CALCULATION OF THE FISSION-FRAGMENT YIELDS OF PRE- ACTINIDE NUCLEI ON EXAMPLE OF nat Pb ISOTOPES

Transcription

1 CALCULATION OF THE FISSION-FRAGMENT YIELDS OF PRE- ACTINIDE NUCLEI ON EXAMPLE OF nat Pb ISOTOPES 1 V. T. Maslyuk, 1 O. A. Parlag, 1 O. I. Lendyel, 1 T. I. Marynets, 1 M. I. Romanyuk, 2 O.S. Shevchenko, 2 Ju.Ju. Ranyuk, 2 A.M. Dovbnya 1 Insttute of Electron Physcs, Natonal Academy of Scences of Ukrane, Uzhgorod, Ukrane 2 Natonal Scence Center Kharkov Insttute of Physcs and Technology, Ukrane Emal: nuclear_photo@meta.ua ABSTRACT In the frame of proposed statstcal method the calculaton of fsson-fragment yelds (mass and charge spectra) for pre-actnde nucle on example of nat Pb sotopes had been conducted. The role of neutron s shells wth N = 5 and N = 82 n the realzng of sngle- and double-humped shape of the fsson-fragment yelds, respectvely for neutron-defcent and neutron- profcent Pb sotopes was nvestgated. Comparson of theoretcal and expermental data of nat Pb fsson was performed wth the nvolvement of the transformaton to the ensemble of long- and short-lved nucle fragments. Keywords: Pre-actnde, nucle, nat Pb sotopes, fsson, fragments yeld, mass/charge spectrum, tme transformaton, 2-, 1- humped shape, 1. Introducton Medum-heavy or pre-actnde nucle wth mass number 195 <A <21 are nterestng, but relatvely poorly explored objects of physcs of atomc nucle fsson. In contrast to heavy nucle wth Z 2 /A 36 fssonablty parameters, the fsson of pre-actnde nucle s exotcal and requres addtonal exctaton energy. Nevertheless, data about the peculartes of fsson that s stmulated as a rule by the nteracton wth hgh-energy nuclear partcles, as well as fssonfragment yelds are mportant for subcrtcal assembles or transmutator reactors. It s well known that such objects contan the pre-actnde nucle at the base of electron-neutron converters. These nucle could be as a part of nuclear fuel s components or recoverable radoactve waste. On the other hand, data of nucle fsson are mportant for understandng the stablty of atomc nucle and the nature of nuclear forces. Although the data on the mass or charge fsson-fragment yelds of that nucle currently are very lmted, the exstng experments of nucle fsson, stmulated by the nteracton wth charged partcles [1-3] or bremmstrahlung radaton [4, 5] for a wde energy range 5 MeV GeV should be emphaszed. Theoretcal studes of the peculartes of energetc structure and medum heavy nucle fsson are sgnfcantly lmted and concerns to the calculaton of transent confguratons and statstcal characterstcs of pre-actnde nucle n the fsson reactons, see [6, 7]. 1

2 Ths work contnued prevous nvestgatons [8] and s devoted to systematzaton of medum heavy nucle fragments on example of Pb sotopes fsson (mass and charge yelds) n the frame of prevously proposed statstcal method [9]. Partcularly, the possble asymmetry of fsson fragment yelds, the role of shell effects and the nterpretaton of expermental data where these yelds have a symmetrcal (one- humped) form s dscussed. 2. Theoretcal understandng In ths artcle the calculaton of the mass/charge yelds of output nucleus fsson fragments wth atomc mass А and charge Z s carred out n the frame of proposed statstcal method based on the followng assumptons [9]: The character of the mass/charge yelds s determned by condton of thermodynamc ensemble s adjustment that contans clusters of fsson fragments; The nucle fragments fsson ensemble s a constant pressure ensemble, and ts thermodynamc parameters: P (pressure) and T (temperature) s determned by state of ntal nucleus; In computng of confguratonal entropy the statstcal non-equvalence of nucleons wth dfferent bndng fracton of each fsson fragments should be consdered; The nuclear partcles emsson (fsson neutrons, gamma radaton) mantans the constancy of the thermodynamc parameters of the nuclear fragments ensemble, namely, system s thermostat; Durng the fsson neutrons emsson the nternal energy of fragment nucle changes are n process, owng to executon of work whle reducng the volume of the ntal nucleus at constant pressure; Inasmuch as atomc nucle are small systems, t should be apprecated the fluctuatons of thermodynamc parameters n the calculaton of the equlbrum confguratons of the nuclear fragments ensemble. In ths case the probablty of realsaton, for example, the -th of two-nuclear fragment set s determned n the usual way va the sobarc dstrbuton functon: ( PV ) / T Z p f ( V ) exp /, (1) where the statstcal sum Z p s defned as Z exp ( PV ) / T.. U (A,Z ), (2) j 1,2 N p N n j j, j, p k, V k k 2

