  By means of the exact Path Integral Monte Carlo method we have performed a
detailed microscopic study of $^4$He nanodroplets doped with an argon ion,
Ar$^+$, at $T=0.5$ K. We have computed density profiles, energies, dissociation
energies and characterized the local order around the ion for nanodroplets with
a number of 4He atoms ranging from 10 to 64 and also 128. We have found the
formation of a stable solid structure around the ion, a "snowball", consisting
of 3 concentric shells in which the 4He atoms are placed on at the vertices of
platonic solids: the first inner shell is an icosahedron (12 atoms); the second
one is a dodecahedron with 20 atoms placed on the faces of the icosahedron of
the first shell; the third shell is again an icosahedron composed of 12 atoms
placed on the faces of the dodecahedron of the second shell. The "magic
numbers" implied by this structure, 12, 32 and 44 helium atoms, have been
observed in a recent experimental study [Bartl et al, J. Phys. Chem. A 118,
2014] of these complexes; the dissociation energy curve computed in the present
work shows jumps in correspondence with those found in the nanodroplets
abundance distribution measured in that experiment, strengthening the agreement
between theory and experiment. The same structures were predicted in Ref.
[Galli et al, J. Phys. Chem. A 115, 2011] in a study regarding
Na$^+$@$^4$He$_n$ when n>30; a comparison between Ar$^+$@$^4$He$_n$ and
Na$^+$@$^4$He$_n$ complexes is also presented.
