by: ( E+ EF ) f be ( ε EF f ) c ( ) 1 + b/3 ε EF f (1 + b/3) E EF f 1 1 N LH, ( E) = σ () ε ε + I()d ε ε Tm EF () f a 1 KT ( )( s 1) Texp( / T)dT s + ε T m I() ε = a= sa s+ 1 s+ 1 s + 1 KT ( ) T+ Tm exp( ε/ T)dT ( s 1) s a In eqs. () and (), ε and E are the prompt neutron energy in CMS and LS, respectively. EF f is the average fission fragment kinetic energy per nucleon obtained from momentum conservation, b is the anisotropy parameter and s is a parameter driving the shape of P(T) (see for details Ref.[]). The spectrum in LS for a fragment pair is given by: r 1 N( E) = N L( E) + N H( E) (5) r + 1 r + 1 where r=ν L /ν H.. The contribution of scission neutrons can be also taken into account by a Weisskopf-Ewing evaporation spectrum [9]. The prompt fission neutron multiplicity corresponding to a fragment pair or to a FF is obtained from energy conservation: TXE < E γ > <ν pair >= (6) < ε>+< S > where <Eγ> is the average gamma rays energy, <ε> is the average prompt neutron energy in CMS (1 st order momentum of prompt neutron spectra of complementary fission fragments) and <S n > is the average neutron separation energy from fragments calculated using the mass excesses [1, 11]. n.5. FRAGMENTS DISTRIBUTIONS USED IN THE PbP TREATMENT The fission fragments distributions used in PbP model are [1]: 1) The charge distribution p(z,a) taken as a narrow Gaussian function [1]: 1 pz ( ) = exp( ( Z Z p ) / c) c= ( σ + 1/1) (7) πc

7 7 I. Visan, G. Giubega, A. Tudora 7 where Z p is the most probable charge obtained from UCD corrected with a possible charge polarization Z. ) The TKE distribution for each pair of fragments: 1 ( TKE TKE( A)) patke (, ) = ( ) exp ( σ ( )) TKE TKE A σ A π where TKE(A) and σ TKE (A) are experimental data. 3) The fragment mass distribution Y(A) (usually experimental data). Others distributions needed in the PbP calculations are obtained from the experimental one-dimensional ones Y(A), TKE(A) and σ TKE (A) as: a) The double distribution: Y ( A, TKE) = Y( A) p( A, TKE) (9) b) Y(TKE) distribution: TKE) Y ( A, TKE) / () Y ( = Y( A) (1).6. AVERAGED QUANTITIES To obtain average quantities as a function of fragment mass (e.g. <ν>((a)), as a function of TKE (e.g. <ν>(tke) or total average ones (e.g. <ν p >) the corresponding multi-parametric matrices are averaged over different distributions [1] as following: Average quantities as function of fragment mass: < q> ( A) = qz (, ATKE, ) pz (, AY ) ( ATKE, )/ pz (, AY ) ( ATKE, ) (11) ZTKE, ZTKE, These quantities (especially ν(a)) are very sensitive to the TXE partition, their comparison with existing experimental data being a crucial test of the TXE partition. Average quantities as function of TKE: < q > ( TKE) = q( Z, A, TKE) p( Z, A) Y ( A, TKE)/ p( Z, A) Y ( A, TKE) (1) Z, A Z, A Quantities like <ν>(tke), <ε>(tke) can be compared with experimental data, too. The model description of different experimental <ν>(tke) data sets exhibiting in some cases different inverse slopes dtke/dν is still under debate. Total average ones: < q>= q( Z, A, TKE) p( Z, A) Y( A, TKE)/ p( Z, A) Y( A, TKE) (13) Z, A, TKE Z, A, TKE Total average prompt emission quantities, especially <ν> tot and spectra as a function of E n are required in evaluated nuclear data libraries for applications.

