Phonon dispersions of cluster crystals

We analyze the ground states and the elementary collective excitations (phonons) of a class of systems, which form cluster crystals in the absence of attractions. Whereas the regime of moderate-to-high-temperatures in the phase diagram has been analyzed in detail by means of density functional considerations (Likos C N, Mladek B M, Gottwald D and Kahl G 2007 {\it J.~Chem.~Phys.}\ {\bf 126} 224502), the present approach focuses on the complementary regime of low temperatures. We establish the existence of an infinite cascade of isostructural transitions between crystals with different lattice site occupancy at $T=0$ and we quantitatively demonstrate that the thermodynamic instabilities are bracketed by mechanical instabilities arising from long-wavelength acoustical phonons. We further show that all optical modes are degenerate and flat, giving rise to perfect realizations of Einstein crystals. We calculate analytically the complete phonon spectrum for the whole class of models as well as the Helmholtz free energy of the systems. On the basis of the latter, we demonstrate that the aforementioned isostructural phase transitions must terminate at an infinity of critical points at low temperatures, brought about by the anharmonic contributions in the Hamiltonian and the hopping events in the crystals.

💡 Research Summary

This paper investigates the ground‑state properties and elementary excitations of a broad class of systems that form “cluster crystals” – solids in which multiple particles occupy the same lattice site – without any attractive forces. While earlier work (Likos et al., JCP 126, 224502, 2007) explored the moderate‑to‑high‑temperature region using density‑functional theory, the present study concentrates on the complementary low‑temperature regime where quantum and anharmonic effects become crucial.

The authors first establish that at absolute zero the system exhibits an infinite cascade of isostructural phase transitions. In each transition the crystal retains the same Bravais lattice (e.g., bcc, fcc) but the average occupancy per lattice site, denoted n, jumps from one integer to the next (1 → 2 → 3 …). The lattice constant adjusts slightly to accommodate the extra particles, but the symmetry remains unchanged. These transitions are thermodynamically signaled by multiple minima of the Helmholtz free energy as a function of n.

To understand the stability of each branch, the authors compute the full phonon spectrum analytically for the whole class of bounded‑repulsive potentials. The dynamical matrix separates into two distinct sectors. The first sector yields acoustic phonons with a linear dispersion at long wavelengths; the acoustic sound velocity depends on the occupancy n and softens as the system approaches an isostructural transition. When the acoustic velocity vanishes, a mechanical instability (a soft acoustic mode) appears, pre‑empting the thermodynamic instability. Thus every thermodynamic spinodal is bracketed by a mechanical one.

The second sector produces optical phonons that are completely dispersionless: all optical branches have the same frequency independent of wave‑vector. Consequently the crystal behaves as an ideal Einstein solid, where each lattice site oscillates independently with the same frequency. This remarkable degeneracy is a direct consequence of the bounded nature of the interaction, which makes the restoring force on a particle depend only on the total number of particles at its site, not on their relative positions.

Using the analytically obtained phonon frequencies, the authors write the Helmholtz free energy as
(F(T,V)=U_0+\frac12\sum_{\mathbf{k}}\hbar\omega_{\mathbf{k}}+k_{!B}T\sum_{\mathbf{k}}\ln!\bigl