Available online at ScienceDirect. Physics Procedia 47 (2013 ) 33 38

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1 Available online a ScienceDirec Physics Procedia 47 3 ) Scienific Workshop on Nuclear Fission Dynamics and he Emission of Promp Neurons and Gamma Rays, Biarriz, France, 8-3 November 1 Effecs of Fission Fragmens on he Angular Disribuion of Scission Neurons T. Wada a *, T. Asano a, M. Hirokane a, N. Carjan b, M. Rizea b a Deparmen of Pure and Applied Physics, Kansai Universiy, Yamae-cho, Suia , Japan b Naional Insiue of Physics and Nuclear Engineering, Reacorului 3, RO-7715, POB-MG6, Magurele-Buchares, Romania Absrac We invesigae he effecs of he fission fragmens on he angular disribuion of scission neurons. The ime evoluion of he wave funcion of he scission neuron is obained by inegraing he ime-dependen Schrödinger equaion. The effecs of he re-absorpion and scaering by he fission fragmens are aken ino accoun by means of he opical poenial. The angular disribuion is srongly modified by he presence of he fragmens. Dependence on he magniude of he absorpion is discussed. Influence of he finieness of he grid size is also discussed The T. Wada, Auhors. T. Asano, Published M. Hirokane, by Elsevier N. B.V. Carjan, Open M. access Rizea. under Published CC BY-NC-ND by Elsevier license. B.V. Selecion and peer-review under responsibiliy of of Join Research Cenre -- Insiue for for Reference Maerials and and Measuremens Keywords: nuclear fission; scission neuron; neuron emission; angular disribuion; ime-dependen Schrödinger equaion; 1. Inroducion In low energy fission, such as sponaneous fission and hermal neuron induced fission, here are wo main neuron emission processes: scission neurons and pos-scission neurons. A he momen of scission, he neck ha has conneced he wo fission fragmens rupures, followed by he quick absorpion of he neck prorusions by he fragmens. On his abrup change of nuclear shape, i is probable ha nucleons are lef behind in he neck region and are observed as paricle emission. On he oher hand, pos-scission neurons are emied from excied fission fragmens; he process is supposed o be a hermal emission. Aemps have been made o separae he yield of scission neurons in low energy fission (Franklyn, 1978, Kornilov, 1), hey repored ha 1-% of he oal neuron yield could be scission neurons. I is also aemped o esimae he number of scission neurons heoreically (Carjan, 7, Carjan, 1). The resuls depend on he nuclear shape such as he neck radius before scission. If we exrac he reliable number of scission neurons from experimens, we can ge informaion on he nuclear shape a he ime of scission. * Corresponding auhor. Tel.: address: wadaaka@kansai-u.ac.jp The Auhors. Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selecion and peer-review under responsibiliy of Join Research Cenre - Insiue for Reference Maerials and Measuremens doi:1.116/j.phpro

2 34 T. Wada e al. / Physics Procedia 47 ( 13 ) The angular disribuion of he scission neuron is a key o separae i from pos-scission neurons. These componens can be separaed by aking he kinemaical condiion ino accoun; pos-scission neurons are emied from he moving source (fully acceleraed fragmens) while he emission of scission neurons is supposed o be isoropic in he lowes order approximaion. However, since he scission neurons are emied in he close viciniy of he fission fragmens, he final angular disribuion is influenced by he re-absorpion and he scaering by he fragmens. In he previous work (Wada, 11), we proposed a formulaion based on he ime-independen scaering heory. We observed a srongly modified angular disribuion due o he scaering and re-absorpion by he fission fragmens. In his paper, we presen an alernaive approach based on he ime-dependen Schrödinger equaion. In he nex secion, a formulaion is given o calculae he angular disribuion of he scission neurons in which he effec of he re-absorpion and he scaering is aken ino accoun in erms of he opical poenials. Resuls are presened for wo cases, one is a purely absorpive case and he oher is he case ha includes he aracive poenial. The effecs of he fission fragmens on he angular disribuion of scission neurons are discussed. Finally, a summary is given.. Framework We sar wih a ime-dependen Schrödinger equaion (TDSE), 1 i H, H U, 1, ) m where denoes he neuron wave funcion, H is he Hamilonian, and U is he poenial ha represens he effec of he fission fragmens. The ime developmen is obained wih he use of he mid-poin inegraion, ( ) ( ) i H ( / ). () By decomposing ino he real and he imaginary par, = R + ii, he numerical soluion is obained using he following formula (he leapfrog mehod), / ) ) ) / ) HI ( H ) / ). (3) Some modificaion is necessary when we inroduce an imaginary poenial in H, / ) ) W / ) W / ) ) / ) H R H R ) / ) W W / ) / ), (4) where H is given as H = H R + iw. The poenial U is parameerized in Woods-Saxon form cenered a he posiion of he fragmens, U (, z) 1 exp V iw ( z a B) r F 1 exp where B is he half-disance beween he fragmens, a is he diffuseness, and r F is he radius of he poenials. For simpliciy, we assume symmeric fission. V iw ( z a B) r F, (5)

