  Let $R$ be a local ring, and let $M$ and $N$ be finitely generated
$R$-modules such that $M$ has finite complete intersection dimension. In this
paper we define and study, under certain conditions, a pairing using the
modules $\Ext_R^i(M,N)$ which generalizes Buchweitz's notion of the Herbrand
diference. We exploit this pairing to examine the number of consecutive
vanishing of $\Ext_R^i(M,N)$ needed to ensure that $\Ext_R^i(M,N)=0$ for all
$i\gg 0$. Our results recover and improve on most of the known bounds in the
literature, especially when $R$ has dimension at most two.
