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Cluster Description of Light Nuclei as a Three-Body Problem.

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Annalen der Physik. 7. Folge, Band 36, Heft 6, 1979, S. 479-480
J. A. Barth, Leipzig
Wurze Mitteilnng
Cluster Description of Light Nuclei
as a Three-Body Problem
By A. OSMAN
International Centre for Theoretical Physics, Trieste, Italy;
Physics Department, Faculty of Science, Cairo-University, Cairo, Egypt
Cluster-Beschreibung leichter Heme als Dreikdrperproblem
The alpha cluster model is used in describing the structures of the bound states of
the 6Li, sBe and 12C nuclei. This assumption of the alpha cluster model considers the
fact that the alpha particle is more tightly bound with an unusual stable structure. With
this model, each one of the three light nuclei 6Li, sBe and 12C is presented as composed
of three particles. The bound states of each of these nuclei are very useful for studying
each one of them as a three-body problem.
I n the present calculations, we follow the FADDEEV
[l] formalism and the LOVELACE
[2] forinulations. With the Faddeev equations, which introduces an exact solution of the
three-body problem, separable potentials are used which in turn reduce the Faddeev
equations to coupled integral equations in one continuous variable [3, 41. The separable
two-body interactions include both attraction and repulsion, each one of it includes
spin-dependent central forces together with tensor forces. The potential functions of the
different parts of the two-body interactions are of the Yamaguchi, Gaussian, Tabakin,
Mongan and of the Reid forms. The different parameters of the different potentials are
obtained by fitting the corresponding phase shifts.
Using separable potentials, a separable expansion for the T matrix is given by
where
With the above-mentioned potentials, the three-body ground-state energies are
calculated with a very high accuracy, using only a very limited number of summation
terms. In the present calculations we apply the well-behavedSchmidt-Hilbert theory of
integral equations. We use a 24-point Gaussian integration. The integral equation is
converted into a matrix eigenvalue equation, where the eigenvalues are given as a
functionof the energy. The three-body bound-state energies are those values of the energy
for which the matrix eigenvalue takes the value one. The binding energy values calculated here are the nuclear binding energies. Corrections due to Coulomb energies should
be added by treating the pure Coulomb T matrix [5]. The results of the actual three-body
480
A.
OSlUN
binding energies of the 6Li, 9Be and lacnuclei using the different two-body interactions
are listed in Table 1. The present theoretically calculated values for the ground-state
energies are quite reasonable and in good agreement with the experimentally observed
values.
Table 1 Calculated binding energies in (MeV)
Nucleus
Yamaguchi
Gaussian
Tabakin
Mongan
Reid
Exp.
6Li
sBe
5.812
1.932
7.491
5.297
1.425
5.723
1.972
1.726
7.411
1.718
1.693
7.319
1.829
1.701
7.398
4.53
1.57
7.28
12C
I am very grateful t o Professor ABDUSSALAM,Professor PAOLO
BUDINI,the International Atomic Energy Agency and UNESCO for hospitality a t the International Centre
for Theoretical Physics, Trieste, Italy. I am deeply indebted t o Professor LUCIANO
FONDA
for his kind hospitality at the ICTP, Trieste, Italy, where most of this work is
done. Thanks are also due t o the Centro di Calcolo dell’universita di Trieste for the use
of the facilities.
References
[l] L. D. F A ~ D E ESoviet
V,
Phys.-JETP 12, 1014 (1961).
Phys. Rev. 136, B 1225 (1964).
[2] C. LOVELACE,
[3] A. OSMAN,
Phys. Rev. C 17, 341 (1978).
€41 A. OSMAN,
ht. J. Theoret. Phys. (N.Y.) 16,81 (1977).
[5] A. OSMAN,
Phys. Rev. C 4, 302 (1971).
Bei der Redaktion eingegangen am 22. November 1978.
Anschr. d. Verf.: Prof. Dr. AHMEDOSMAN
Physics Department, Paculty of Science
Cairo University, Cairo, Egypt
Q
Chefredakteur: Professor Dr. G. Richter, DDR-1199 Berlin-Adlershof, Rudower Chaussee 5. Anzeigen Inland: DEWAG LEIPZIG, 705 Leipzig, OststraBe 105, Ruf 7974303. Ausland: Interwerbung GmbH., - Geeellachaft fur Werbung und Auslandsmessen der DDR, DDR-1157 Berlin-Karlahorst, Hermann-Duncker-Str. 89, Ruf 509 09 81. Fiir die Anzeigenpreise gelten die Festlegungen gemiiB Preiskatalog Nr. 286/1 vom l. 7.1975. Verlag Johann Ambrosius Barth, DDR-701 Leipzig,
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