Methodology of Projection of Wave Functions of. Light Nuclei on Cluster Channels on

Transcription

1 Advanced Studies in Theoretical Physics Vol no HIKARI td Methodology of Proection of Wave Functions of ight Nuclei on Cluster Channels on the Exaple of Quantu B{- 7 i}-syste D.A. Tursynbayeva R.S. Kabatayeva M.A. Zhusupov N.A. Burkova K.A. Zhaksybekova F.B. Belissarova A.S. Taukenova and G.B. Alibekova Departent of Methodology of Teaching of Matheatics Physics and Coputer Science Abay Kazakh National Pedagogical University Alaty city Kazakhstan Departent of Theoretical and Nuclear Physics Al-Farabi Kazakh National University Alaty city Kazakhstan Copyright 06 D.A. Tursynbayeva et al. This article is distributed under the Creative Coons Attribution icense which perits unrestricted use distribution and reproduction in any ediu provided the original work is properly cited. Abstract In the present paper the authors investigate the cluster structure of B nucleus by a ethod of proection of its three-body wave function on the cluster channel 7 i +. An estiation of the wave function of B nucleus in the three-body todel on the cluster channel 7 i t + has been obtained. It is shown that the account of only one configuration in the wave function of В nucleus does not describe copletely the cluster structure of this nucleus. Keywords: light nuclei cluster structure any-particle shell odel wave function B 7 i proection Introduction The proection includes several steps of transforations knowledge and ability of which are necessary for investigations of the light nuclei. For estiation of the wave function of В nucleus let s proect the wave function of this nucleus in the

2 90 D.A. Tursynbayeva et al. three-body t-odel on the cluster channel 7 i +. The nuclei under consideration have the following quantu nubers of spin parity and isospin in the ground state (fig. ) []: 3 B gs.. ; 7 3 i gs.. ; 0 ; ; 0. B t R r r R y y 7 i t Figure Relative Jacobi coordinates for the channel В 7 i + According to the any-particle shell odel [] the wave function of the ground state of the В nucleus has the configuration (s) 4 (p) 7 that is it contains N = 7 quanta of excitation when decaying by the channel: В 7 i ((s) 4 (p) 3 ) + ((s) 4 ). There are N = 4 quanta for the relative otion of nuclei in the final state and the wave function of the relative otion has the shell R4 for. The radial wave function of B nucleus has two coponents: R40 S and R4 D. For the realization of the procedure of proection it is necessary to know the forulas of transition fro the set of y -coordinates when 7 i nucleus and -particle are given to the set of Rr -coordinates when the В nucleus coordinates are given (fig. ) and vice-versa: r r r r r t 4 3 () R 4 r 4 r r t y r r. t r Then the transition fro one Jacobi coordinates to another is realized by the forulas: r R 3 y r R r y 7 R y. 4 ()

3 Methodology of proection of wave functions 9 Wave functions et s write down the relative radial wave function of the B t nucleus [3]: Then the total wave function of B t l r R N r R C r R B exp. (3) nucleus has the for: 00 3 t 00 B t M M M Y r Y R 00 M t t t N C exp r R. (4) For the proection of the wave function of B t channel 7 it nucleus on the cluster one needs to calculate the overlapping integral: 7 7 y r R d. i B (5) i B et s write down the total wave function of the 7 i nucleus in the two-body odel: 3 7 t i t M M t t t t i ai i Ae Y M where the coefficients of expansion of the relative function are taken fro [4]. Now let s substitute the expressions (4) and (6) into the expression (5): y tm 3 00 M tm 3 M M t 00 M t ai c r d R AC i e Y M Y00 r Y ; R d (7) l i (6) 3 Method of diagonalization of the squared for et s diagonalize the squared for on the exponent in the expression (7). Firstly let s transfor the for with account of the transforations (): 9 6 h ai c d c d y c d y (8)

4 9 D.A. Tursynbayeva et al. Change of variables and new denotations: x y y. y 9 f ai c d f c d 4 6 f3 c d. 7 4 (9) Then with account of the expression (9) the expression (8) will take the for: h f x f f yx f f f y. (0) 3 3 For the diagonalization it is necessary to put the coefficient at the crossing ter to be equal to zero: f3 f f3 0 =. () f With account of the expression () let s find the coefficient at the third ter: f 3 f3 f f f. () 4 f Now let s substitute the expression () into the expression (0): f3 q f h fx qy. (3) 4 f et s express R through x and y. For this let s use the expression R y 4 fro () and substitute x y fro (9) in it: R x y. (4) Transforation of the spherical functions For the transforation of the expressions for M table forulas fro [5] then one obtains: Y R and Y l M let s use the Y R l Y x Y y Y00 x Y y 4 4 (5) Further let s use the forulas for the Clebsch-Gordan coefficients [5] and the expressions of the spherical function will take the for: Y R Y x Y y 4 Y Y x Y y f3 where (6) f M M M

