  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 $p/q\in[0,1)$ 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.
