Volume 162B, number 4,5,6 PHYSICS LETTERS 14 November 1985 VALIDITY OF THE SELF-CONSISTENT CRANKING APPROXIMATION IN THE SU(3)-U(5) PHASE TRANSITION M.C. CAMBIAGGIO 1, j. DUKELSKY and G.R. ZEMBA Departamento de Ftsiea, Comisibn National de Energ'ta Atbmica, Avda. del Libertador 8250, 1429, Buenos Aires, Argentina Received 20 June 1985 The validity of the self-consistent cranking approximation is studied in a soluble bosonic model which presents a first-order phase transition from an SU(3) (axially deformed) to an U(5) (spherical) limit. It is shown that the results are good not only in the SU(3) limit but also throughout the transition and even in the U(5) limit. The self-consistent cranking model (SCC) has been extensively used to describe the rotational spectra of wen-deformed nuclei [1-5]. It has been very success- ful in explaining the backending phenomenon as a band crossing, although recently doubts have been raised as to whether the cranking model can be ap- plied in regions of band crossing [6-9]. The method can be derived as a first approximation to an angular momentum projection before variation [10-12] un- der certain conditions: (i) "strong deformation", i.e., large angular momentum fluctuations (¢1 A,[2 [~b)>> 1; (ii) good signature, exp(inJx)l¢) cc I¢); (iii) nearly axial symmetry, (¢1,I2 I¢) ~ I(I + 1). Here Jx, Jy, Jz are the three components of the angular momentum operator and I¢) is the intrinsic wave function. The physical meaning of the first condition is not clear be- cause large angular momentum fluctuations do not necessarily imply large deformation parameters of the corresponding mean field. It is evident that well- deformed nuclei will satisfy condition (i) but this will also be the case with large particle numbers, even for small deformation parameters. How much may the deformation be decreased before the cranking approx- imation turns meaningless is still an open question. The purpose of the present letter is to look for an ans- wer to this question, studying the validity of condi- tion (i) in connection with the deformation. 1 Fellow of the Consejo Nacional de Investigaciones Cientificas y T6cnicas, Argentina. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) With this objective in mind we consider the interac- ting boson model (IBA) [13] which is specially suited to this investigation because it has a rotational [SU(3)] limit and a vibrational [U(5)] limit and it is very easy to follow the transition from one to the other [14]. Within this model it is easy to obtain the exact solu- tions not only in both limits where they are analytical, but also for any intermediate case by means of a sim- ple diagonalization. Moreover, the validity of the mod- el is supported by an excellent agreement with many experimental spectra in spherical, transitional and de- formed nuclei [15]. The many-boson hamiltonian considered is [ 14] /t= e(1 - X)/] d - KX02 "O2 , (1) where Q2"Q2 = ~ (-)UQ2uQ2-~,, (2) # and t" 2 1 1 2 O2tz = (¢0')'2 + ~/2~/0)# -- ~ %/ff'(~/2~/2)tz " (3) The parameter X varies between X = 0 [U(5) limit] and X = 1 [S.U(3)limit]. h d is the d-boson number op- erator and 7/T (7 l) creates (destroys) one boson with angular momentum l. The interaction strengths e and K are so chosen as to locate the phase transition at X -~ 0.5. The corresponding values are e = 1 MeV, K = 0.015 MeV. The exact results for different values of the parameter X are obtained through the code PHINT [16]. 203