^{1}

^{*}

^{2}

A
semimicroscopic analysis of a set of experimental data of elastic α + ^{12}C scattering was performed at several laboratory energies. The
Woods-Saxon parameters were adjusted to obtain the best χ^{2} fit to the scattering data. The energy systematics of
the positions of Airy minima was constructed, and it was shown that their
positions depend linearly on the inverse center of mass energy. The parameters
of the model potential have been determined unambiguously. It has been shown
that the energy dependence of the volume integrals satisfies the dispersion
relation and agrees well with the results obtained within a phenomenological
analysis. Also, it has been shown that the found positions of the Airy minima
satisfy the rule of the quadratical dependence of the position of the Airy
minima on the reduced mass of the colliding nuclei.

The essence of the potential approach, that has been applied to describe elastic scattering and direct reaction in nucleus-nucleus collisions at energies in the range extending up to 100 MeV per nucleon, is that the system of two interacting nuclei at the given energy in the elastic channel can be described by a model-dependent wave function that is found by solving the single-particle Schrödinger equation with an effective potential [

The new and important point in this work is the calculation of the dispersion relation between the real and imaginary parts of the dynamical polarization potential. This relation is a result of the causality principle: a scattered wave cannot be emitted before the interaction has occurred [

The model wave function may be written in terms of the complete set of internal wave functions of the projectile (a) and target (A) nuclei as [

where

This equation is the basis of the optical model formulation that its essence is the construction of the effective potential model can be written

where

potential. Strength parameters

One of the approaches to construction of effective potential is microscopic approach: attempt to understand interaction of two nuclei in terms of the motions of individual nucleons and their interactions. Static component, that is the matrix element of the real effective nucleon-nucleon interaction, is the ground states wave functions of colliding nuclei

In this relation, already, dual nucleon-nucleon interaction,

where

The radial functions of the central component,

The radial part of the various components of interaction is shown by the sum of the Yukawa’s potentials. But this model of interaction is averaged over the energy and density and is not dependent on them explicitly. Obviously, it is very difficult to abandon the energy and density dependence. The density dependence is included in effective nucleon-nucleon interactions (central direct and exchange components in this case) in the form of a density-dependent factor multiplied by the M3Y model interaction (indices being suppressed):

In the present study, we use the parametrization

in the CDM3Y6 version, where the parameters involved were fitted to the properties of cold nuclear matter [

where

and

Here

where

The density matrix, usually, is made with the aid of the modified Slater approximation (indices are suppressed)

where

It was shown that, in the absence of the density dependence of nucleon-nucleon forces, the result of the calculation with the aid of (12) and (13) is virtually coincident with the result of the exact calculation. The use of the different version of the approximation in (13) leading to a modest distinction. Upon the inclusion of the density dependence, however, the deviation in both versions from the result of the “exact” calculation becomes sizable (smaller for the first, but greater for the second version) especially at low energies. All versions of the calculation of nuclear density matrices are rather cumbersome. In order to simplify the calculations, an approximate representation of the density matrix is used in the form [

where an empirical nuclear density for

For

Other approach, that is widely used to construct effective potential, is called dispersion semimicroscopic approach. A semimicroscopic approach combines a microscopic calculation of the mean-field potential and a phenomenological construction of the dynamical polarization potential.

In this model, the dynamical polarization potential is constructed on the basis of physically justified combinations of the volume and surface forms whose geometric parameters are assumed to be independent of energy.

Furthermore, in this model the imaginary part of the central component DPP is represented as the sum of the volume,

where the first term is the mean-field potential being calculated, the second term is the real part of the phenomenological dynamical polarization potential (dispersive correction), and the third term is its imaginary (absorptive) part. The absorption includes the volume

where

(

In this study, the Coulomb component

Keeping simplicity and convenience of application, such model effectively has smaller number of parameters and allows reducing ambiguity of the analysis.

The dispersive analysis of the volume integrals was performed here on the basis of the difference dispersion relation [

where the dependence

corresponding to the mean-field potential and the dispersion correction.

In order to analysis of experimental data, in this work, we used angular distributions for elastic

The values found for the remaining parameters and values of

in a given angular distribution) are given in

In this work, three tests were used to avoid ambiguities in determining the potential: due to of rainbow structure, it is possible to construct energy systematics of the Airy minima positions. This systematics showed and confirmed the inverse-energy law and made it possible to select potential (see

The curve introduced by Goncharov and Izadpanah [

. Found parameters and integral volumes of potential for α + ^{12}C system

E_{lab} (MeV) | V (MeV) | W (MeV) | (MeV∙fm^{3}) | (MeV∙fm^{3}) | (mb) | |||
---|---|---|---|---|---|---|---|---|

104 | 0.186 | 0.236 | 15.28 | 16.94 | 339.5 | 118.4 | 825 | 8.5 |

120 | 0.120 | 0.20 | 16.76 | 7.55 | 340.4 | 120.8 | 821 | 3.2 |

139 | 0.559 | 0.466 | 14.89 | 7.49 | 287.7 | 108 | 757.9 | 10.4 |

145 | 0.158 | 0.230 | 16.79 | 7.9 | 325.5 | 121.3 | 788.7 | 4.1 |

166 | 0.163 | 0.391 | 15.72 | 14.8 | 314.5 | 119.7 | 753 | 6.5 |

172.5 | 0.129 | 0.142 | 17.49 | 4.17 | 318.4 | 123 | 769.3 | 5.1 |

240 | 0.485 | 0.357 | 13.78 | 6.21 | 262.9 | 99.4 | 650.5 | 3.4 |

Ratios of the differential cross sections for elastic scattering α + ^{12}C system

Positions of the Airy minima for α + ^{12}C system

Dispersion analysis of volume integrals of dispersive potentials for the α + ^{12}C system

liding nuclei, third test, has been shown in

The farside components calculated by us have been shown in

A semimicroscopic analysis of a set of experimental data of elastic

Estimated positions and approximating second- order polynomials curves of first Airy minima versus the reduced mass

Calculated farsaid components for α + ^{12}C system

Found radial dependence of the total real and imaginary part of potential for α + ^{12}C system at various laboratory energies

laboratory energies. In this way, the microscopic calculation of static component of potential combines with the phenomenological calculation of the dynamical component of potential. In this model, the dynamical polarization potential is constructed on the basis of physically justified combinations of the volume and surface forms whose geometric parameters are assumed to be independent of energy.

The energy systematics of the positions of Airy minima was constructed, and it was shown that their positions depend linearly on the inverse center of mass energy. The energy dependence of the volume integrals was performed and has been shown that the dispersion relation is satisfied and agrees well with the results obtained within a phenomenological analysis. Using systematics of the positions of Airy minima with respect to reduced mass of colliding nuclei introduced by Goncharov and Izadpanah, has been shown that the found positions of the Airy minima satisfy the rule of the quadratical dependence of the position of the Airy minima on the reduced mass of colliding nuclei.

Using these three systematics, the parameters of the model potential at various energies have been determined unambiguously.

The obtained imaginary components are shown the meaningness of aberrations from its Woods-Saxon forms obtained using the phenomenological approach. This aberration occurs in the distances about 1 fm and one can be due to ignore of the dependence energy in this study.

We are grateful to C.А. Goncharov for enlightening comments and discussions.