3 U (A j j,, Z j, ) - value of the bndng energy of the j-th nucleus fragment from the -th cluster; the symbol <...> means that the summaton n (1) s conducted for the -th set that contans two fsson fragments wth the numbers of protons and neutrons that satsfy followng condton: j 1,2 p n ( N N ) n A, (3) j, j, ) p n where n - number of fsson neutrons, N j, / N j, - number of protons/neutrons for the j-th fragment. The value - s the number of possble realzatons of -th set of nucle fragments, calculated at protons/neutrons statstcal nonequvalence condton of varous nucle fragments: p n A!/ n!/( ( N! N!), (4) j 1,2 j, j, The sobarc term PV (see Equaton (1) was chosen n the form PV P( V vn), where V s the ntal nucleus volume, P s the nucleon gas pressure and v s the averaged value of volume related to a sngle fsson neutron wth total number of n. The sobarc constant Pv value was estmated wthn 4-5 МеV and was evaluated from the condton that the total fsson neutron number n does not exceed 3 neutrons per fsson. Then, the dstrbuton functon F(A 1 ) of a sngle fsson fragment wth mass A 1, or the same, F(Z 1 ), wth charge Z 1 has to be obtaned by a followng procedure: Formng the whole ensemble of post-scsson fragment clusters, usng for nucleons conservaton condtons (1), The ntal (not normalzed) values F(A 1 ) are obtaned as the sum of probabltes of two-fragment set, contanng the fsson fragment wth the mass A 1, see Equaton (1). Ths procedure s smlar to the method of hstograms and must ncludes cumulatve chans; the same procedure s vald for F(Z 1 ); The Monte Carlo procedure must be appled to smulate the statstcal fluctuatons of the thermodynamcal parameters of the fsson fragments ensemble; The next step ncludes the normalzaton procedure and determnaton of the fnal values of F(A 1 ) and F(Z 1 ). These functons must satsfy the followng normalzaton equatons: F ( A1 ) F( Z1) 2%, where <A 1 >, <Z 1 > have the same meanng as n (2). A 1 Z 1 It should be noted that proposed statstcal method contans no adjustable parameters, but only those that can be obtaned from experment. It provdes an 3

4 opportunty to nvestgate the features of mass and charge dstrbutons dependng on the length of the cumulatve fsson fragments chan, wth or wthout consderaton of presence of short-, long-lved or stable fsson fragments that are not observed by methods of semconductng gamma spectrometry. On the other hand, ths calculaton method s useful for verfyng of the avalable mass formula systematc for nucle and for tabulaton of ther bndng energes values. 3. Results and dscusson Fg. 1 shows the calculaton results of the mass spectra and charge yelds of fsson fragments for 2 lead sotopes 19 Pb 21 Pb as a basc component of nat Pb, obtaned under temperature (exctaton energy) of ntal nucleus wth atomc mass А and charge Z, T = 1 MeV, sobarc constant Pv = 4.5 MeV, and no more 3 emtted neutrons per fsson act was permtted. The length of cumulatve chan dd 24 4 Y, % Y, % A A A Z 55 a) b) A 12 -, A 5,8 11,7 17,6 23, ,2 Z 45 9,85 19,9 4 29, , A с) d) Fg.1. (Color onlne) The values of mass - a), c), and charge, - c), d) spectra of lead sotopes fsson fragments are presented n the form of 3D (a, b) and contour (c, d) dagrams. 4

