Bar-shape and eigenshape parameters from the zero-energy wave functions

It is shown that the Biedenharn-Blatt formula for the shape parameter for the coupled-channel case as an integral involving the zero-energy wave function actually gives the bar-shape parameter Pb and not the eigenshape parameter Pe as stated by the authors. We show that the zero-energy bar-wave functions and the zero-energy eigenwave functions are identical, and thus we obtain, by two different methods, the formula Pb-Pe=q2/(atr03), where at is the triplet scattering length, r0 the effective range, and q the energy derivative of the eigenmixing parameter at zero energy. The term q2/(atr03) results from the k2 behavior of the eigenmixing parameter at low energies, the bar-mixing parameter having a k3 dependence, and the order in which two particular limits are taken. We verify our new formula by considering potentials derived from the Reid hard-core potential model for the 3S1-3D1 state of the neutron-proton system, with the tensor potential taken to have a variable strength.