Debye temperature of iron nanoparticles: Finite-size effects in the scaling regime

The Debye temperature ${\mathrm{\ensuremath{\Theta}}}_{\mathrm{D}}$ characterizes the vibrations of a solid and marks transition between quantized classical behaviors nuclear motion. While thermodynamical theories suggest that for nanoparticles should be lower than bulk limit increase with increasing nanoparticle size, various experimental measurements have reported intriguing variations including values above possibly decreasing size. In this paper we theoretically determined iron up to about 7 nm diameter at atomistic level detail, using complementary approaches based on equilibrium heat capacity or mean-square atomic displacement. Both methods consistently indicate steady finite-size effects in scaling regime, no evidence higher varying opposite ways, but they also produce marked quantitative differences. Further comparison melting ${T}_{m}$ indicates two quantities correlate each other through ${T}_{m}\ensuremath{\propto}{\mathrm{\ensuremath{\Theta}}}_{\mathrm{D}}^{\ensuremath{\alpha}}$, an exponent $\ensuremath{\alpha}$ close 3 deviates from expected value 2 predict pure ingredients.