[en] In this paper, we introduce an improvement of the Debye model for the phonon spectrum in a quantum box at low temperatures. This improvement is a direct consequence of the consideration of the discrete phonon spectrum in some special quantum boxes, names in this paper, quantum cubic cells (QCC in which the highest significant level, q_T, is smaller than 100). We show that a discrete Planck spectrum may occur in nanometric QCC around the temperature 1K. Furthermore, we demonstrate that the total energy density and the heat capacity become functions of the product of cell size by temperature P_q, which are clear and measurable quantum effects in solid QCC. The limits of this quantum regime of the cubic cell are set as P_q is an element of [0.1, 1] and a reciprocity rule for the cell size and temperature is given. The thermodynamic functions for the phonon gas in QCC are recalculated taking into account their dependency on P_q. The calculation of the phonon average velocity, which plays a key role in the Debye model, is also reconsidered and simplified for quasi-isotropic cubic crystals. Finally, the theoretical formula of the Debye temperature for quasi-isotropic cubic solids is corrected (at low temperatures) and put in a form which depends on a single elastic constant. We show that this correction reduces the errors between the calculated and experimental Debye temperatures of some elements which crystallize in the cubic system. (author)