8 Prompt emission modeling in the fission process EXAMPLES OF POINT-BY-POINT RESULTS In the figures below are presented a few examples of different quantities provided by the PbP model calculations. An example of primary result is the multi-parametric matrix ν(a,tke) of 5 Cf(SF) [13], plotted in Fig.3.1 (multiplicity as a function of A H for given TKE values) and Fig.3. (multiplicity as a function of TKE for given mass numbers) that describe well the existing experimental data. In Fig. 3.3 [1] is presented an example of PbP model calculation of the average energy of prompt neutrons in CMS for 5 Cf(SF) (obtained by averaging the matrix ε(z,a,tke) over Y(A,TKE) and over Y(A)) in comparison with experimental data of Bowman (plotted with open black circles) and Nifenecker (solid gray squares). PbP calculations of <ν>(tke) obtained by averaging ν(a,tke) over Y(A) and over Y(A,TKE) in comparison with experimental data taken from EXFOR [1] are given in Fig.3. [1] for the case of 5 Cf(SF). Average model parameters as a function of TKE obtained also by averaging the corresponding matrices according to eq.(1) are plotted with different symbols in Fig 3.5 [1] for the case of 39 Pu(n th,f). As it can be seen they exhibit regular behaviours than can be fitted well. Appropriate fits are also plotted with lines. Examples of ν(a) results are given in Fig.3.6 [6] for the case of 35 U(n,f) and Fig.3.7 [, 7] for the case of 37 Np(n,f). As it can be seen in Fig.3.6 the PbP calculations of the ν(a) using a parameterization of E*(A) resulted from modeling at scission [6] describe very well the experimental data [1], including the interesting behaviour of the multiplicity increase with E n, mainly for heavy fragments. In the case of 37 Np(n,f) (Fig.3.7) the PbP calculations at two incident energies (En =. MeV and En = 5.5 MeV) were performed using two TXE partition methods (one based on the experimental ν H/ ν pair (A H ) parameterization plotted with red circles and blue stars and one based on modeling at scission plotted with magenta circles and cyan stars). As it can be seen, the ν(a) results using both TXE partitions are close to each other and describe well the experimental data including the ν-increase with E n for heavy fragments only. The PbP model provides also results of the prompt gamma rays. In Fig.3. is given as example the PbP result of the average prompt gamma-ray energy as a function of fragment mass in comparison with the existing experimental data (open symbols) for the case of 35 U(n th,f). As it can be seen both E γ (A) results [15], obtained by using the TXE partition from modeling at scission (blue stars) and from the E * H/TXE parameterization (red circles) are close to each other and give a very good description of the experimental data.

9 I. Visan, G. Giubega, A. Tudora 9 FF pair multiplicity TKE=16 MeV TKE=17 MeV TKE=1 MeV 5 Cf(SF) TKE=19 MeV TKE= MeV Zacharova Bowman PbP AH Zacharova PbP Zacharova Bowman PbP Zacharova Bowman PbP Fig Cf(SF): Fission fragment pair multiplicity for a given TKE versus the heavy fragment mass number. 5 Cf(SF) FF pair multiplicity 1 Zackharova Bowman PbP AH=135 AH=16 AH=17 AH=156 AH=159 AH= TKE(MeV) Fig.3. 5 Cf(SF) Multiplicity of a given pair of fragments vs. TKE.

10 1 Prompt emission modeling in the fission process Cf(SF) Bowman et al. Phys.Rev.19(1963)133 Nifenecker et al. Nucl.Phys.A 19 (197) 5 <ε> SCM (MeV) PbP calc. using Y(A) PbP calc. using Y(A,TKE) in both calculations the multi-parametric matrix ε(z,a,tke) was the same TKE(MeV) Fig.3.3 PbP calculations of average energy of prompt neutrons in CMS for 5 Cf(SF) (red and green circles) in comparison with experimental data (black and gray symbols). Average prompt neutron multiplicity Cf(SF) Vorobyev 1 RUSLIN <ν H >, <ν L > Budtz-Jorgensen, Knitter, 199 Bowman 1963 USABRK Vorobyev, PNPI 1 RUSLIN Sing Shengyao 19 CPRAEP Sh.Zeynalov 11 (preliminary), van Aarle (199) Calculations PbP (Y(A)) PbP 1 (Y(A,TKE)), PbP HF, LF groups most prob fragm. (1) TKE (MeV) Fig.3. <ν>(tke) for 5 Cf(SF) obtained by averaging over Y(A,TKE) (red symbols) and over Y(A) (blue circles) in comparison with experimental data (different black and gray symbols).

11 I. Visan, G. Giubega, A. Tudora 11 <Sn> (MeV) <Er> (MeV) <C> (MeV) Pu(n th,f) PbP Y(A/TKE) Tsuchiya <Er>= TKE-.513 TKE PbP Y(A/TKE) Wagemans <Er>= TKE-.5 TKE PbP Y(A/TKE) Tsuchiya <Sn> = TKE+9.957E- TKE PbP Y(A/TKE) Wagemans <Sn>= TKE+.5596E- TKE PbP Y(A/TKE) Tsuchiya <C> = TKE+.73E- TKE PbP Y(A/TKE) Wagemans <C>= TKE+.13 TKE TKE (MeV) Fig Pu(n th,f) Average model parameters obtained from the PbP treatment (symbols) and their appropriate fits (lines). Prompt neutron multiplicity using E* H /TXE parameterization En=.5 MeV En=5.5 MeV ν exp (A) Mueller and Naqvi GERKFK En=.5 MeV En=5.55 MeV 35 U(n,f) FF range: Z/A exp. Z (Wahl) A Fig U(n,f) PbP results of prompt neutron multiplicity as a function of fragment mass number obtained by using the E* H/ TXE parameterization.

