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

Abstract

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 Na 139 + . 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 T 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.

Advances in thermodynamic modelling of nanoparticles

“…experimentally that vibrations of nanoparticles are still ruled by continuum mechanics laws down to sizes of a few nanometers [44,45]. At frequencies higher than the Debye frequency, the 'lattice' is unable to 'see' the vibrations because the wavelength of the vibrations is smaller than the lattice parameter; therefore, the vibrations become independent from the lattice.…”

Effect of Size, Temperature, and Structure on the Vibrational Heat Capacity of Small Neutral Gold Clusters

“…It is surely expected that such a vibrational spectrum depends on the material, size, and shape of clusters and nanoparticles [22]. Most importantly, the vibrational properties of atomic clusters are a fingerprint of their structures and can be used to investigate their thermodynamic behavior at low temperatures [23].…”

“…The acoustic vibrations are more important at low temperatures, because they dominate the heat capacity [20][21][22][23][24][25][26]. Low-frequency modes of harmonic systems can be related to the small amplitude of acoustic waves, which are experimentally observed in all elastic bodies.…”

“…The vibrational properties of clusters and small particles have been studied very intensively [19][20][21][22][23][24][25][26], and are vital for understanding and describing the atomic interactions in the cluster [27][28][29][30][31][32]. Thermal properties like heat capacity and thermal conductivity as well as many other material properties are strongly influenced by the vibrational density of states (VDOS).…”

Vibrational Heat Capacity of Gold Cluster AuN=14 at Low Temperatures

“…As one of the precious metals, gold has good corrosion resistance and extremely high stability and has been widely researched for biomedical applications [6,7]. The vibrational properties of clusters and small particles have been studied very intensively, [8][9][10][11][12][13][14][15] and are vital for understanding and describing the atomic interactions in the cluster [16][17][18][19][20][21]. Thermal properties like heat capacity and thermal conductivity as well as many other material properties are strongly influenced by the vibrational density of states (VDOS).…”