PhysicsLetters B 277 (1992) 27-32 North-Holland PHYSICS LETTERS B Microscopic interpretation of potential energy surfaces -A- O. Castafios, P O. Hess Instttuto de Ctenctas Nucleates, UNAM, A P 70-543, 04510 Mextco D F, Mextco J P Draayer * and P. Rochford 1 Department of Physlcs and Astronomy, Loutszana State Umverstty, Baton Rouge. LA 70803, USA Received 11 February 1991, revised manuscript recetved24 October 1991 Starting from a m~croscoplc hamlltoman of the (pseudo) symplect~c model, an elementary treatment using coherent states ts proposed for denying an analyticform for the potentml energysurface (PES) of the geometriccollectwe model The method is apphed to 23sU and the result comparedto the correspondingPES generahzedcollectivemodel In the last 10-15 years various algebrmc mlcro- scop~c collective models, hke the symplectlc model, have been shown to be effective for describing low- lying collective states of hght (ds-sheU) nuclei [ 1- 3 ] Most recently a simple shell-model hamfltoman was introduced which successfully describes collec- tive features ofhght and heavy deformed nuclei, and m particular their E2 transmon probablhtles without the use of effectxve charges [3,4] The symmetry properties of this hamdtoman are mamfest most s~m- ply m the Sp(3, 0~) ~ U(3) ~ SU(3) ~ SO(3) group chain of the symplectxc model, and as a consequence the theory can be viewed as an extension of the EUlott SU(3) [5] (hght nuclei)and pseudo-SU(3) [61 (heavy nuclei) models that includes multiple 2hto lntershell excltauons of the monopole (l=0) and quadrupole (l= 2 ) type On the other hand, there are also various geomet- rical models, such as those of the Frankfurt school, that are phenomenologteal m nature and that have been applied with good success to many nuclei [ 7- 10 ] Understanding how these microscopic algebraic and phenomenologlcal geometric models are related to each other is important for gaming deeper insight ~r Supportedby projectUNAM-DGAPA IN 10-3091 and the US NaUonal Science Foundation under the joint US-Mexicoco- operauve ScienceProgram,Grant No INT88-01337 i Funded by the US National ScienceFoundaUon, Grant No PHY89-22550 into the structure of nuclei In ref [ 1 ] an attempt was made to estabhsh a connection between the al- gebraic and geometric approaches by assuming a power series m the quadrupole operator for the mter- action potential Although such an interaction ~s mi- croscopic, ~t is graded by the generahzed collective model and therefore deviates significantly from tra- dmonal one- plus two-body interactions because it includes three, four, and for a non-zero equlhbrlum gamma deformation, which reqmres at least a qua- dratlc dependence on cos(37), on up to six-body parts In ref [ 11 ] a very different approach was pro- posed Upon a detailed mvestxgatlon of the relatlon- sh~p of the rotor and SU(3) models, a one-to-one mapping of the irreducible representation (lrrep) la- bels (2,/t) of SU (3) to the deformatxon parameters fl, 7 of the collective model was estabhshed Speclfi- caUy, wxthm a single major shell the elgenvalue of the quadrupole-quadrupole operator, Q-Q, which pro- vldes a dxrect measure offl 2, ~s given for angular mo- mentum L"=0 + states by the expectation value of the second order SU (3) Caslmlr operator, C2, whde the elgenvalue of (Q×Q).Q, which measures f13cos(3y), is related to the expectauon value of the third order Caslmlr lnvarlant, 6"3 Since for any par- txeular nucleus the SU (3) lrreps allowed by the Pauh principle can be easily determined, this mapping yields a set of points that ldent~fies a sub-plane of the 0370-2693/92/$ 05 00 © 1992Elsevier SciencePubhshers B V All rights reserved 27