5 Methodology of proection of wave functions 93 5 Calculation of the integral with respect to variable et s write down without account of Clebsch-Gordan coefficients algebra the separate integral fro the expression (7) with account of forulas (6): fx qy y e Y M x Y. M y Y x Y y d (7) et s use the change of variables (9) x y and taking into account that x 4 d dx x dx d substitute it into the expression (7): x M M Ml 4 4 Y M x Y. y Y M y Y y (8) qy fx y e e x dx d Y x Y x Y y Y x et s consider separately the integral of the spherical functions fro the expression (8): I x d x Y M x Y x Y M y x d x Y x 4 4 yy x d Y y Y Y d (9) y x M x M y y x. Further let s use the table forulas [5] and the expression (9) will take the for: M I x M 4 y Y. M y Y y (0) 4 Now let s use the table forula [5] for the product of two spherical vector functions and obtain for the expression (0): M I x M 4 y M M Y M y. () M 4 Having transfored it one obtains: M 3 I x M 4 y M M. Y M () y 4 4 M Now let s return to the expression (8) with account of the forula (): qy fx 4 y e M e xdx 4 y M fx M M YM y e xdx. (3) M

6 94 D.A. Tursynbayeva et al. Then the expression (3) with account of the table integrals will take the for: 3 y y e qy 33 5 M f M 3 M M YM y f M (4) Now let s write the expression (7) with account of the forula (4): y AC i tm 3 4 i M t 00 M t M t M M y (5) 6 Transforation of the algebra of the Clebsch-Gordan coefficients et s substitute the expression (4) into the forula (5) and represent the obtained expression in the following for: qy y AC i e I I (6) 4 i where 33 I 5 M t M M t M M (7) f M 00 t M t 3 y I = M 3 00 M M 3 M t t 3 f M t 00 M t M M Y (8) M M y M Having transfored the product of the Clebsch-Gordan coefficients in the expressions (7) and (8) let s write down the final for of the wave function (6) with account of the last two expressions: qy y AC i e I I 4 i where 3 f I 33 (9) 5 t t M f 3 y I M M M M 4M M 3 Y M 3 y. (30)

7 Methodology of proection of wave functions 95 The quantu nubers can take the following values: 00 M M M. The expression (9) and the case for the expression (30) give us the contribution into the S -coponent of the wave function. The case for the expression (30) is the D -coponent of the wave function. et s consider the separately M 3 M 00 f y M y I Y. (3) Then the radial S -coponent of the wave function will take the for: qy 33 y S AC i e. 5 t t M 3 M 4 i f f (3). M M. y I 3 M 3 M Y (33) M M 4M M y 3 f Then the radial D -coponent of the wave function will take the for: qy y D AC i e 3 3 M 3 M Y M y. 4 i f (34) Then the coponents of the radial part of the wave function are obtained fro the expressions (3) and (34) without angular functions Y M y since the wave function of В nucleus has the for: where M y 4 B i M M i Y R y. 7 7 M S40 33 AC y e qy i 5 3 i 4 f 4 f (35) D4 AC y e 4 4 f f3 q f f 3 4 f qy i (36) 3 i f 9 f ai c d f c d 4 6 f3 c d. 7 4

8 96 D.A. Tursynbayeva et al. 7 Nuerical calculations In the figure there are represented the S- and D-coponents of the radial wave functions of the В nucleus calculated by the forulas (35) and (36) in the range of 0 to 5 feri. The radial wave function of the В nucleus in the three-body todel is proected on the cluster channel 7 i t} +. 5x0-5 0x0-5 5x0-5 0x0-5 50x0-6 R x0-6 y f Figure The radial part of the wave function of В nucleus: dashed line S-coponent solid line D-coponent The aplitudes of the S- and D-coponents of the radial wave function of the В nucleus are turned out to be sall by values this eans that in this case the account of only one configuration in the wave function of the В nucleus exactly the t} configuration having the Young schee [443] is not able to describe well the wave function of this nucleus. Since the weight of this coponent in the wave function of the ground state of the В nucleus in the any-particle shell odel is not ore than 40 % [] the account and contribution of the coponents with Young schees [44] [433] is turned out to be iportant. A calculation with account of such configurations presents an interest and is the subect of the future investigations for the authors. Acknowledgeents. This investigation was financially supported by the Ministry of Education and Science of the Republic of Kazakhstan grant 3094/GF4. References [] D.R. Tilley et al. Energy levels of light nuclei A = 7 Nuclear Physics A 708 (00) no. 3.

9 Methodology of proection of wave functions 97 [] А.N. Boyarkina Structure of p-shell Nuclei Moscow Moscow State University 973. [3] S.B. Dubovichenko A three-body odel of the B nucleus Journal of Experiental and Theoretical Physics 3 (0) no [4] S.B. Dubovichenko Astrophysical S factors of radiative 3 He 4 He 3 H 4 He and H 4 He capture Physics of Atoic Nuclei 73 (00) no [5] D.А. Varshalovich A.N. Moskalev V.K. Khersonskii Quantu Theory of Angular Moentu eningrad Science 975. Received: February 7 06; Published: February 4 06