Volume 34B, number 6 PHYSICS LETTERS 29 March 1971 INTRABAND BE2-RATIOS BY ANGULAR PROJECTED K = S. FRAUENDORF, D. JANSSEN and L. MI)NCHOW Zentralinstitut fiir Kernforschung, Rossendorf bei Dresden, DDR Received 10 January 1971 0 STATES The matrix elements of the Hamiltonian and the electric quadrupole operator with angular momentum projected states are parametrized to get expressions for the spectrum and the ratio BE2(I--* 1-2)/ (BE2(2 ~ 0) depending on few parameters only. By appropriate choice of these parameters the experi- ments can be reproduced with satisfying accuracy. One possibility to describe rotational bands is to project angular momentum eigenstates from a deformed internal state. In the case of axialsym- metric t~I,)the states of a K = 0 band are 7f llM)=N-½ f dOsinOdloM(O) R(O) I,~o) (1) 0 R(O) being the rotation operator and N a normali- zation constant. If ]~o ) is strongly deformed it is possible to confine its influence on matrix ele- ments to very few parameters chosen to fit the experimental data. By this method the energy levels of the ground state bands of even-even nu- clei have been calculated very accurately [1-3]. Recently E2-transition probabilities in the ground state band have been measured by multi- ple coulomb excitation and Dopplershift experi- ments [4-6]. In this note we derive analogous expressions for the excitation energy and the ratio BE2(I~I-2)/BE2(2--. 0). We will show that these expressions describe the experimental findings consistently whereas a pure stretching model in some cases leads to contradictions. The matrix element of a spherical tensor operator T~q with the angular momentum projected states (1) is [7] (IM[TLII'~) = (I'M'Lq JIM) 4~-~)/4 (2I+ 1)N(I)N(I') (2) ?r × ~ (rOLKIIK)f dO sin OtL (O)n(O) dlog(O) K 0 ff N(/) =fdOsinOn(O) d I (0) (3) 0 oo n(o) -- <~ol R(o) I%> (4) t~(o) : <% rr~- n(o) 1%~t n(o). For strongly deformed nuclei the overlap inte- gral n(O) may be approximated by a Gaussian, e.g. only the lowest order of an expansion of the exponent into powers sin 20 is kept. We use sin 20 instead of 0 2 as usual because it has the proper symmetry. n(O) = exp (-a 1 sin 2 O) (5) This approximation implies a statistical distri- bution for the large number of sharp angular momentum states in the decomposition of ~6o'" For the spatial case of the deformed BCS-solu- tion the approximation (5) has been shown nu- merically to be satisfied [1]. Consistent with (5) we keep only the first and second term of the ex- pansion of tKL(O). For the scalar Hamiltonian that means h(O) = h o + h 1 sin 20. (6) The transformation properties of tensor operators lead to symmetry relations to be obeyed by t. £, (O). Each term of an expansion should fulfill these K conditions. Considering the operator of electric quadrupole radiation this reads. t2 (O) =qK (O) = (5oK+doK (O))(qo +ql sin2 0)(7) The approximations discussed result in the fol- lowing relations determining the spectrum and the ratio BE2(I-~I-2)/BE2(2 ~ 0) [N2(I) N2 (0)~ = (8) 469