Abnormal negative thermal expansion of sodium: A first-principles discovery at high pressures

Negative thermal expansion coefficients ($\alpha$) are found in the fcc and cI16 (a distorted bcc structure) phases of sodium at pressures above 90 GPa from first principles. At room temperature, the most significant negative $\alpha$ of about $-4.5\times {10}^{-6}{\mathrm{K}}^{-1}$ is predicted in fcc sodium at 120 GPa, which represents the first prediction of a robust negative $\alpha$ in a simple metal. Anharmonic lattice dynamics calculations were carried out to rationalize this intriguing behavior. We find that the negative thermal expansion of fcc sodium originates from the transverse acoustic modes with large negative Grüneisen parameters. Furthermore, full phonon spectra of fcc sodium at elevated temperatures are calculated from perturbation theory. The phonon-phonon couplings in fcc sodium are found to increase abnormally under high pressures.

Figure 1

(a) Enthalpy differences between the competing phases of sodium from 20 to 150 GPa. (b) Thermal expansion coefficient of sodium at ambient pressure. The solid line is obtained from our calculations, and the dots represent experimental measurements [13]. (c) Thermal expansion coefficients of sodium at pressures of 50 GPa (bcc), 70 GPa (bcc), 80 GPa (fcc), 120 GPa (fcc), 130 GPa (cI16), and 140 GPa (cI16). (d) Grüneisen parameters at 300 K as a function of pressure.

Figure 2

(a) Phonon dispersions and (b) Grüneisen dispersions of fcc sodium at 80 GPa at 0 K. (c) Phonon dispersions and (d) Grüneisen dispersions of fcc sodium at 120 GPa at 0 K. TA, TA', and LA modes are given in red, green, and blue, respectively.

Figure 3

Phonon spectra of fcc sodium at (a) 50 K and (b) 200 K under a hydrostatic pressure of 80 GPa. Phonon spectra of fcc sodium at (c) 50 K and (d) 200 K under a hydrostatic pressure of 120 GPa are shown. The green curves give the phonon dispersions at 0 K.

Figure 4

Temperature-dependent frequency shifts of the TA and the LA phonon modes at the (a) X point, (b) L point, and (c) K point.