  We introduce a perturbation expansion for athermal systems that allows an
exact determination of displacement fields away from the crystalline state as a
response to disorder. We show that the displacement fields in energy minimized
configurations of particles interacting through central potentials with
microscopic disorder, can be obtained as a series expansion in the strength of
the disorder. We introduce a hierarchy of force balance equations that allows
an order-by-order determination of the displacement fields, with the solutions
at lower orders providing sources for the higher order solutions. This allows
the simultaneous force balance equations to be solved, within a hierarchical
perturbation expansion to arbitrary accuracy. We present exact results for an
isotropic defect introduced into the crystalline ground state at linear order
and second order in our expansion. We show that the displacement fields
produced by the defect display interesting self-similar properties at every
order. We derive a $|\delta r| \sim 1/r$ and $|\delta f| \sim 1/r^2$ decay for
the displacement fields and excess forces at large distances $r$ away from the
defect. Finally we derive non-linear corrections introduced by the interactions
between defects at second order in our expansion. We verify our exact results
with displacement fields obtained from energy minimized configurations of soft
disks.
