Accuracy of multipole expansion of density distribution in calculating the potential for deformed spherical interacting pair

Abstract

The interaction potential for a deformed-spherical pair is calculated, and the error in using the truncated multipole expansion is evaluated for different numbers of terms of the expansion considered. It was found for the internal region of the nuclear part that three terms are sufficient, but for the surface and tail region up to five terms are necessary, while for the Coulomb potential three terms were found to be sufficient.In recent years, nuclear reactions involving deformed nuclei have become an important topic of research in nuclear physics ͓1͔. One type of these reactions is the fusion reaction, which is an important intermediate step in the production of superheavy nuclei by heavy ion collisions. The nuclear potential between the interacting nuclei plays an important role in describing the reaction process.A model that is commonly used in deriving a heavy ion ͑HI͒ potential is the double folding model ͓2͔. The basic input into the folding calculation is the nuclear densities of the colliding nuclei. If one or both have deformed density distribution, the use of this model to derive Coulomb or nuclear interaction potentials becomes very difficult, since the six-dimensional integral cannot be simplified to fewer dimensions. In this case, one usually simplifies the folding model by expanding the density distribution of the deformed nuclei using a multipole expansion ͓3͔. This method is useful and reduces the amount of calculation except in some cases, where the nucleon-nucleon ͑NN͒ force is density dependent.In the multipole expansion one may take a finite number of terms ͑usually three terms͒ and neglect the others ͓3͔. Since this method is used frequently in deriving the real part of the HI potential for deformed-deformed and spherical-deformed pairs of nuclei, it is interesting to test its accuracy. In the present paper, we estimate the accuracy of the multipole expansion in deriving the heavy ion potential. We include both quadrupole and hexadecapole deformation parameters, and determine the number of terms in the multipole expansion needed to guarantee a very small percentage error.We limit ourselves here to the interaction potential between deformed target and spherical projectile ͑see Fig. 1͒. The HI potential in this model is divided into a direct part U d , and an exchange part U ex ,They are stated in Refs. ͓2͔ and ͓3͔ as follows:where R is the separation distance between the interacting nuclei, and ␤ is the relative orientation angle of the target nucleus symmetry axis measured with respect to the separation vector R.The deformed density T (r,) has the formwhere R() is the half density radius expressed by the relation R͑ ͒ϭR o ͓1ϩ␦ 2 Y 20 ͑ ͒ϩ␦ 4 Y 40 ͑ ͔͒. ͑5͒ ␦ 2 and ␦ 4 are the quadrupole and hexadecapole deformation parameters. The multipole expansion of the target nuclear density distribution has the form T ͑ r͒ϭ ͚ lm lm ͑ r ͒Y lm ͑ ,͒, ͑6͒FIG. 1. The coordinate system.

Global calculation of α-decay half-lives with a deformed density-dependent cluster model

“…It is also proved in Ref. [40] that the multipole expansion is accurate and stable for the spherical-deformed nuclear pair. In our calculations, we use the standard proton-proton Coulomb interaction and the effective M3Y interaction which is derived from the Reid nucleon-nucleon potential [41].…”

“…The double-folding potential in the deformed case involves a complex six-dimensional integral. For the spherical-deformed interacting pair, the double-folding potential is solved numerically by using the multipole expansion method in which the density distribution of the daughter nucleus is expanded as [39,40] ρ(r, θ ) = l=0,2,4...…”

“…In this article, we present detailed formulas of the deformed DDCM and extend our calculation to both odd-A nuclei and odd-odd nuclei in a wide mass region Z = 52−110. In the deformed DDCM, the microscopic deformation-and orientation-dependent α-core potentials are evaluated numerically from the double-folding model by the multipole expansion method [37][38][39][40]. Such nonspherical double-folding potentials are difficult to calculate by common Fourier transformation techniques and have rarely been used in the calculation of α-decay half-lives [37,38].…”

Effect of in-medium NN cross section and finite range force on the reaction cross section for a deformed target nucleus

“…The inclusion of both finite range force and proper density dependence in NN produces complications in numerical calculation of R , even for a spherical-spherical interacting pair. For a deformed target density distribution, one cannot use multipole expansion [6,8] to simplify the numerical calculations since the dependence of NN on the density is not linear [5].…”

Accuracy of the Multipole Expansion of Density Distribution in the Presence of Octupole Deformation

“…The Coulomb interaction 13 between two ions is also calculated by a similar procedure after replacing the matter density distributions by charge distributions. The numerical calculations of the folding model is simple when both the interacting nuclei are spherical (the density distribution depends on the magnitude of the vector r. 14 When one of the two interacting nuclei or both is deformed, 15,16 the deformed density distribution is a function of both magnitude of the radius vector r and its direction. This angular dependence of the deformed density complicates too much the numerical calculation of heavy ion potential compared with the case of spherical nuclei.…”