Vertex constants are calculated with an effective nucleon-nucleon interaction and translationinvariant harmonic-oscillator shell-model wave functions consistently obtained with the same interaction. Such calculations are presented for the ~3C --~ ~ZC + n vertex constant for which an accurate experimental value is available, and for the 8B --~ 7Be+p vertex constant, which is crucial for the determination of the 7Be(p, y)SB reaction at astrophysical energies. Calculations confirm the strong sensitivity of vertex constants to the choice of the effective nucleon-nucleon interaction. A comparison of vertex constants obtained in the shell-model approach and in microscopic twoand three-cluster models with identical nucleon-nucleon potentials shows that the main contribution to the cluster-model vertex constants comes from the shell-model parts of the cluster-model wave functions. The reliability of model calculations of the 7Be(p, T)aB astrophysical S factor is discussed. (~) 1997 Elsevier Science B.V. the remaining nucleons [ 1 ]. It can be determined experimentally. At present, accurate experimental values of these constants are available in a few cases only. The properties of nuclear vertex constants and methods for experimentally determining them are reviewed in Ref. [ 1 ].
Nuclear vertex constants are interesting for two main reasons. (i) Their study provides additional information about nuclear structure and about nucleon-nucleon (NN) forces. (ii) Knowledge of them is necessary in investigations of nuclear reactions, including reactions of astrophysical interest. Particularly, in the important case of the 7Be(p,y)8B reaction, the 8B ~ 7Be + p vertex constant determines the absolute cross section at astrophysically relevant energies for which this non-resonant capture takes place at large distances.
A systematic calculation of vertex constants for one-nucleon removals is presented in Refs. [2,3] for lp-shell nuclei with translation-invariant harmonic-oscillator shellmodel wave functions. The incorrect asymptotics of the shell-model wave functions does not influence the vertex-constant calculations because the virtual decay amplitude can be expressed as a matrix element of the nuclear interaction between the external nucleon and the residual nucleus. This interaction falls rapidly off outside the nuclear interior and cuts off the contributions of the asymptotic part of nuclear wave functions. Analytical expressions for vertex constants obtained in Refs. [ 2,3 ] with the shell-model basis depend on two types of parameters: (i) weights of SU(3) configurations in the initial and final nuclear wave functions and (ii) effective oscillator matrix elements of the two-body virtual decay which depend on the NN-interaction choice, on the nucleon separation energy and on the oscillator radius. Both kinds of parameters can be either calculated microscopically or taken from phenomenological considerations. In Refs. [2,3], the weights of the SU(3) configurations are taken from Ref. [4], where they were obtained with NN-interaction matrix elements adjusted so as to reproduce in the whole the natural-parity spectra of the lp-shell nuclei. The effective oscillator matrix elements of the two-body virtual decay are calculated with several NN interactions available in the literature. From a microscopic point of view, this procedure is not selfconsistent since the NN interaction controlling the virtual decay differs from the NN interaction determining the wave functions.
Nuclear vertex constants can also be determined from the asymptotic form of nucleoncore or cluster-core wave functions. This procedure is only possible with models where the shape of this asymptotic form is exact, i.e. is an exponential decrease in the neutral case or a Whittaker decrease in the charged case, depending on the cluster separation energy. This condition is satisfied, for example, in cluster models. In microscopic cluster models [5][6][7], such as the resonating-group method (RGM) or the equivalent generatorcoordinate method (GCM), correct asymptotic forms are obtained when the cluster separation energy is accurately reproduced by the model. The simpler potential cluster model also possesses this property. In these cases, the vertex constant is obtained from the asymptotic normalization coefficient. In principle, a direct calculation from a matrix element of the nucleon-core interaction, as in the shell-model approach, is also possible in these models.