5 not exceed 1 nucle, the values of bndng energes complaton was chosen from [1]. The calculaton shows a tendency toward shape changng of the yelds of fsson fragments from sngle-humped (symmetrc) to double-humped dependences when passng from neutron-defcent (as at A = 19) to neutron-profcent Pb sotopes, whch occurs n the vcnty of lead sotopes Pb. Note that ths tendency s more pronounced for the mass spectra of fsson fragments and lowest values of temperature T. It s nterestng to nvestgate the reasons of the sotopes fsson-fragment yelds asymmetry that formed nat Pb, and role of 5 and 82 nuclear flled shells n ths process. The calculaton shows that the entropy term S = ln ( ), where defned n (4) s responsble for the symmetrzaton of the fsson fragment yelds wth the rse of the nuclear temperature Т. Ths s due to the fact that S reaches a maxmum p p n n f N1, N2,, N1, N12 and role of ths component s mportant at hgh temperatures, T. However, for neutron-defcent nucle (e.g. 19 Pb), the mentoned symmetrzaton effect of fsson fragment yelds s enhanced by contrbuton of fsson fragment yelds that resde near to the flled neutron shells at N = 5, as the Zr, 96,97 Mo. In case of 21 Pb fsson, for example, the double-humped form of fsson fragments s caused by presence n fsson fragments the neutron-profcent sotopes 89,91,93 Kr, 95,96,97 Sr and Pd, Ru wth neutron numbers close to non-flled shell N = 82, those role s more sgnfcant than contrbuton of entropc term S, responsble for the symmetrc fsson of nucleus. In other words, the asymmetry or the double-humped nature of mass and charge spectra (as follows from calculatons) are caused as a result of the competton of unflled neutron shells ether N = 5 or N = 82 under formaton of heavy and lght fsson fragments, respectvely, for nat Pb base sotopes. The same reason s crucal at pre-actnde of the mass/charge fsson fragments spectra for other pre-actnde fsson nucle, as Ta, Re, In, Au and Hg. The expermental results of nat Pb photofsson confrmed the sngle-humped form of nuclear fsson-fragment yelds dstrbuton, and there s a tendency of peak expandng of gamma quantum energy: from 6 MeV to 2.5 GeV [4, 5]. Ansotropy, partcularly, double-hamped form of nat Pb sotopes fsson-fragment yelds was not observed n ths experment and the cause s to be sought n the condtons of experment. Thus, the nat Pb fsson fragments yelds measurements was carred out only after.5-1 hour after rradaton, the gamma-actve nucle was dentfyng only, short-lved sotopes was dsregarded. These facts sgnfcantly affected the results of nvestgatons. In ths respect t s nterestng to nvestgate the propertes of sotopc fsson fragments spectra for ntal nucle from 19 Pb to 21 Pb. Fgure 2 shows the results of such calculatons, for example, for Ru sotope yelds, had been obtaned for the same calculated parameters as n Fgure 1. For understandng of the role of statstcal fluctuatons that are typcal for small systems, the sotopc spectra (on Fgure 2) were obtaned by usng the Monte Carlo procedure. The fluctuaton range was up to 2% for all Т and P and 1 statstcal tests were curred out. Obvously, that maxma of Ru sotopes yelds 5

6 (peaks n Fgure 2) was dsplaced from stablty feld to unstable n - transformng nucle when passng from neutron-defcent ( 19 Pb) to neutron- profcent ( 21 Pb) sotopes. Ths tendency s much more expressed for the lght nuclear fragments. Specfed regulartes on Fgure 2 are typcal for the other sotopes of nat Pb fsson fragments ensemble. Ru sotopes, Z= Y,% Pb 2Pb 21Pb Average charges, Z(A) Pb 2Pb 21Pb mass, a.m.u A a) b) Fg. 2 (Color onlne) a) mass spectra for Ru sotopes yelds and b) feasble charge dstrbuton of Ru nucle n case of the dfferent lead sotopes fsson. It allows to suggest a model of the fsson fragments yeld shapes evoluton of nat Pb sotopes over tme, that s recorded n measurements wth the t delay. m Understandably that result of such nvestgatons depends on the rato of long- and short-lved nucle n the ensemble of fsson fragments. For example, n case of 28 Pb sotope the fsson fragments mass spectrum shape can vary from doublehumped (Fgure 1), when durng the experment we can observe long - and shortlved fsson fragments, that are possble at t m 1 sec; to the sngle-humped one, when t m 1 hour and long-lved nucle fragments are fxed only. Last case meets the expermental condtons [4] and ts results could be explaned by the dynamcs of nuclear reactons. Thus, the sotopes 13,15 Ru had been expermentally observed n maxmum of fsson fragments yelds for 28 Pb [4, 5], could appear as result of the - transformatons chans: 13,15 Nb 13,15 Mo 13,15 Tc 13,15 Ru, That occur durng ~ 1 mn., that s less than n expermental condtons, where t m ~.5-1 hour. Reducton of double-humped mass and charge spectra structure (Fgure 1), can arse even durng less tme. Consderng that maxmum of the lghtfsson fragment yelds are formed by the Rb sotopes durng 1 sec owng to chan of - transformaton accordng to the scheme: 97 Rb 97 Sr 97 Y 97 Zr, 6

