Calculation of hydrodynamic mass for atomic impurities in helium

We present a simple numerical procedure for calculating the irrotational hydrodynamic flow in a helium solvation structure around a spherical solute in linear motion through superfluid helium. The calculation requires only the radial helium density around the impurity as input. From the resulting irrotational flow, the helium contribution to the effective mass of the solute is calculated. The results for alkali cations are compared to recent many-body Variational Monte Carlo (VMC) calculations by M. Buzzacchi, D. E. Galli, and L. Reatto (Phys. Rev. B., {\bf 64} 094512 (2001)). The helium contribution to the effective masses calculated by the two methods are 12.9(4.6) versus 9.4 u for Li$^+$, 48.2(5.6) versus 52.1 u for Na$^+$, 69.6(4.8) versus 70.1 u for K$^+$, and 6.4(8.8) versus 6.8 for Cs$^+$, with the VMC result listed first (with one $\sigma$ statistical error estimate) and the hydrodynamic result listed second. For the cases of Na$^+$ and K$^+$, the hydrodynamic calculation treated the first helium solvation shell as a rigid solid, as suggested by the VMC calculations; treating the first solvent layer as part of the superfluid lead to considerable underestimate, $\approx 50%$, of the mass increase in both cases. In all cases, the agreement of the two results are in within the error estimate of the VMC calculation, demonstrating the accuracy of hydrodynamic treatment of helium motion even on the atomic length scale.