Pöschl Teller model for the total cross-section of neutron scattering from 232 Th

Transcription

1 Indian Journa of Pure & Appied Physics Vo. 50, May 01, pp Pösch Teer mode for the tota cross-section of neutron scattering from 3 Th J Joseph Jeremiah & B M Jyrwa* Department of Physics, North Eastern Hi University, Shiong 793 0, Meghaaya *E-mai: bjyrwa90@hotmai.com Received 14 November 011; revised 19 January 01; accepted 15 March 01 Neutron scattering cross-section of 3 Th have been investigated using an attractive potentia. On appying the modified Pösch-Teer mode, the tota cross-section of the n+ 3 Th in the energy range 5-0 MeV has been cacuated. It was compared with the avaiabe experimenta data and evauated data of JENDL-4.0, ENDF/B-VII.0 and CENDL-3.1 as we as with the theoretica vaue from TALYS-1. Nucear Reaction Program, EMPIRE:.19 Nucear Reaction Mode Code and are found to be in reasonaby good agreement. This supports the vaidity of the present cacuation. Keywords: Tota cross-section, Anguar distribution, Pösch-Teer potentia, Evauated data of JENDL, CENDL, ENDF, TALYS, EMPIRE, Neutron scattering cross-section 1 Introduction Scattering of nuceus is an extremey usefu study as an introduction to nucear theory. It gives us information about the strength and range of nucear forces, structure of the nuceus and magnetic properties of the nuceus. It is aso usefu to investigate the nature of the wave function after scattering 1,. Schrödinger time independent equation provides an essentia too for studying and investigating the phenomenon of scattering 3,4. The eastic and reaction cross-section can be cacuated from the knowedge of nuceon-nuceus potentia v(r) and then the properties of interacting nucei can be determined. Thus, it is necessary to examine the appropriate form of the potentia, so that the wave equation gives cross-section that can be compared with the experimenta data. When the nucear potentia has been chosen, the corresponding observabe quantities can be cacuated from it. So a that can be done is to find out the optica potentia that gives a good fit to a given set of data and to cacuate for a series of tria parameters unti a satisfactory fit is obtained. Now that a arge number of optica mode anaysis have been done for many different types of interactions 5, it is usuay possibe to make quite a good guess for the parameters of the first tria potentia, we have started with parameters obtained from Ref. (6). Optica Mode cacuations pay roe in evauating nucear data for scattering and reaction. So reiabe nucear parameters are in demand to achieve correct parameterization and evauate the covariance of the parameters, as we as quantities ike cross-section, anguar distribution and poarization. In the optica mode anaysis, two basic approaches have so far been foowed such as (i) Phenomenoogica 4 (ii) Microscopic approximation 5. In the Phenomenoogica approach, we define empirica form of the nucear potentia with the associated parameters. The optica potentia parameters are suitaby varied so that the cacuated cross-section agrees best with the experimenta data. The parameters have been determined by adopting the minimization of χ fit. Phenomenoogica methods mediate between experimenta data and basic theory. In the present work, we use Pösch-Teer potentia based on the framework of the Optica Mode. With this approach, we cacuated the systematic contents of measurements. The methodoogy was formuated by means of simpe concepts and assumptions we within the genera framework of quantum mechanics. In nucear interactions, experiments provide us with cross-sections and transition probabiities in the form of energy eves. These quantities are determined by the asymptotic behaviour of the wave functions. For scattering processes, a reevant information is contained in the eements of the scattering matrix. The atter and quantities reated to it wi form the genera framework for the phenomenoogica method to be discussed in this work. The present cacuation has been carried out to study neutron scattering crosssection of 3 Th. Cacuation of Observabes for an Assumed Potentia We have used Pösch-Teer potentia 7 taking into account where the form factor differs from that used in Optica Mode Potentia 4. The Schrödinger

