Microscopic theory of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">Be</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>8</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>(<i>α</i>,γ<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>C reaction in a three-cluster model

The generator-coordinate method is applied to the study of the three-\ensuremath{\alpha}-particle system. The choice of the basis functions allows us to calculate the \ensuremath{\alpha}${+\mathrm{}}^{8}$Be phase shifts. We perform a comparison between the full three-cluster approach and two of its approximations. Distortion effects are shown to be important in the spectroscopic properties of $^{12}\mathrm{C}$. We present the $^{8}\mathrm{Be}$(\ensuremath{\alpha},\ensuremath{\gamma}${)\mathrm{}}^{12}$C S factor derived in the three-\ensuremath{\alpha}-particle model. At low temperatures, the reaction rate is found to be slightly different from earlier studies.