[ Ntwlear Physics A!49 (1970) 11 -32; (~) North-Holland Publishin.q Co., Amsterdam 1.D.2 I Not to be reproduced by photoprint or microfilm without written permission from the publisher SEMI-MICROSCOPIC THEORY OF TWO-PHONON QUADRUPOLE- OCTUPOLE VIBRATIONS IN SPHERICAL NUCLEI A. RADUTA lnstit~ate for Atomic Physics, Bucharest, Romania and A. SANDULESCU t ,~tax Plank htstitut ]'fir CTtemie, MIainz, Germany and Niels Bohr Institute Copenhagen, Denmark and P. O. LIPAS University of Helsinki, Finland Received 6 February 1970 Abstract: A microscopic description is given for the quintet of states (I -, 2-, 3 -, 4-, 5 -) obtained by combining one quadrupole phonon and one octupole phonon. To the conventional shell- model Hamiltonian with pairing, quadrupolc and octupole forces is added a quadrupole-octu- pole interaction term H2.3 of thc form [Q2Q.~Q3]o. The multipole-moment operators QL are those defined through the ordinary pairing-multipolc theory. The quadrupole-octupole coupling emerging directly from that theory is here neglected, as the quadrupole and octupole operators are assumed to commute. The spectral decomposition method of Do Dang and Klein is used to treat H2.3 as a perturbation on the RPA. Expressions are derived for the splitting of the quintet and for the relevant B(E3) values. Also the effects of H23 on the lower phonon states are cal- culated. The theory is applied to ~t'~Sn and compared with experiment anti phenomenology. 1. Introduction In recent years the vibrational spectra of even spherical nuclei have received much attention. ]'he 2~ and 3 states assigned as ene-quadrupole- and one-octupole- phonon states have been studied extensively; for review see refs. J,2). Also some attempts have been made to interpret the observed triplet 0 +, 2 +, 4 + as a two-qua- drupole pbonon state and the observed I , ~ and second 3- states as members of the quintet obtained from the combination of one quadrupole phonon and one octt,- pole phonon. The first attempts, based on Bohr's collective model, to interpret the triplet was ,made by Kerman and Shakiq 3). They added to the harmonic ltamiltonian third-order terms in the collecti~e coordit:atcs. A similar attempt to interpret the quintet has been made by Lipas ~). These phenomenological models give a reasonable lit to the vibra- tional spectra. However, without fourth-order terms, they cannot fit all the data [levels, transitions, qqadrupole moments -~)]. t Present address: Institute for Atomic Physics Bucharest, Romania. 11