2 90 INDIAN J PURE & APPL PHYS, VOL 50, MAY 01 equation appropriate to the eastic scattering of reduced mass µ and incident kinetic energy E in the centre of mass system is given by: µ h ( E v r ) ψ + ψ = m ( ) 0 µ E = Eab = Eab m1 + m m1 v(r)s the potentia. From partia wave anaysis, we have (1) () u ( r) Ψ( r ) = P (cos θ) (3) r where u ( r ) is the radia wave function for the partia wave. Essentiay u ( r ) satisfies the equation: d u µ + ( ) dr ( 1) + E v( r) u = 0 h r th (4) For convenience, we make the transformation µ E ρ = kr where k =, is the wave number and the h ( ap at ) reduced mass µ =, where a p and a t are the ( a + a ) p t mass of the projectie and target, respectivey. Thus, the radia equation becomes: d u v ρ + ( ) ( 1) + 1 u = 0 dρ ρ ρ The optica potentia 4 is, generay, written as: (5) where α is the width parameter and the pot of this form is shown in Fig. 1. We have the Schrödinger equation for each vaue to get the tota anguar momentum, since the spin of the incident partice is couped, and the eigen vaue.σ of corresponding to these two spin orientations are and (+1), so consequenty the Schrödinger equation for each vaue, resuts into a pair of equations for the corresponding radia wave functions. + ( ρ) ( ρ) ( ρ) ( ρ) dρ E E E d u V f W f W g + 1 v i v i s and CsoVso h( ρ) ( + 1) + u ( ) 0 ρ = E ρ (8) d u ( ρ) V f ( ρ) W f ( ρ) W g( ρ) + 1 v i v i s dρ E E E CsoVsoh( ρ) ( + 1) + ( + 1) u ( ) 0 ρ = (9) E ρ Essentiay u + and u are the wave functions for the two spin orientations. The soutions of the Esq (8 and 9) have both the rea and imaginary components, since the potentia considered is a compex one. At sma vaue of ρ, within the nucear potentia, the wave function has an osciatory form with short waveength, whie outside the nucear potentia the wave function exhibits increased wave ength as expected. The radia Schrödinger equation is soved by dividing the range of ρ into an interior region (ρ ρ m ) and an externa region (ρ ρ m ). The soution of the Schrödinger equation was carried out using Fox- Goodwin method 8, The initia vaues for sma ρ were obtained by a series expansion in powers of ρ. This v( r) = Vv f ( r) + iwv f ( r) + iws g( r) +. scsov soh( r) (6) where the four terms V v, W v, W s and V so are the rea voume potentia, the imaginary voume potentia, the imaginary surface potentia and the spin orbit potentia, respectivey. The quantities f(r), g(r) and h(r) are the form factors. The rea potentia has a Pösch-Teer 7 from factor. f ( r) = V v cosh ( αr) (7) Fig. 1 Form of Pösch-Teer potentia

3 JEREMIAH & JYRW: PÖSCHL TELLER MODEL 91 was carried out to a matching radius ρ m. Beyond this matching radius, the nucear fied is negigibe. The wave function (=0) for spin up is shown in Fig.. The phase shift δ of the th partia wave is computed by fitting the radia wave function R (ρ) for ρ a, which may have an anaytic form and can aways be found anayticay. The boundary condition for ρ=a is that 1 dr k shoud be continuous. R dρ γ is the ratio of the sope to the interior wave function, j, n, j & n are the spherica Besse function and Neumann function 9 aong with their derivatives, respectivey. + + k j ( ka)cos δ n ( ka)sin δ + + = γ j ( ka)cos δ n ( ka)sin δ k j ( ka)cos δ n ( ka)sin δ = γ j ( ka)cos δ n ( ka)sin δ + (10) (11) down. Transmission coefficient for spin up is given in Fig. 4. The tota cross-section is given by: π + σ = {( + 1)[1 R( η )] + [1 R( η )]} (14) T k = 0 The tota cross-section for the energy range 5-0 MeV is shown in Fig. 5. The shape-eastic anguar distribution is given by: d σ E = A( θ ) + B( θ ) dω (15) where the scattering ampitudes A(θ ) and B(θ ) are defined by: i + A( θ ) = ( 1)(1 ) ( ) P (cos ) k + η + η θ = 0 (16) δ The parameter can be used to evauate the Scattering matrix η and Transmission coefficient T. + + η ( ) = exp( iδ ) & η ( ) = exp( iδ ) (1) T & T 1 = η = η (13) Consequenty, we used the Scattering Matrix and Transmission Coefficient to cacuate the physica quantities or observabe 10 ike differentia crosssections and tota cross-section. The behaviour of the Scattering matrix for spin up is given in Fig. 3. We have a simiar case for spin Fig. 3 Scattering matrix for spin [j=+(1/)] Fig. Wave function (=0) for spin [j=+(1/)] Fig. 4 Transmission coefficient for spin [j=+(1/)]

