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The complex optical potential and inelastic form factor given by the foldingmodel for the $\alpha+^{12}$C scattering are used to calculate the($\alpha,\alpha'$) cross sections for the known isoscalar states of $^{12}$C inan elaborate coupled channel approach. The strengths of the form factors forthese states are then fine tuned against the ($\alpha,\alpha'$) data to deducethe corresponding $E\lambda$ transition strengths. A significant $E2$transition strength has been obtained for the $2^+_2$ state from the presentanalysis of the ($\alpha,\alpha'$) data measured at $E_\alpha=240$ and 386 MeV.The $E\lambda$ transition strengths of the $0^+_2,\ 3^-_1,\ 0^+_3,$ and $1^-_1$states were also carefully deduced, and some difference from the results ofearlier analyses has been found and qualitatively understood.

physical review. c. nuclear physics/physical review. c, nuclear physics

Folding model analysis of the inelastic $\alpha+^{12}$C scattering at  medium energies, and the isoscalar transition strengths of the cluster states  of $^{12}$C