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Three-Dimensional Distribution of the ISM in the Milky Way Galaxy: IV. 3D Molecular Fraction and Galactic-Scale HI-to-H$_2$ Transition

Three dimensional (3D) distribution of the volume-density molecular fraction, defined by fmol_rho=rho_H2/(rho_HI + rho_H2), is studied in the Milky Way Galaxy. The molecular front appears at galacto-centric distance of R \sim 8 kpc, where the phase transition from atomic to molecular hydrogen occurs suddenly with fmol_rho dropping from \sim 0.8 to 0.2 within a radial interval as narrow as \sim 0.5 kpc. The front in fmol_rho is much sharper than that for surface density molecular fraction. The front also appears in the vertical direction with a full width of the high-fmol_rho disk to be \sim 100 pc. The radial and vertical fmol_rho profiles, particularly the front behaviors, are fitted by theoretical curves calculated using the observed density profile and assumed radiation field and metallicity with exponential gradients. The molecular fraction was found to be enhanced along the Perseus and some other spiral arms. The fmol_rho arms imply that the molecular clouds are produced from HI gas in the spiral arms and are dissociated in the interarm region. We also show that there is a threshold HI density over which the gas is transformed into molecules.