Optimal Fermi-averaging t-matrix for ($\pi^+$,$K^+$) reactions in $\Lambda$ quasi-free region

We propose an optimal Fermi-averaging for an elementary $\pi^+$+$n$$\to$$K^+$+$\Lambda$ t-matrix under the on-energy-shell condition, in order to describe ($\pi^+$,$K^+$) reactions on a nuclear target in the framework of a distorted-wave impulse approximation. We apply it to calculate $\Lambda$ quasi-free spectra from $^{12}$C($\pi^+$,$K^+$) reactions at $p_\pi$= 1.20 GeV/c and 1.05 GeV/c, and compare them with experimental data. The results show that the calculated spectra are in excellent agreement with the data, because the energy-dependence originates from the nature of the optimal Fermi-averaging t-matrix.