Charge-dependent spectral softenings of primary cosmic-rays from proton to iron below the knee

In most particle acceleration mechanisms, the maximum energy of the cosmic rays can achieve is charge dependent. However, the observational verification of such a fundamental relation is still lack due to the difficulty of measuring the spectra of individual particles from one (kind of) source(s) up to very high energies. This work reports direct measurements of the carbon, oxygen, and iron spectra from ~ 20 gigavolts to ~ 100 teravolts (~ 60 teravolts for iron) with 9 years of on-orbit data collected by the Dark Matter Particle Explorer (DAMPE). Distinct spectral softenings have been directly detected in these spectra for the first time. Combined with the updated proton and helium spectra, the spectral softening appears universally at a rigidity of ~ 15 teravolts. A nuclei mass dependent softening is rejected at a confidence level of > 99.999%. Taking into account the correlated structures at similar energies in the large-scale anisotropies of cosmic rays, one of the most natural interpretations of the spectral structures is the presence of a nearby cosmic ray source. In this case, the softening energies correspond to the acceleration upper limits of such a source, forming the so-called Peters cycle of the spectra. The results thus offer observational verification of the long-standing prediction of the charge-dependent energy limit of cosmic ray acceleration.