Beta-expansions of rational numbers in quadratic Pisot bases

We study rational numbers with purely periodic R'enyi $\beta$-expansions. For bases $\beta$ satisfying ${\beta }^2=a\beta +b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $\gamma (\beta )=1$, i.e., that all rational numbers $p\mathrm{/}q\in [0,1)$ with $\gcd (q,b)=1$ have a purely periodic $\beta$-expansion. A simple algorithm for determining the value of $\gamma (\beta )$ for all quadratic Pisot numbers $\beta$ is described.