Using the renormalization group (RG) theory and Monte Carlo simulation (MCS), the semi-infinite mixed spin-1/2 and spin-5/2 Blume–Capel system has been studied. Depending on the ratios (bulk–surface) R and D, four qualitative types of phase diagrams are determined and classified; each type is characterized by the presence or not of ordinary, extraordinary, surface and special phase transitions. To show the critical behavior of this system at the vicinity of the first-order transitions at low temperature using RG theory, we have plotted the variations of the bulk and surface free energy derivatives. We have also given the associated fixed points and the critical exponents showing the different classes of universality at the surface. In addition, the distribution of the different types of phase diagrams in the (R, D) plane is given exhibiting the existence of a critical value R_C = 0.57 delimiting two types on each side. The MCS method is also employed to validate the results obtained by the RG theory. To confirm the critical behavior of the system near the first-order transitions at low temperature using MCS, we have presented the variations of the bulk and surface magnetizations per site. Finally, this work is completed by discussions and comparisons.

使用重归一化组（RG）理论和蒙特卡洛模拟（MCS），研究了半无限混合的spin-1 / 2和spin-5 / 2 Blume-Capel系统。取决于比率（体表）R和D，确定并分类了四种定性类型的相图；每种类型的特征在于是否存在普通，非常规，表面和特殊相变。为了使用RG理论显示该系统在低温下一阶跃迁附近的临界行为，我们绘制了体能和表面自由能导数的变化图。我们还给出了相关的固定点和临界指数，这些指数在表面上显示出不同的通用性类别。此外，给出了（R，D）平面中不同类型相图的分布，显示出存在临界值R _C。 = 0.57，每侧限定两种类型。MCS方法也用于验证通过RG理论获得的结果。为了使用MCS确认系统在低温下靠近一阶跃迁的临界行为，我们介绍了每个部位的体积和表面磁化强度的变化。最后，通过讨论和比较完成了这项工作。