Simply connected symplectic 4-manifolds with $b_2^{+}=1$ and $c_1^2=2$

In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^{+}=1$ and $c_1^2=2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a rational surface ${\mathbf CP}^2 \sharp 7{\bar{{\mathbf CP}}^2}$ admits an exotic smooth structure.