12 1 Prompt emission modeling in the fission process 3 Prompt neutron multiplicity Np(n,f) En=. MeV J.Phys.G 1 Nucl.Phys.A 11 En=5.5 MeV J.Phys.G 1 Nucl.Phys.A 11 <ν p > =.751 EXFOR Mueller 1 GERKfK En =. MeV En = 5.55 MeV EXFOR Naqvi GERKfK En =. MeV En = 5.5 MeV <ν p > = A Fig Np(n,f) PbP results of prompt neutron multiplicity as a function of fragment mass number U(n th,f) PbP Z/A TXE part. modelling at scission PbP 3Z/A TXE part. parameteriz. <Eγ> of FF (MeV) Experimental data of Pleasonton et al., A Fig U(n th,f) PbP results of prompt gamma rays energy as a function of fragment mass number (full circles and stars) in comparison with experimental data (open squares).

13 I. Visan, G. Giubega, A. Tudora 13. CONCLUSIONS The PbP model was successfully applied to many spontaneous and neutron induced fissioning systems, such as 5 Cf(SF), 36- Pu(SF),, Cm(SF), 33,3,35,36,3 U(n,f), 39 Pu(n,f), 37 Np(n,f), Pa(n,f), 3 Th(n,f), being able to provide prompt emission results validated by the excellent description of all existing experimental data. The evaluation of total average prompt neutron multiplicity and spectra required in all fission applications are also based on the Point-by-Point model. The uncertainties of PbP results are less or inside the experimental data errors. The input parameters of the PbP model are experimental data or are provided by independent models. These facts assure a good prediction and made the PbP model a powerful tool for evaluation purposes. REFERENCES 1. A. Tudora, Ann.Nucl.Energy 53, 57 51, (13).. RIPL 3, Reference Input Parameter Library of IAEA, Segment 1, Nuclear Masses and Deformations, database of Audi and Wapstra (mass excesses) and of Moller and Nix (shell corrections), (11), available online 3. RIPL-3, Reference Input Parameter Library of IAEA, Electronic file from segment, Optical Model Parameters, ID 1 Becchetti-Greenless, (11) C. Morariu, A. Tudora, F.-J. Hambsch, S. Oberstedt, C. Manailescu, Modelling of the total excitation energy partition including fragment deformation and excitation energies at scission, J. Phys. G 39, 5513 (15pp), (1). 5. RIPL 3, Reference Input Parameter Library of IAEA, Segment 1 Deformations, database HFB-1 (S. Goriely, M. Samyn and J.M. Pearson Phys. Rev. C 75, 613, 7), (11). 6. I. Visan, G. Giubega, A. Tudora, The 6th Annual International. Conference. on Sustainable Development through Nuclear Research and Education, Proceeding ISSN , Piteşti, România, May, (13). 7. C. Manailescu, A.Tudora, F.-J. Hambsch, C. Morariu, S. Oberstedt, Possible reference method of total excitation energy partition between complementary fission fragments, Nucl. Phys. A 67 1 (11).. F.-J. Hambsch, Anabella Tudora, G.Vladuca, S.Oberstedt, Prompt fission neutron multiplicity for 5 Cf(SF) in the frame of the multi-modal fission model, Annals of Nuclear Energy 3, pp (5). 9. I.Visan, A.Tudora, Study of model parameters influence on prompt fission neutron spectrum, Journal of Nuclear Research and Development, No 1, (11). 1. Anabella Tudora, B. Morillon, F.-J. Hambsch, G. Vladuca, S. Oberstedt, A refined model for 35 U(n,f) prompt fission neutron multiplicity and spectrum calculation with validation in integral benchmarks, Nucl.Phys.A 756, pp (5). 11. Anabella Tudora, Experimental prompt fission neutron sawtooth data described by Point-by- Point model, Annals of Nuclear Energy 33, pp (6). 1. C. Wagemans, The Nuclear Fission Process, CRC Press, Bo ca Raton, USA, (1991). 13. Anabella Tudora, Multi-parametric prompt neutron and fission fragment experimental data described by Point-by-Point model, Annals of Nuclear Energy 35 (1), pp. 1 1 (). 1. EXFOR Experimental Nuclear Reaction Data Base available online (1). 15. A. Tudora, C. Morariu, F.-J. Hambsch, S. Oberstedt, C. Manailescu, Prompt gamma-ray energy in the frame of prompt emission models, Physics Procedia 31, 3 5 (1).