4 9 INDIAN J PURE & APPL PHYS, VOL 50, MAY 01 and B i (17) + 1 ( θ ) = ( η η ) P (cos θ) k = 0 Shape-eastic anguar distribution with respect to ange for 15. MeV is shown in Fig Resuts and Discussion The tota cross-section of n+ 3 Th is shown in Fig. 5 and shape-eastic anguar distribution is shown in Fig. 6. The tota cross- section and shape-eastic anguar distribution obtained by using the Pösch- Teer potentia show a reasonabe agreement with that of the experimenta data obtained from EXFOR data base 11,1, the nucear reaction program 13 TALYS-1., the nucear reaction mode 14 code EMPIRE:.19 and the Evauated nucear data fies ENDF (Ref. 15), CENDL ( Ref. 16), JENDL (Ref. 17). Fig. 5 Tota cross- section of n+ 3 Th for the energy range 5-0 MeV Fig. 6 Shape-eastic anguar distribution of n+ 3 Th with respect to ange for 15. MeV EXFOR is the exchange format designed to aow transmission and exchange of nucear reaction data between the Nucear Reaction Data Centres (NRDC). In addition to storing the data and its bibiographic information, experimenta information is aso compied. The source of the data and history of the data set are aso incuded. It was originay conceived for the exchange of neutron data, was deveoped through discussion among personne from centres situated in Sacay, Vienna, Livermore and Brookhaven. Athough users may obtain data from the centres in the EXFOR format, other centre-to-user formats have been deveoped to meet the needs of the users within each centre s own sphere of responsibiity. The EXFOR format, as outined, aows a arge variety of numerica data tabes with expanatory and bibiographic information to be transmitted in a format: that is machine-readabe (for checking and indicating possibe errors); that can be read by personne (for passing judgement on and correcting errors). The data presenty incuded in the EXFOR exchange fie incude: a compete compiation of experimenta neutron-induced reaction data, a seected compiation of charged-particeinduced reaction data, and a seected compiation of photon-induced reaction data. TALYS is a computer code system for the anaysis and prediction of nucear reactions. TALYS, namey is a nucear data too: Either in a defaut mode, when no measurements are avaiabe, or after fine-tuning the adjustabe parameters of the various reaction modes using avaiabe experimenta data, TALYS can generate nucear data for a open reaction channes, on a user-defined energy and ange grid, beyond the resonance region. The nucear data ibraries that are constructed with these cacuated experimenta resuts provide essentia information for existing and new nucear technoogies. Important appications that rey directy or indirecty on data generated by nucear reaction simuation codes ike TALYS are: conventiona and innovative nucear power reactors (GEN-IV), transmutation of radioactive waste, fusion reactors, acceerator appications, homeand security, medica isotope production, radiotherapy, singe-event upsets in microprocessors, oi-we ogging, geophysics and astrophysics. The basic objective behind its construction is the simuation of nucear reactions that invove neutrons, photons, protons deuterons, tritons, 3 He- and apha-partices, in the 1 kev-00 MeV energy range and for target nucides of mass 1 and

