The half inverse problem is to construct coefficients of the operator in a whole interval by using one spectrum and potential known in a semi interval. In this paper, by using the Hocstadt--Lieberma and Yang--Zettl's methods we show that if $p(x)$ and $q(x)$ are known on the interval $(\pi/2\pi)$, then only one spectrum suffices to determine $p(x)$, $q(x)$ functions and $\beta,h$ coefficients on the interval $(0,\pi)$ for impulsive diffusion operator with discontinuous coefficient.