  We study the quantitative pointwise behavior of the solutions of the
linearized Boltzmann equation for hard potentials, Maxwellian molecules and
soft potentials, with Grad's angular cutoff assumption. More precisely, for
solutions inside the finite Mach number region, we obtain the pointwise fluid
structure for hard potentials and Maxwellian molecules, and optimal time decay
in the fluid part and sub-exponential time decay in the non-fluid part for soft
potentials. For solutions outside the finite Mach number region, we obtain
sub-exponential decay in the space variable. The singular wave estimate,
regularization estimate and refined weighted energy estimate play important
roles in this paper. Our results largely extend the classical results of Liu-Yu
\cite{[LiuYu], [LiuYu2], [LiuYu1]} and Lee-Liu-Yu \cite% {[LeeLiuYu]} to hard
and soft potentials by imposing suitable exponential velocity weight on the
initial condition.