5 JEREMIAH & JYRW: PÖSCHL TELLER MODEL 93 heavier. TALYS has been extensivey used for both basic and appied science. EMPIRE is a moduar system or nucear reaction codes, comprising various nucear modes, and designed for cacuations over a broad range of energies and incident partices. A projectie can be any nuceon or Heavy Ion. The energy range starts just about the resonance region, in the case of a neutron projectie, and extends up to few hundred MeV for Heavy Ion induced reactions. A comprehensive ibrary of input parameters cover nucear masses, optica mode parameters, ground state deformations, discrete eves and decay schemes, eve densities, fission barriers, moments of inertia and Gamma- ray strength functions. A dimensions in the main code are set up through the parameter statements contained in the separate fie, which were incuded wherever appropriate, making any adjustment of the code to the actua probem and/or computer straightforward. The code has a moduar structure. Each modue performs a we defined task and communicates with other modues through a set of goba COMMONS which are incuded in most of the sub-routines. The ENDF system was deveoped for the storage and retrieva of evauated nucear data to be used for appications of nucear technoogy. These appications contro many features of the system incuding the choice of materias to be incuded, the data used, the formats used and the testing required before a ibrary is reeased. An important consequence of this is that each evauation mush be compete for its intended appication. If the required data are not avaiabe for some particuar reactions, the evauator shoud suppy them by using systematic or nucear modes. ENDF/B ibrary maintained at Nation Nucear Data Centre (NNDC) contains the recommended evauation for each materia. ENDF/B data sets are revised or repaced ony after extensive review and testing. This aows them to be used as standard reference data during the ifetime of the particuar ENDF/B ibrary version. The ENDF formats and ibraries are decided by the Cross-Section Evauation Working Group (CSWEG), a cooperative effort of nationa aboratories, industry, and universities in the U S and Canada, and are maintained by the Nationa Nucear Data Centre (NNDC). ENDF format provides representations for neutron cross-sections and distributions, photon production from neutron reactions, a imited amount of charged-partice production from neutrons reactions, photo-atomic interaction data, therma neutron scattering data, and radionucide production and decay data (incuding fission products). JENDL-4.0, the fourth version of the Japanese Evauated Nucear Data Library has been produced in cooperation with the Japanese Nucear Data Committee, which contains neutron-induced reaction data for 406 nucides, in the neutron energy range from 10-5 ev to 0 MeV. Two nucear mode codes were deveoped in order to evauate the crosssections of fission products and minor actinides. Couped-channe optica mode parameters, which can be appied to wide mass and energy regions, were obtained for nucear mode cacuations. Therma cross- sections of actinides were carefuy examined by considering experimenta data or by the systematic of neighbouring nucei. The purpose is to provide a Japanese standard ibrary for fast breeder reactors, therma reactors, fusion neutronics and shieding cacuation and other appications. CENDL-3.1, the updated Chinese Evauated Nucear Data Library is based on the recent evauation of nucear data by the China Nucear Data Centre (CNDC) in cooperation with the China Nucear Data Coordination Network (CNDCN). This incudes the evauated data for the reactions induced by neutron with energy up to 0 MeV on 40 nucides from H-1 to Cf-49. A these data were obtained according to the evauation of experimenta data and theoretica predictions. The vaidity of the most important nucides in this ibrary has been verified by performing the benchmark integra experimenta test. From genera properties of nucei and microscopic optica mode cacuations, one can estimate the aowed range for the parameters determined in a phenomenoogica approach. The fitting procedure basicay consists of the probem of minimizing the quantity used in statistics 18 caed χ is frequenty defined as: N exp th 1 χi χ i χ = exp N m i = 1 χi (18) where χ i is any physica quantity such cross-section, differentia cross-section. N is the number of data and is the number of parameters. This is purey a phenomenoogica approach as the parameters may be adjusted freey, starting from the experimenta data. The parameters used in the mode are presented in Tabe 1.

