Z. Physik A 274, 41-49 (1975) 9 by Springer-Verlag 1975 Exchange Symmetry of the Effective Interaction in the Landau-Migdal Theory of Finite Fermi Systems R. Bauer, K. Ebert, P. Ring*, W. Theis**, E. Werner**, and W. Wild Physik-Department der Technischen Universit~it Mtinchen, Garching, Germany Received May 27, accepted May 28, 1975 Abstract. We investigate the particle exchange symmetry of the phenomenological effective particle-hole interaction used in the theory of Finite Fermi Systems. We find that experi- mental data and consistency relations support an interaction which is antisymmetric at the surface and in the exterior region of the nucleus and nonsymmetric in the interior. Besides, we show that the number of adjustable parameters of the effective interaction can be reduced substantially by the use of generalized Ward identities without any loss of agreement with experimental data. 1. Introduction In the past few years many papers have been published which applied the Theory of Finite Fermi Systems (FFS) [1-6] developed by Migdal [7] and extensions of this theory [8] to low-energy nuclear structure problems. In addition, it had been shown that the parametrization of the effective interaction obtained from a fit to experimental data is in accordance with requirements following from subsidiary conditions being of the form of generalized Ward identities (GWI) [9-12]. This makes it possible to reduce substantially the number of free interaction parameters by con- sidering the parameters which are determined from the GWI no longer as adjustable quantities, but rather as already determined by the internal structure of the theory. The success of this procedure makes it natural to look out for other properties of the effective inter- action which might eventually be derived from micros- copic features of the considered system. One such feature, the behaviour of the renormalized effective particle-hole (ph) interaction F~; 23 under the exchange 1+-,4 (or 2~--~3) will be investigated in this paper. Before we discuss the motivation for this investigation in detail, we want to make a remark concerning a possible misunderstanding: the parametrization of the * Present address: Lawrence Berkeley Laboratory, Calif., USA. ** Permanent address: Institut ftir Theoretische Physik der Tech- nischen Universit~it Hannover. Germany. effective interaction as proposed by Landau and Migdal does in principle already include all effects due to the Pauli principle: the effective quasiparticle- quasihole (qp-qh) interaction is defined through an infinite series of diagrams which, of course, incor- porates automatically all contributions of exchange diagrams. It would therefore be illegitimate to perform an antisymmetrization in the sense of shell model calculations, i.e. to consider the matrix elements of the usual Migdal interaction as the socalled direct terms to which the exchange terms must be added in order to ensure the requirements of the Pauli principle. The fact that the Pauli principle is included in the definition of the effective interaction F~;2 3 manifests itself in the following way: If F(];2 3 is completely unsymmetrical under the exchange 2~3, then the four force param- eters f f', g and g' of the Migdal interaction are in- dependent of each other; if, however, F ~ is antisym- metric (F1~;23=-F1~;32) then, as will be shown in Section 2, two linear relations between f f', g and g' exist, thus reducing the number of free parameters. Therefore, antisymmetry is already contained as a possible special case in the definition of F ~ and should not be imposed by an a posteriori antisymmetrization. The essential feature of the Migdal interaction is its dependence on the local density. This density depend- ence is to be interpreted as a kind of local density