  In the present paper, we study the collective states of even even nuclei in
gamma rigid mode within the sextic potential and the Minimal Length (ML)
formalism in Bohr Mottelson model. The eigenvalues problem for this latter is
solved by means conjointly of Quasi-Exact Solvability (QES) and a Quantum
Perturbation Method (QPM). Numerical calculations are performed for 35
nuclei:(98 108)Ru, (100 102)Mo, (116 130)Xe, (180 196)Pt, (172)Os, (146 150)Nd,
(132 134)Ce, (154)Gd, (156)Dy and (150 152)Sm. Through this study, it appears
that our elaborated model leads to an improved agreement of the theoretical
results with the corresponding experimental data by reducing the rms with a
rate going up to 63% for some nuclei. This comes out from the fact that we have
combined the sextic potential, which is a very useful phenomenological
potential, with the formalism of the ML which is based on the generalized
uncertainty principle and which is in turn a quantum concept widely used in
quantum physics. Besides, we investigate the effect of ML on energy ratios,
transition rates, moments of inertia and a shape phase transition for the most
numerous isotopic chains, namely Ru, Xe, Nd and Pt.
