Temperature dependence of the momentum distribution of positronium in ${\mathrm{MgF}}_2,$ ${\mathrm{SiO}}_2,$ and ${\mathrm{H}}_2\mathrm{O}$

Temperature dependence of the momentum distribution of delocalized Bloch-positronium in solids is studied. The momentum distribution of the positronium, which is proportional to the energy integration of the spectral function weighted with the Bose distribution, is expressed in terms of the effective mass and the deformation potential of the crystal for positronium. A simple formula for the shape of the positronium peak in the 1D-ACAR spectrum is derived and applied to the analysis of the experimental data for ${\mathrm{MgF}}_2,$ ${\mathrm{SiO}}_2,$ and ${\mathrm{H}}_2\mathrm{O}$ in wide temperature ranges. An extraordinary broadening of the peak shape is observed for ${\mathrm{MgF}}_2.$ It is interpreted as an effect of the umklapp phonon scattering. The diffusion constants of Ps in these materials are also estimated.