Isoscalar dipole resonance: Form factor and energy weighted sum rule

A collective formalism first proposed by Deal to describe isoscalar dipole excitations in electron scattering is extended to include excitations of these modes by hadron scattering. Distorted-wave Born approximation calculations are performed using this collective isoscalar $L=1\mathrm{}$ form factor to describe the excitation of the $1^{-}$, $T=0\mathrm{}$ state at $E_x=7.12\mathrm{}$ MeV ($1\hslash \omega$ character) in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and a newly proposed candidate for the isoscalar giant dipole resonance ($3\hslash \omega$ character) at $E_x=21.3\mathrm{}$ MeV in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ excited in inelastic $\alpha$ scattering. A good description of the cross sections of the $1^{-}$, 7.12 MeV state in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and the $E_x=21.3\mathrm{}$ MeV bump in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ is obtained in this model exhausting 4.2% and 130% of the energy-weighted sum rule, respectively.