ADIABATIC POTENTIALS FOR HEAVY ION COLLISIONS

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ADIABATIC POTENTIALS FOR HEAVY ION COLLISIONS

M. Rayet, G. Reidemeister

To cite this version:

M. Rayet, G. Reidemeister. ADIABATIC POTENTIALS FOR HEAVY ION COLLISIONS. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-259-C6-260. �10.1051/jphyscol:1971659�. �jpa-00214877�

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JOURNAL DE PHYSIQUE Colloque C6, supplbment au no 11-12, Tome 32, Novembre-Dicembre 1971, page C6-259

ADIABATIC POTENTIALS FOR HEAVY ION COLLISIONS

M. RAYET and G. REIDEMEISTER

Theoretical Nuclear Physics, University of Brussels, Belgium

RbumB. - Des potentiels adiabatiques sont obtenus en minimisant la valeur moyenne du Hamiltonien nucleaire contenant une force effective a deux corps. Des resultats sont donnes pour la diffusion elastique a-a dans une approximation atomique et pour la diffusion ' 6 0 - 1 6 0 dans le mod61e molCculaire a deux centres.

Abstract. - Adiabatic potentials are derived by minimization of the expectation value of the nuclear Hamiltonian containing an effective two-body force. Results are given for a-a scattering in an atomic approximation and for 1 6 0 - 1 6 0 scattering in the molecular two-centre model.

The problem of defining a potential between two interacting ions becomes ambiguous as soon as one considers the nucleons forming the total system of the two ions as not distinguishable. The system consisting of two ions of masses A, and A, is described by the Hamiltonian

where the summations on i and j run from 1 to A ⁼A ,

+

^{A ,}and where Tc.,. is the centre of mass kinetic energy for the system of mass A. The internal energy of this system is given by the expectation value

where the wave function

+

is wanted to be the best possible approximation to the lowest eigenstate of H, and is in any case completely antisymmetrized bet- ween all A nucleons. In order to get an equation for the relative motion between the ions, it is necessary to make II/ an explicit function of a variable p des- cribing this motion and to take the expectation value (2) on all variables except p. This can be done by the resonating group method but at the expense of very tedious calculations.

In this work we investigate two methods for deriving a potential where the dynamical variable p is replaced by some distance parameter d contained in the wave- function $. The calculation of the expectation value (2) by integration over all variables then results in an expression E(d). Our <( adiabatic B potential is defined as E(d)

-

E(m) where E(m) is the sum of the two internal energies plus the relative kinetic energy at large distance.

In the two methods described here, $ is an antisymmetrized product (Slater determinant) of single particle wave functions. In addition to d, II/ contains other parameters which are determined by minimiza-

tion of the energy expectation value (2) for each value of d.

In the first method, $ is constructed from harmonic oscillator orbitals centered around two points,

+

^d

and

-

d on the z axis.. This approach corresponds to the model proposed by Brink-Margenau [I] for the study of 4 N nuclei. As an example we consider the simple case of the a-a system. The two a particles are described by the lowest states of harmonic oscillators centered around

+

d and - d, each being multi- plied by the four possible spin-isospin states. The system being axially symmetric, we can restrict the two oscillators to have the same axial symmetry.

The variational parameters are the two usual size and shape parameters defining an axially deformed harmonic oscillator. We choose for the two-body potential in H the phenomenological nuclear potential V1 of Volkov [2].

The minimization of the energy (2) for different values of d results in an attractive potential showing a minimum of - 6 MeV at an interdistance 2 d = 2.8 fm.

between the two a particles and increasing slightly to- 3.9 MeV at d = 0. The outer part (d R, 2.8 fm.) of this potential is the sort of potential which can be expected from a phenomenological analysis of the elastic a-a scattering [3] but the inner part does not at all show the strong repulsion which must be introduced in such analysis. This reflects the fact that the a-a interaction at short distance is strongly non- local as a result of the Pauli principle 131. The mini- mization of the energy curve E(d) in the inner region however is interesting to show how the He4 nuclei take on a prolate deformation when they are slowly pushed near to each other, reaching near d = 0 the strong prolate deformation of the 8Be nucleus [2].

In the second method, the Slater determinant $ is built with molecular orbitals of the double-centre oscillator potential [5, 61, the centres being separated

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971659

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C6-260 M. RAYET A N D G. REIDEMEISTER

by the distance 2 d. We investigate the case of two colliding ^{1 6 0}ions. All orbitals are taken which completely fill the s and p shells of the two separated nuclei. Brink and Boeker's nuclear force B1 is used here [4]. The problem being again axially symmetric, only two variational parameters are used, which describe the two-centered spheroidal equipotentials.

In the figure we show two energy curves. The first is minimized for each d with respect to the parameters of the orbitals, the other is calculated with fixed parameters corresponding to a free ^{1 6 0}nucleus.

These two curves have to be compared with the slow ^>>

and (t fast >> collision curves of Greiner and Pruess [7]

obtained in the crude approximation where the indi- vidual energy curves are summed. Another calculation based on the eigenenergies of the same potential and including liquid drop, model corrections has been done [8]. It is.interesting to note that we do not find any difference between the two processes (slow and fast) until small distances 2 d

-

5 fm. for which the two nuclei overlap strongly. At small distances the chosen configuration gives a 4 p-4 h state in 32S. For the completely varied calculation we point out that

160 +j60 _BI Interaction Completely minirnlzed

I

^'\. ^{. . .}^{. . .}^.^.Optical potential (Real part)

Interdistance 2d ( f m )

another configuration gives a lower energy (a few MeV) for small distances 2 d < 1.5 fm. Another interesting feature of this calculation is that it gives a shallow interaction energy between the two ions

(-- 20 MeV). This is in good agreement with the

shallow optical potential deduced from experimental data on the elastic scattering ^{1 6 0}-I- 1 6 0 [9, 101.

References

[l] BRINK (D. M.), Proc. Int. School of Physics ⁽⁽Enrico [6] HOLZER (P.), MOSEL (U.) and GREINER (W.), Nuel.

Fermi ^)),Course 36, Academic Press, 1966, 247. Phys., 1969, A 138, 241.

[2] VOLKOV (A. B.), NUCZ. Phys., 1965, 74, 33. 171 PRUESS (K.) and GREINER (W.), Phys. Letters, 1970, [3] AFZAL (S. A.) et al., Rev. Mod. Phys., 1969, 41, 247. 33B, 197.

[4] BRINK (D. M.) and BOEKER (E.), NUCZ. Phys., 1967, [8] MOSEL (U.) et al., Phys. Letters, 1970, 33B, 565.

A 91, 1. [91 SIEMSSEN (R. H.) et al., Phys. Rev. Letters, 1967, 19, [5] DEMEUR (M.) and REIDEMEISTER (G.), A m . de Phy- 369.

sique, 1966, 1 , 181. [lo] CHATWIN (R. A.) et al., Phys. Rev., 1969, 180, 1049.