7 the long-lvng sotopes of Zr are formed, whch was observed n [4]. The results of such sotopc transformatons are schematcally llustrated n Fg. 3, whch presents the evoluton of the mass spectra shape, for example, for 28 Pb sotope fragment fsson yelds versus the possble - - decay chan length. Y Y Y A A A a) b) c) Fg. 3 Change of shape of the mass fsson fragments yelds of the 28 Pb had been obtaned wthout, - (a) and consderng the possblty of emsson 3, - (b) or 5, - (c) - partcles durng the conversons of sotopes. Summng up, the research results allow us to expect the yeld and peculartes of the pre-actnde materals fsson products transformatons that are mportant for nuclear-power engneerng, especally, for subcrtcal assembles reactors. The analyss shows that for a wde number of pre-actnde nucle the competton of unflled neutron shells N = 5 and N = 82 s mportant and determnes the peculartes of fsson-fragment yelds from asymmetrc (double -humped) to sngle-humped dependence. 4. Acknowledgements Ths work was carred out wthn the framework of the Program of the Natonal Academy of Scences of Ukrane, Project К-5-27/214. REFERENCES 1. Drake T.E., Pa H.L., Nascmento I. The electrofsson of 28Pb and 29B Nuclear Physcs A259 (1976) O.A. Zhukova, A.V. Ignatjuk, M.G. Itks dr. Delene doaktndnyh jader. Reakcja (p,f) v oblast jenergj protonov 24-3 MeV. Jadernaja fzka 1977, t.26, 3, s. (n russan) 3. M. C. Dujvestjn and A. J. Konng J. P. M. Bejers, et al Proton-nduced fsson at 19 MeV of nat W, 197 Au, nat Pb, 28 Pb, and 232 Th Phys. Rev. v. 59, # , Haladhara Nak, Gunyun Km, Ashok Goswam, et al Mass yeld dstrbutons of fsson products from photo-fsson of nat Pb nduced by 5 7 MeV bremsstrahlung J Radoanal Nucl Chem (21) 283:

8 5. H. Nak, S. Ganesan, Manwoo Lee and Gunyun Km Yelds of Fsson Products n the Gamma Induced Fsson of Preactndes and Actndes Journal of the Korean Physcal Socety, Vol. 59, No. 2, August 211, pp. 1848_ M. Bender, P. Bonche, T. Duguet and P.-H. Heenen Confguraton mxng of angular momentum projected self-consstent mean-feld states for neutrondefcent Pb sotopes Phys. Rev. C 69, 6433 (24) 7. L-Gang Cao, H. Sagawa, G. Colo Effects of tensor correlatons on low-lyng collectve states n fnte nucle Phys. Rev. C 83, (211) 8. O.S. Shevchenko, Yu. M. Ranyuk, A.M. Dovbnya et al, Investgaton of nat Pb photofsson Uzhhorod Unversty Scentfc Herald. Seres Physcs. Issue , c. 9. V. T. Maslyuk, O. A. Parlag, O. I. Lendyel et al New Statstcal Methods for Systematzng the Nucle Fsson Fragments: Post-Scsson Approach Journal of Modern Physcs, 213, 4, N. Wang and M. Lu, Nuclear mass predctons wth radal bass functon approach, Phys. Rev. C 84, 5133(R) (211). 8