POTENTIAL REPRESENTATION METHOD FOR WOODS-SAXON POTENTIAL

Transcription

1 iauliai Math. Semin., 5 (13), 2010, 4145 POTENTIAL REPRESENTATION METHOD FOR WOODS-SAXON POTENTIAL Arvydas Juozapas JANAVIƒIUS, Donatas JURGAITIS iauliai University, P. Vi²inskio str. 19, LT iauliai, Lithuania; A.Yanavy@gmail.com, pletra@cr.su.lt Abstract. In this paper, the analytical solutions of the Schrödinger equation for Woods-Saxon potential are obtained for the bound states (negative energies case for attractive potential). Taking into account that analytical solutions in the case of positive energies are also found, we can nd the connection between scattering experiments, parameters of scattering potential and one-nucleon energy levels of nucleus. Key words and phrases: analytical solutions, dierential equation Mathematics Subject Classication: 00A71, 34E10, 00A Introduction The method of potential representation for a solution of the Schrödinger equation was proposed in [7], [8] for nucleus scattering calculations, using optical model [5]. The main feature of that method is an expression of the wave functions or solutions of the Schrödinger equation like functions of the potential [7]. In this case, analytical solutions of Schrödinger equation for nucleus scattering which give a possibility to express experimental results analytically by parameters of nucleus interaction potentials, were obtained. Using this method, a new theorem [8] for multiplicative perturbation theory was presented. According this theorem, when a new potential to the Hamiltonian is added, the solution can be obtained multiplying a previous solution by the function depending on this potential. This theorem was applied [6]

2 42 A. J. Janavi ius, D. Jurgaitis for the Coulomb and short range potentials, using a modied method of undetermined Lagrange coecients [4]. The presented theory can be used for a wide class of potentials, but in our investigations we mostly apply Woods- Saxon potential which is very useful for nuclear reactions rates calculations in reactors, and for investigation of astrophysical phenomenon [1]. In [7], we presented the analytical solution u(r) of the radial Schrödinger equation d 2 ( u dr 2 + k 2 L(L + 1) r 2 V (r) = V 0 ( 1 + exp { r R a ) cv (r) u = 0, k 2 = ce, c = 2m 2, }) (1) for the Woods-Saxon potential V (r) with potential parameters V 0, a, R, where m is a mass of nucleon (proton or nucleon). Using the Woods-Saxon potential V (r) instead of a variable and substituting u(r) = φ(v (r)) exp{ ikr} into (1), we obtain, for the case L = 0, the following dierential equation V Vd 2 d 2 φ(±k, V ) dv 2 + (Vd 2 + V d(v 2ikaV 0 )) dφ(±k, V ) dv ca 2 V0 2 φ(±k, V ) = 0, V d = V (r) V 0. (2) Note that in quantum mechanics scattered waves with orbital quantum number L = 0 are called s-waves. Assuming that a solution of (2) satises the standard boundary condition [7] we can express it in the following way lim φ(±k, V ) = 1, (3) r φ(±k, V ) = V n d n. (4) n=0 After substituting (4) into (2), the following recursion relation d n+1 = (ca2 V0 2 + n(2n + 1 ± 2ika)V 0)d n (±k, V ) (n 1)nd n 1 (n + 1)(n + 1 ± 2ika) V 2 0 was obtained [7]. The solution (4) satises the boundary condition (3) when d 0 = 1, and can dene the useful expression of scattering matrix S(k) L=0 = φ(+k, V p) φ( k, V p ), lim V (r) = V p, (5) r 0

3 Potential representation method for Woods-Saxon potential 43 by which we dene scattering cross sections [7], or scattering probabilities [8] of the colliding nucleus. The solutions (4) expressed by Taylor's series, according to the Poisson integral representation [2], converge and exist in the region 0 V (r) V p. The calculated values of scattering matrix for scattering of neutron from dierent nuclei in a wide range of energies were compared [7], [8] with optical model [7] of the program Local Optical Program with Automatical Parameters Search (F. G. Perey, Krakow Computing Center, 1978) results. This program was used for analysis of scattering experiments, and very good coincidences have been obtained. But the presented analytical solution [6] and the expression of scattering matrix (5) was obtained only for s-waves when L = Analytical solution for bound states We shall consider solutions of the radial Schrödinger equation (1) for Woods- Saxon potential for negative discrete energies E < 0 (proper values or eigenvalues) with L = 0, 1, 2,... by changing k 2 in equation (1) by k 2. The eigenfunctions of equation (1) of bound states with discrete energies E < 0, representing a particle (one nucleon) located in the denite region, must satisfy the following boundary conditions lim u(r) = 0, lim r 0 and must be represented by [9] u(r) = 0, r u(r) = r L+1 exp{ kr}w(r). (6) Substituting (6) into equation (1) and using a new variable y = r a = R ( ) a + log V0 V (r) 1, we obtain the dierential equation ( F (y) y d2 w(y) dy 2 + 2(L + 1 kay) dw(y) dy cv 0 a 2 yw(y) { = 0, F (y) = 1 + exp R a ) 2ka(L + 1)w(y) } exp{y}. (7) The last dierential equation is obtained by using the fact that, for bound states E < 0, and substituting (6) into (1). From the obtained equation d 2 w(r) dr 2 + 2(L + 1 kr) dw(r) dr (2(L + 1)k + crv (r))w (r) = 0,

