First application of the Trojan horse method with a radioactive ion beam: Study of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow/><mml:mn>18</mml:mn></mml:msup><mml:mi mathvariant="normal">F</mml:mi><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mn>15</mml:mn></mml:msup><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:math>reaction at astrophysical energies

Measurement of nuclear cross sections at astrophysical energies involving unstable species is one of the most challenging tasks in experimental nuclear physics. The use of indirect methods is often unavoidable in this scenario. In this paper the Trojan horse method is applied for the first time to a radioactive ion beam-induced reaction studying the ${}^{18}\mathrm{F}(p,\ensuremath{\alpha}$)${}^{15}\mathrm{O}$ process at low energies relevant to astrophysics via the three-body reaction ${}^{2}\mathrm{H}{(}^{18}\mathrm{F},{\ensuremath{\alpha}}^{15}\mathrm{O})n$. The knowledge of the ${}^{18}\mathrm{F}{(p,\ensuremath{\alpha})}^{15}\mathrm{O}$ reaction rate is crucial to understand the nova explosion phenomena. The cross section of this reaction is characterized by the presence of several resonances in $^{19}\mathrm{Ne}$ and possibly interference effects among them. The results reported in literature are not satisfactory and new investigations of the ${}^{18}\mathrm{F}{(p,\ensuremath{\alpha})}^{15}\mathrm{O}$ reaction cross section will be useful. In the present work the spin-parity assignments of relevant levels have been discussed and the astrophysical $S$ factor has been extracted considering also interference effects.