  We determine hadronic matrix elements relevant for the mass and width
differences, $\Delta M_s$ & $\Delta \Gamma_s$ in the $B^0_s - \bar{B^0_s}$
meson system using fully unquenched lattice QCD. We employ the MILC
collaboration gauge configurations that include $u$, $d$ and $s$ sea quarks
using the improved staggered quark (AsqTad) action and a highly improved gluon
action. We implement the valence $s$ quark also with the AsqTad action and use
Nonrelativistic QCD for the valence $b$ quark. For the nonperturbative QCD
input into the Standard Model expression for $\Delta M_s$ we find $f_{B_s}
\sqrt{\hat{B}_{B_s}} = 0.281(21)$GeV. Results for four-fermion operator matrix
elements entering Standard Model formulas for $\Delta \Gamma_s$ are also
presented.