4 44 A. J. Janavi ius, D. Jurgaitis introducing the variable y instead of r, we obtain equation (7). We can represent the solution w(y) by power series w(y) = b n y n, (8) n=0 which must be terminated [9] at some nite maximum power N. Only in this case, eigenfunctions which exist in the region 0 y can be found. Substituting the last expression in equation (7) and expanding in the power series (7), after equating to zero coecients of y n, we obtain the recurrent relations e 0 n(n + 2L + 1)b n = n = 0, 1, 2,..., N, where F (y) = N e m y m, m=0 n 1 m=0 2ak(m + 2k + 2)e n m 1 b m + cv 0 a 2 b n 2, (9) { e 0 = 1 + exp R } a, e m = exp { R a m! }, (10) m = 1, 2,..., N. Taking in the care that in equation (9) must be b 1 = 0, b 2 = 0, we obtain that the power series begins from constant term b 0. Now all expansion coecients for solution (8) can be obtained from (9), (10). For getting eigenfunctions (6) from the power series (8), they must be abbreviated, and the coecients b N, for freely chosen N = 2, 3, 4..., must be equal zero. In this case, we transform solution (6) to eigenfunctions u NL (r) with eigenvalues E NL dened by the quantum numbers N, L. However, calculation of eigenvalues k 2 by terminating of power series (8) at some nite maximum power [9] in analytical form from recurrent relations (9) is complicated, and some additional investigations must be provided. 3. Conclusions In this paper, the analytical expressions of eigenfunctions (8) for Woods- Saxon potential are obtained using potential representation method. In this case, eigenfunctions can be expressed by potential and are represented by polynomials with coecients satisfying presented recurrent relations. We showed that a potential representation method for positive energies and negative energies essentially diers. The analytical expressions of wave functions u(r) = φ(v (r)) exp{ ikr}, for any positive energy E > 0 and for L = 0,

5 Potential representation method for Woods-Saxon potential 45 are presented like analytical solution [7] by expanding in the power series the interaction potential V (r). Solutions (4) and (8) satisfy the standard boundary conditions [9] of the Schrödinger equation and, represent the wave functions by the interaction potential. Improvements of possibility to get the analytical solutions of the Schrödinger equation with Woods-Saxon potential in common case is presented in [3]. References [1] C. A. Bertulani, A potential model tool for direct capture reactions, Computer Physics Communic., 156, (2003). [2] Ph. Dennery, A. Krzywicki, Mathematics for Physicists, Dover Publications, Inc., Mineola, New York, [3] Jian-You Guo, Zong-Qiang Sheng, Solution of Dirac equation for the Woods- Saxon potential with spin and psiaudospin symmetry, Physics Letters A, 338, 9096 (2005). [4] R. S. Guter, A. R. Janpolski, Dierential Equations, High School, Moscow, 1976 (in Russian). [5] P. E. Hodgson, Optical model of elastic scattering, Atomizdat, Moscow, 1966 (in Russian). [6] A. J. Janavi ius, D. Jurgaitis, Separation of the scattering matrix with short range potential from the background Coulomb eld, Lith. Physical Collect., 26, (1986) (in Russian). [7] A. J. Janawiczius, K. Kwiatkowski, Schrödinger's equation in potential representation for Saxon-Wood's potential for the case of s-wave, Report No. 1001/PL, Institute of Physics Jagellonian University, Institute of Nuclear Physics, Krakow, 1978 (in Russian). [8] A. J. Janawiczius, R. Planeta, General solution of Schrodinger equation in potential representation, Report No. 1018/PL, Institute of Physics Jagellonian University, Institute of Nuclear Physics, Krakow, 1978 (in Russian). [9] E. Merzbacher, Quantum Mechanics, John Wiley and Sons, New York, Received 23 September 2009