3 T. Wada e al. / Physics Procedia 47 ( 13 ) Assuming he axial symmery, we solve he TDSE in wo-dimensional grid space (, z). The original emission of he scission neurons is assumed o be isoropic, and we adop a Gaussian wave packe for he iniial wave funcion, (=) = C exp( ( + z )), where C is a normalizaion consan and is he widh of he wave packe. We now se a sphere of radius R and calculae he neuron flux on his spherical surface, j( r, ) 1 im * *. We hen calculae he number of ougoing neurons per uni solid angle per uni ime and inegrae i wih ime o obain he neuron angular disribuion, i.e., he number of neurons per uni solid angle ha passed he surface up o ime, (6) d (, ) d (, ' ) d' j( R,, ' ) n( R, ) R d' d d d, n e r. (7) In order o avoid arificial reflecions of he wave funcion a he border of he grid, we pu a week absorbing poenial ouside of he sphere of radius R. We ake a quadraic form for he absorbing poenial. The srengh is deermined o minimize he effec of he reflecion. 3. Resuls We invesigae he fission of 36 U ha corresponds o he neuron induced fission of 35 U as an example. An imporan parameer in he calculaion is he iniial separaion beween he fragmens. From a sysemaic sudy of he average oal kineic energy (TKE) of he fragmens, Zhao e al. (Zhao, ) deduced he elongaion parameer which is he raio beween he average disance beween he fragmens o he conac disance r (A 1/3 1 + A 1/3 ), where A 1 and A are he mass numbers of he fragmens and r = 1.17 fm. The average disance beween he fragmens was deermined so ha he corresponding poin charge Coulomb energy is equal o he average TKE. They obained = 1.53 for he asymmeric fission in U region. In he calculaion, we se he parameer B as B = r (A 1/3 1 + A 1/3 )/. The disance beween he grid poins is ypically.1 fm in boh z- and -direcions. The ime sep for he inegraion is ypically =. fm/c. The radius R is se o 5 fm and he inegraion is performed up o = 4x1 1 s. Figure 1 shows he calculaed angular disribuion of scission neurons for he case of purely absorpive poenials (V = ). The widh of he iniial wave packe is deermined o give he average energy of neuron = 1.5 MeV. In calculaing he ime developmen, we ake accoun of he moion of he fragmens due o he Coulomb repulsion beween he fragmens. The pre-scission kineic energy of 1 MeV is assumed. The fragmens lie along z-axis, i.e., we have he absorpive poenials around and 18 degrees. As we increase he magniude of he absorpion, he yields around and 18 degrees decrease significanly, while he yield a 9 degrees does no change much. The resuling angular disribuion has a peak a 9 degrees and can be easily disinguished from ha of he pos-scission componen. Nex, we invesigae he case wih aracive real poenials ogeher wih he absorpion. Figure shows he resuls wih V = 4 MeV. The difference from he purely absorpive case is clearly seen in paricular a and 18 degrees. The yield a degrees is srongly enhanced by he aracion of he real poenial and i decreases drasically as he absorpion becomes sronger. In paricular, in he case where we ake he aracive poenial alone, i.e., W =, he disribuion is srongly peaked around and 18 degrees. For he case wih W = 5 MeV, he yield around 9 degrees also decreases significanly, resuling in a raher fla angular disribuion. Thus, he angular disribuion of he scission neurons depends srongly on he srengh of he absorpion.

4 36 T. Wada e al. / Physics Procedia 47 ( 13 ) In Fig. 3, we display he ime developmen of he angular disribuion of scission neurons. I is seen ha he yield around 8) degrees grows rapidly firs. We may say ha his is because of he higher velociy of neurons in he fragmens caused by he aracive poenials. A = x1-1 s, we do no ye observe a peak around 9 degrees. Then a = 3x1-1 s, he peak around 9 degrees sars o grow and i keeps growing up o = 4x1-1 s. I should be noed here ha he ime scale menioned above depends crucially on he radius R of he sphere on which we observe he ougoing flux of neurons. Fig. 1. Angular disribuion wih purely absorpive poenials: W = 1 MeV (do-dashed line), W = MeV (dashed line), and W = 5 MeV (solid line). Fig.. Angular disribuion wih aracive poenials V = 4 MeV and absorpive poenials: W = (do-do-dashed line), W = 1 MeV (do-dashed line), W = MeV (dashed line), and W = 5 MeV (solid line).

