A Correspondence between Phenomenological and IBM-1 Models of Even Isotopes of Yb

Abstract

This paper studies the nonterminal complexity of weakly conditional grammars, and it proves that every recursively enumerable language can be generated by a weak conditional grammar with no more than \textit{seven} nonterminals. Moreover, it is shown that the number of nonterminals in weakly conditional grammars without erasing rules leads to an infinite hierarchy of families of languages generated by weakly conditional grammars. Energy levels and the reduced probability of $E2$-- transitions for Ytterbium isotopes with proton number $Z=70$ and neutron numbers between 100 and 106 have been calculated through phenomenological ($PhM$) and interacting boson ($IBM-1$) models. The predicted low-lying levels (energies, spins and parities) and the reduced probability for $E2$-- transitions results are reasonably consistent with the available experimental data. The predicted low-lying levels ($gr$--, $\beta_1$-- and $\gamma_1$-- band) by produced in the $PhM$ are in good agreement with the experimental data comparison with those by $IBM-1$ for all nuclei of interest. In addition, the phenomenological model was successful in predicted the $\beta_2$--, $\beta_3$--, $\beta_4$--, $\gamma_2$-- and $1^+$-- band while it was a failure with $IBM-1$. Also, the $3^+$-- band is predicted by the $IBM-1$ model for $^{172}Yb$ and $^{174}Yb$ nuclei. All calculations are compared with the available experimental data.