We discuss resonance properties in three-body systems, with examples on \(^{12}{\mathrm C}\) and \(^{24}\mathrm{Mg}\). We use a microscopic cluster model, where the generator coordinate is defined in the hyperspherical formalism. The \(^{12}{\mathrm C}\) nucleus is described by an \(\alpha +\alpha +\alpha \) structure, whereas \(^{24}\mathrm{Mg}\) is considered as an \(^{16}\mathrm{O}+\alpha +\alpha \) system. We essentially pay attention to resonances. We review various techniques which may extend variational methods to resonances. We consider \(0^+\) and \(2^+\) states in \(^{12}{\mathrm C}\) and \(^{24}\mathrm{Mg}\). We show that the r.m.s. radius of a resonance is strongly sensitive to the variational basis. This has consequences for the Hoyle state (\(0^+_2\) state in \(^{12}{\mathrm C}\)) whose radius has been calculated or measured in several works. In \(^{24}\mathrm{Mg}\), we identify two \(0^+\) resonances slightly below the three-body threshold.

我们以\（^ {12} {\ mathrm C} \）和\（^ {24} \ mathrm {Mg} \）为例讨论三体系统的共振特性。我们使用微观集群模型，其中生成器坐标在超球形形式中定义。的\（^ {12} {\ mathrmÇ} \）核是由描述\（\阿尔法+ \阿尔法+ \阿尔法\）结构，而\（^ {24} \ mathrm {镁} \）被认为是一个\（^ {16} \ {mathrmø} + \阿尔法+ \阿尔法\）系统。我们基本上要注意共鸣。我们回顾了可能将变分方法扩展到共振的各种技术。我们考虑\（^ {12} {\ mathrm C} \）中的\（0 ^ + \）和\（2 ^ + \）状态，\（^ {24} \ mathrm {Mg} \）。我们表明，共振的均方根半径对变化基础非常敏感。这对Hoyle状态（\（^ {12} {\ mathrm C} \）中的\（0 ^ + _ 2 \）状态）产生了影响，该半径已通过多次工作进行了计算或测量。在\（^ {24} \ mathrm {Mg} \）中，我们确定了两个\（0 ^ + \）共振，略低于三体阈值。