Vibrational Spectrum, Caloric Curve, Low-Temperature Heat Capacity, and Debye Temperature of Sodium Clusters: The Na₁₃₉⁺ Case

The vibrational properties of atomic clusters are a fingerprint of their structures and can be used to investigate their thermodynamic behavior at low temperatures. In this work, we report a theoretical study, using density functional theory, on the vibrational spectrum and density of states (VDOS) of the cationic sodium cluster Na₁₃₉⁺. Our study focuses on the most stable isomer, which corresponds to a truncated icosahedron. This isomer displays an electronic density of states that is in good agreement with photoelectron spectroscopy data previously published. After validation of the sodium cluster structure, its vibrational frequency spectrum was obtained in the harmonic approximation through a diagonalization of the dynamical matrix. The calculated vibrational frequencies were used to evaluate the cluster caloric curve and heat capacity at low temperatures. An excellent agreement was obtained between the calculated caloric curve and experimental data recently reported down to 6 K. A fit to the bulk Debye model of the calculated and measured cluster thermal energy yields a large variation at low temperatures of the equivalent Debye temperature as compared with a weaker temperature dependence found in bulk materials. Moreover, a further analysis shows that the calculated heat capacity of the 139-atom cationic sodium cluster does not follow the bulk Debye <i>T</i>³ law at very low temperatures, due to the discreteness of the cluster frequency spectrum, and to the finite value of its acoustic gap (lowest frequency value). These results, indicating a finite size effect on the cluster vibrational spectrum, reflect the difference in the VDOS between clusters and bulk, and confirm the limitation of the bulk Debye model (and Debye temperature) to describe the low-temperature thermal behavior of metal clusters in the size range of around 139 atoms. The calculated vibrational frequency spectrum also provides the temperature dependence of the total vibrational excitation for the 139-atom sodium clusters, indicating that at 6 K, ∼ 92% of them are in their vibrational ground state