Evaluation of angular integrals using matrix algebra: Applications to the α-decay integralsKll′mand the scattering of deformed nuclei

In many physical problems one is faced with the calculation of matrix elements of the form 〈lm|f(\ensuremath{\theta})|l\ensuremath{'}m〉. We use matrix algebra to derive a powerful technique for the numerical evaluation of such integrals. We obtain two different integration formulas: one which may be used if f has no particular symmetry and a second which exploits reflection symmetry of this operator. Our final results are essentially generalizations of the Gauss quadrature formula especially suited to the evaluation of matrix elements of the above form. We demonstrate this with an application to the angular part of the barrier transmission coefficient for the \ensuremath{\alpha} decay of a heavy nucleus. The derivation involves a unitary transformation to the eigenstates of the Legendre polynomials ${\mathrm{P}}_{1}$ or ${\mathrm{P}}_{2}$ in a truncated angular momentum space. The same transformation also relates the eigenchannels of the adiabatic treatment of the scattering of deformed nuclei to the corresponding physical channels.