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

Path Integral Monte Carlo study confirms a highly ordered snowball in 4He nanodroplets doped with an Ar+ ion