6 94 INDIAN J PURE & APPL PHYS, VOL 50, MAY 01 E ab Tabe 1 Parameters used in the mode Vv Wv Ws Vso χ Concusions The present cacuations are found to be in exceent agreement with the experimenta data, which means that the parameters are appropriate for cacuation of transmission coefficients and various observabe quantities ike cross- sections and anguar distributions. The present cacuation of neutron scattering crosssection using the modified Pösch-Teer approach, gave us the motivation to cacuate neutron scattering with other ong ived actinides used in Acceerator Driven Sub-critica 19 System (ADSS) and to extend to higher energies up to 00 MeV in future, the minimized χ indicates that the parameters chosen are quite suitabe for reproducing the experimenta data with the minimum errors. 3 Th has been chosen because it is a ong ived actinide used in Advance Heavy Water 0 Reactor (AHWR) and aso in ADSS. The study of neutron scattering from 3 Th is vita for designing the AHWR and ADSS. It is necessary to have knowedge about accurate nucear data of heavy and ong ived actinides. Therefore, the semi-empirica cacuations and modes are usefu for understanding the neutron scattering from a ong ived actinide ike 3 Th. Energy range up to 0 MeV is very reevant in the case of ADSS since 80-90% of the neutrons eaving the spaation target have energies in this range. Since empirica data are not enough, evauators depend more in semi-empirica cacuations and mode to fit the cross- sections. In India, AHWR and ADSS in the word are important from the fact of 3 Th utiization as more efficient use of 38 U in the generation of nucear power. India has a program to deveop subcritica systems and coupe to externa source of neutrons with a view to augment the thorium utiization programme 1. Acknowedgement We are extremey gratefu to Dr H Naik from Radio- Chemistry Division, BARC for giving us constructive criticism and usefu suggestions. We are highy indebted to him. References 1 Jackson D F, Nucear Reactions (Methuen, London), Fröbrich P & Lipperheide R, Theory of Nucear Reactions (Carendon Press), Feshbach H, Porter C E & Weisskopf V F, Phys Rev, 96 (1954) 448; Wat M & Bavrche J, Phys Rev, 93 (1954) 106; Fumer C B, Phys Rev, 15 (196) Hodgson P E, The Optica Mode of Eastic Scattering (Carendon Press, Oxford), Hodgson P E, Introductory Nucear Physics (Oxford University Press), Koning A J & Dearoche J P, Loca and goba nuceon optica modes from 1keV to 00 MeV, Nuc Phys A, 713 (003) Fugge S, Practica quantum mechanics (Springer, nd edition), Schmid W, Theoretica Physics on the Persona Computer (Springer-Verag), Schiff I, Quantum Mechanics (Tata McGraw-Hi Education, 3 rd edition), Bersion O, The Computer Code SCAT-, Proc Workshop in appied Nucear Theory and Nucear Mode Cacuation for Nucear Technoogy Appication, p 319, Word Scientific (1990). 11 Abfaterer W P, Bateman F B, Dietrich F S, Finay R W, Haight R C, Morgan G L, Phys Rev C, 63 (001) Hudson C I, Waker W S & Berko S, Phys Rev, 18 (196) Koning A J, Hiaire S and Duijvestijn M C, TALYS-1.0, Proceedings of the Internationa Conference on Nucear Data for Science and Technoogy, Apri -7, 007, Nice, France, editors Bersion O, Gunsing F, Bauge E, Jacqmin R, and Leary S, DEP Sciences, p 11 (008). 14 Herman M et a., EMPIRE: Nucear Reaction Mode Code System for Data Evauation, Nuc. Data Sheets 108, (007). 15 Chadwick M B et a., ENDF/B-VII.0, Next Generation Evauated Nucear Data Library for Nucear Science and Technoogy, Specia Issue on Evauated Nucear Data Fie ENDF/B-VII.0 Nucear Data Sheets, 107(1), (006). 16 Ge Z G, Zhuang Y X, Liu T J, Zhang J S, Wu H C, Zhao Z X, Xia H H, The Updated Version of Chinese Evauated Nucear Data Library (CENDL-3.1), Proc. Internationa Conference on Nucear Data for Science and Technoogy, Jeju Isand, Korea (010). 17 Shibata K, Iwamoto O, Nakagawa T, Iwamoto N, Ichihara A, Kunieda S, Chiba S, Furutaka K, Otuka N, Ohsawa T, Murata T, Matsunobu H, Zukeran A, Kamada S, and Katakura J: JENDL-4.0: A New Library for Nucear Science and Engineering, J Nuc Sci Techno 48 (1) (011). 18 Scarborough J B, Numerica Mathematica anaysis (Johns Hopkins Press), Ganesan S, Pramana-J Phys, 68 (007) Naik H, Reddy A V R & Mauchanda V K, Yieds of fission products in neutron induced fission on heavier actinides, BARC Report, BARC/008/E/ Banerjee S, Sinha R K & Kaias S, Thorium utiization for sustainabe suppy of nucear energy, Journa of Physics, Conference Series, 31 (011) 0600.