The positron spin-relaxation method permits the detection and investigation of positronium (Ps) in condensed matter with very small formation probabilities. In a set-up that makes use of spin-polarized positrons emitted from a source, the effect of reversals of an external magnetic field applied longitudinally to the positron spin polarization on the Doppler broadening of the annihilation photon line is measured. For the first time, the fractions r of Ps-forming positrons and normalized electron densities at the positron in Ps have been determined for natural type-IIa diamond and crystalline SiC . The two materials show a hitherto not observed anomalous dependence of the Doppler broadening on the magnetic field strength that indicates that a large Ps fraction annihilates before becoming thermalized. Fused quartz , fullerite , and highly orientated pyrolytic graphite showed the `normal' field dependence.