Proceeding of the Third Eurasian Conference "Nuclear Science and its Application”. October 5 - 8 . 2004.
EFFECTS ON THE ENERGY EIGENVALUES
OF THE DIFFUSENESS PARAMETER OF THE WOODS-SAXON
POTENTIAL IN HEAVY-ION FUSION
‘Berkdemir C., ‘Berkdemir A., 2Sever R.
' Erciye s University
2Middle East Technical UniAnkara, Turkey,
ABSTRACT
The interactions between nuclei are commonly described using a potential which consists of the well-known repulsive Coulomb and the attractive nuclear potential which is usually taken to be of Woods-Saxon form. For the elastic scattering problems, the value of the diffuseness parameter of Woods-Saxon potential extracted by fitting precise fusion cross sections is considerably larger than the value of « 0.63 fm usually accepted as a typical value. Having obtained exact solutions of the Schrödinger equation with a standard form of the deformed
140
Proceeding o f the Third Eurasian Conference “Nuclear Science and its Application”, October 5 - 8 , 2004. single-particle Woods-Saxon potential by referring Nikiforov-Uvarov method, energy eigenvalues for different values of this parameter are calculated numerically and effects on the bound states of the energy spectrum of it are discussed in heavy-ion fusion. In addition, the energy eigenvalues of the Woods-Saxon potential are investigated within the framework of complex quantum mechanics formulation.
INTRODUCTION
The interaction between two nuclei is regulated by the well-known repulsive Coulomb potential and the attractive nuclear potential, which is often described by a Woods-Saxon potential:
V( r ) = - V 0 {l + exp[(r - + ))/a] } ' (D
where r denotes the center - of -mass distance between the projectile nucleus of mass n u m b er^
and the target nucleus of mass number AT.The standard Woods-Saxon potential (WS) is defined
by three parameters in the literature for Vo, Ro and a. The latter parameter is the surface diffuseness parameter that is around 0.663 fm for scattering processes [1], The shape of the Woods-Saxon potential given in Eq.(l) is plotted in Fig. 1, by the parameters Vo= 67,4
MeV, Ro=10,1 fin and a = 0,663 fin for 19F + 208Pb for example. For the different values of the
diffuseness parameter, exact solutions of the Schrödinger equation with a standard form of the deformed single-particle Woods-Saxon potential by referring Nikiforov-Uvarov (NU) method can be obtained in an analytical manner and effects on the bound states of the energy spectrum of it can be discussed in heavy-ion fusion [2],
Fig. 1. A schematical representation of the standard Woods-Saxon potential for three different values of the deformation parameter q.
The NU method is based on the solutions of general second order linear differential equation with special orthogonal functions. In this method, for a given real or complex potential, the time- independent Schrödinger equation in one dimension is reduced to a generalized equation of hypergeometric type with an appropriate coordinate transformation.
It is the aim of the present study to investigate via NU-method real and complex Woods- Saxon potentials for which the corresponding energy eigenvalue problem can be solved exactly. The Schrödinger equation is solved by using the NU method so as to ensure the energy eigenvalues of the bound states for real and complex forms of the standard spherical Woods- Saxon potential, which is used widely in analyses of heavy-ion reactions [3],
METHOD AND NUMERIC CALCULATIONS
We will use the Nikiforov-Uvarov method by getting the hypergeometric or confluent hypergeometric form of the Schrödinger equation with the WS potential for s-states only. The
Section II. Basic problems o f nuclear physics
Proceeding o f the Third Eurasian Conference “Nuclear Science and its Application”, October 5 - 8 , 2004. radial part of Schrödinger equation with the deformed WS potential in the spherical coordinates is given by T O f \ d 2R(r) 2 dR(r)2 mr2 d r r dr tv E + Vn
{l
+ qexp[(r -
R0(a'P + A '/3))/o]}
R(r) = 0 (2)At this point, we assume that R(r) = (l/r)F (r)is bounded as >0, consequently, radial Schrödinger equation given in Eq.(2) becomes
A"(r) + —2m h f A + - Vr0 V 1 + qe 2 a r F(r) = 0 (3)
where the conversions r - R0 (a1/ 3 +^ / 3) = r and are done by inserting an arbitrary real constant q within the potential. By introducing the following dimensional parameters, we
apply a transformation to s - - ewhich leads to the generalized hypergeometric type
equation given by NU method:
1 F"(s) + 1 qS + s ( \- q s ) s 2(l - qs) -x(- s q 2s 2 + (2 sq- y q )s + y - s )F(s) = 0(4) mE = - s > 0 mVn = y > 0. l h 2a 2 ’ l h 2a 2
Second order differential equation according to the NU method is given by Ref.[2]
a(s) a (s) (5)
and after the comparison of Eq.(4) with Eq.(5), one can immediately determine the energy eigenvalues En as A'., tv_ 2 ma2 f ma2Vn \ 2 yh 2(n + \)y + n +1V +ma2Vn h 2 (6)
Here, the index n is non-negative integer with co < 0 and Eq.(6) indicates that we deal with a family of the standard Woods-Saxon potential. The index n describes the quantization of the bound states and the energy spectrum. The shape of the Woods-Saxon potential given in Eq.(l) is plotted in Fig. 1, by the parameters Vo = 67,4 Ro = 10,1 fm and a = 0,663 fm for 19F +
208Pb for example. Fig.2 shows the energy eigenvalues as a function of the discrete level n for different values of the parameter a. Some of the initial energy levels for =1 value are
presented by choosing a = 0,663 fm and a = 1,07 fm.
Fig. 2. The variation of the energy eigenvalues with respect to the discrete levels n for the standard Woods-Saxon
potential ( q =1). The curves are plotted
for the two values of the surface diffuseness parameter a.
142
Proceeding o f the Third Eurasian Conference “Nuclear Science and its Application”, October 5 - 8 , 2004. If we take the potential parameters in Eq.(l) as V0 —> V0 and a —» i a}, energy eigenvalues are
obtained as [4]
En 2 h 2 ma,V0 (w + 1)
m 2h2(n + \) 4 a} (?)
Although energy eigenvalues of Woods-Saxon potential are negative, a positive energy spectrum within the framework of complex quantum mechanics formulation is obtained.
CONCLUSION
In this work, the exact solutions of the radial Schrödinger equation with the Woods-Saxon potential for the s-states are obtained by using Nikiforov-Uvarov method. If surface diffuseness parameter is chosen complex in the Woods-Saxon potential, it is found that the energy levels of the single particle are positive on the contrary to expectation.
REFERENCES
1. C. P. Silva et al., Nuclear Physics A 679, 287 (2001).
2. A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics (Birkhauser, Basel, 1988).
3. I. Gontchar, D. J. Hinde, M. Dasgupta, and J. O. Newton, Nuclear Physics A 722, 479c-483c (2003).
4. Berkdemir, C. Berkdemir and R. Sever, submitted to Phys. Lett. A (2004).
Section II. Basic problems o f nuclear physics