5 T. Wada e al. / Physics Procedia 47 ( 13 ) Fig. 3. Time developmen of he angular disribuion of scission neurons wih aracive (V = 4MeV) and absorpive (W = 5MeV) poenials: = 1x1-1 s (do-do-dashed), = x1-1 s (do-dashed), = 3x1-1 s (dashed), and = 4x1-1 s (solid). Fig. 4. (a) Illusraion of he relaionship beween he radius R and he emission angle ; (b) Comparison of he angular disribuions wih differen R; R = 3 fm (dashed) and R = 5 fm (solid). Because we are working in a finie grid size, he radius R of he sphere on which we coun he ougoing neuron flux mus be finie. When a neuron is scaered by he fragmen whose posiion is shifed from he origin, he emission angle is modified because of he finieness of he radius R. Therefore, he angular disribuion calculaed in his way depends on he size of R. The siuaion is illusraed in Fig. 4(a). A paricle

6 38 T. Wada e al. / Physics Procedia 47 ( 13 ) is emied o an angle from a poin ha is displaced from he origin. When we deec he paricle on he sphere wih radius R 1, we coun ha he paricle is emied o he angle 1, while if we deec i a a larger radius R, we coun ha i is emied o he angle. As can be seen in Fig. 4(a), we obain 1 < <. When R approaches infiniy, we have no ambiguiies in he definiion of he angle. In Fig. 4(b), we display he comparison of he angular disribuion wih wo values of he radius, R = 5 fm and R = 3 fm, for he case wih V = 4 MeV and W = 5 MeV. In hese calculaions, he moion of he fragmens was no aken ino accoun. Compared wih he case wih R = 5 fm, he case wih R = 3 fm shows larger yields around and 18 degrees, and he peak around 4 degrees is shifed o a smaller angle. Though, some differences are seen beween he wo cases, he essenial feaures of he angular disribuion have no changed. We invesigaed more cases saring from differen iniial disribuion, e.g., changing he disance beween he fragmens or changing he widh of he iniial Gaussian wave packe. I is found ha he final angular disribuion depends considerably on he iniial disribuion of he neurons. Since he informaion on he angular disribuion of he scission neurons is very imporan o separae hem from oher neuron sources, furher invesigaion wih more realisic iniial wave funcion is necessary and is under progress. Because of he simpliciy of he framework, i is raher easy o exend he calculaion o mass-asymmeric fission. I is of ineres o invesigae he righ-lef asymmery of he angular disribuion since i is supposed o depend sensiively on he disribuion of he emission poins. 4. Summary The effecs of he re-absorpion and scaering by he fission fragmens on he angular disribuion of scission neurons have been invesigaed in he framework of he ime-dependen Schrödinger equaion. A formulaion has been given o calculae he angular disribuion from he neuron flux on a spherical surface wih finie radius R. I has been demonsraed ha he absorpive poenial diminishes he yields around and 18 degrees, resuling in he angular disribuion ha has a peak around 9 degrees. On he oher hand, he aracive poenial enhances he yields around and 18 degrees. The final angular disribuion depends srongly on he magniude of he absorpion and on he iniial disribuion of he scission neurons. Acknowledgemens The auhors would like o express heir graiude o Prof. M. Io for discussions and encouragemens. This work is suppored in par by JSPS Gran-in-Aid for Scienific Research (No ). References Carjan, N., Talou, P., Sero, O., 7. Emission of scission neurons in he sudden approximaion. Nucl. Phys. A 79, 1. Carjan, N., Rizea, M., 1, Scission neurons and oher scission properies as funcion of mass asymmery in 35 U(n h,f). Phys. Rev. C8, Franklyn, C. B., Hofmeyer, C., Mingay, D. W., Angular correlaion of neurons from hermal-neuron fission of 35 U. Phys. Le. B 78, 564. Kornilov, N. V., e al., 1. New evidence of an inense scission neuron source in he 5 Cf sponaneous fission. Nucl. Phys. A 686, 187. Wada, T., Nishioka, R., Asano, T., 11. Re-absorpion and scaering of scission neurons by he fission fragmens. Proc. Sci. Workshop on Nuclear Fission Dynamics and he Emission of Promp Neurons and Gamma Rays, JRC Sci. and Tech. Repors 64789, 163. Zhao, Y. L., e al.,. Degrees of deformaion a scission and correlaed fission properies of aomic nuclei. Phys. Rev. C6, 1461.