2.G.... [ Nuclear Physics A249 (1975) 329--348; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher WAVE FUNCTIONS OF NUCLEAR STATES WITH SMALL SINGLE-PARTICLE SPECTROSCOPIC FACTORS K. G. BERNHARDT and W. R. HERING Sektion Physik der Universitlit Miinchen, D-8046 Garching P. RING Physik-Department der Technischen Universitgit Miinchen, D-8046 Garching and E. WERNER Institut fiir Theoretische Physik der Technischen Universitiit Hannover, D-3000 Hannover Received 11 November 1974 (Revised 2 May 1975) Abslraet: We have analyzed the reactions 2°aPb(d, p)2°gPb and 2°SPb(p, d)2°Tpb leading to frag- mented single-particle states in the final nuclei by using microscopically calculated bound state wave functions. Considerable deviations between these results and those of the conventional well depth procedure occur and are traced back to non-localities in the potential associated with the single-particle component of the bound state wave function which are caused by the mixing of the single-particle and excited-core configurations. Consequences for the deter- mination of spectroscopic factors and the application of sum rules are discussed. 1. Introduction For many years it has been one of the prime goals of nuclear studies to determine the spectroscopic strength of states populated in nuclear reactions and make com- parison with the predictions of nuclear structure calculations. One of the main tools used in this context are sum rules 1) which predict from elementary quantum me- chanics that the total strength of fragmented levels based on a single-particle state j must sum up to 2j+ 1. In fact, sum rules have been the accepted means to determine the relative spectroscopic strength of states of equal quantum numbers and thus to circumvent basic difficulties which troubled the quantitative DWBA analysis of nuclear reactions induced by light ions until rather recently 2). In the course of these DWBA analyses it has been generally assumed that following the prescription (do/dg2),xp = S(da/df2)DwB A, the remaining uncertainties of theory and experiment allow S-values to be determined to an average accuracy of 10-20 % and that, con- sequently, sum rules can be expected to be fulfilled to this accuracy. It was of course recognized long ago that the studies of weakly populated states may be rendered very difficult because of possible interference from non-direct and higher order direct con- tributions to the reaction mechanism. However, even in the case of a clean direct reaction one has to realize that the dependence of (da/df2)owB A on the final state which 329