Mikheyev-Smirnov-Wolfenstein Effects in Vacuum Oscillations

We point out that for solar neutrino oscillations with the mass-squared difference of $\Delta m^2\sim {10}^{-10\mathrm{}}–{10}^{-9\mathrm{}}{\mathrm{eV}}^2$, i.e., in the so-called vacuum oscillation range, the solar matter effects are non-negligible, particularly for the low energy $\mathrm{pp}$ neutrinos. One consequence of this is that the values of the mixing angle $\theta$ and $\pi /2-\theta$ are not equivalent, making it necessary to consider the entire physical range of the mixing angle $0\le \theta \le \pi /2$ when determining the allowed values of the neutrino oscillation parameters.