  We study the question of pure periodicity of expansions in the negative base
numeration system. In analogy of Akiyama's result for positive Pisot unit base
$\beta$, we find a sufficient condition so that there exist an interval $J$
containing the origin such that the $(-\beta)$-expansion of every rational
number from $J$ is purely periodic. We focus on the case of quadratic bases and
demonstrate the following difference between the negative and positive bases:
It is known that the finiteness property (${\rm Fin}(\beta)=\Z[\beta]$) is not
only sufficient, but also necessary in the case of positive quadratic and cubic
bases. We show that ${\rm Fin}(-\beta)=\Z[\beta]$ is not necessary in the case
of negative bases.
