  We examine the impact of dark matter particle resolution on the formation of
a baryonic core in high resolution adaptive mesh refinement simulations. We
test the effect that both particle smoothing and particle splitting have on the
hydrodynamic properties of a collapsing halo at high redshift (z > 20).
Furthermore, we vary the background field intensity, with energy below the
Lyman limit (< 13.6 eV), as may be relevant for the case of metal-free star
formation and super-massive black hole seed formation. We find that using
particle splitting methods greatly increases our particle resolution without
introducing any numerical noise and allows us to achieve converged results over
a wide range of external background fields. Additionally, we find that for
lower values of the background field a lower dark matter particle mass is
required. We define the radius of the core as the point at which the enclosed
baryonic mass dominates over the enclosed dark matter mass. For our simulations
this results in $\rm{R_{core} \sim 5\ pc}$. We find that in order to produce
converged results which are not affected by dark matter particles requires that
the relationship ${M_{\rm{core}} / M_{\rm{DM}}} > 100.0$ be satisfied, where
${M_{\rm{core}}}$ is the enclosed baryon mass within the core and $M_{\rm{DM}}$
is the minimum dark matter particle mass. This ratio should provide a very
useful starting point for conducting convergence tests before any production
run simulations. We find that dark matter particle smoothing is a useful
adjunct to already highly resolved simulations.
