Theoretical assessment of the structure and stability of the $\lambda$ phase of nitrogen

The $\lambda$ phase of nitrogen was reported in 2016 and is one of more than a dozen high-pressure solid nitrogen forms that have been discovered. However, its crystal structure could not be solved unambiguously from powder diffraction alone; rather the reported structure was determined by combining experimental monoclinic lattice parameters with atomic positions from an earlier, computationally predicted structure that had similar unit cell dimensions. Here, we revisit this structure using density functional theory and higher-level fragment-based second-order Møller-Plesset perturbation theory (MP2) and coupled cluster singles, doubles, and perturbative triples [CCSD(T)]. Crystal structure prediction is performed to demonstrate that the reported $P{2}_1/c$ structure is indeed the likeliest candidate for the $\lambda$ phase. Furthermore, we provide further evidence for the structural assignment by demonstrating reasonable agreement between its predicted and experimental structural parameters and Raman spectra. Finally, the thermodynamic stability of the $\lambda$ phase relative to other phases has been uncertain, but the calculations do suggest that it may be the thermodynamically most stable phase for at least part of the pressure range over which it has been observed.

Figure 1

Phase diagram of nitrogen. The $\lambda$ phase has been observed over the conditions highlighted in red, though its thermodynamic stability relative to the other phases remains unclear.

Figure 2

Crystal energy landscape for the low-energy crystal structures at 34 GPa with the B86bPBE-XDM (red $Z$=4/orange $Z=2)$, MP2/CBS $+$ pHF (dark blue $Z$=4/light blue $Z=2)$, and MP2/CBS $+$ AMOEBA (dark green $Z$=4/light green $Z=2)$ levels of theory. Open symbols correspond to further CCSD(T) refinements of the structures. The experimentally inferred molar volume is indicated in purple. Enthalpies at each level of theory are plotted relative to the lowest-energy structure.

Figure 3

Structure overlay (top) and simulated powder x-ray diffraction spectrum (bottom) comparing the experimental [1], CCSD(T)/CBS $+$ periodic HF, and CCSD(T)/CBS $+$ AMOEBA crystal structures.

Figure 4

Comparison of the predicted and experimentally observed [1] equations of state for $\lambda {\mathrm{N}}_2$.

Figure 5

Comparison of the predicted and experimentally observed [1] Raman spectra for $\lambda$ nitrogen in the librational region.

Figure 6

Crystal structures of ${\mathrm{N}}_2$ (top left) phase $\alpha$ [59] (top right) phase $\gamma$ [60] (bottom left) phase $\epsilon$ [61] (bottom right) phase $\iota$ [7].

Figure 7

Comparison of CCSD(T) enthalpies at 0 K for the $\alpha$, $\gamma$, $\epsilon$, $\lambda$, and $\iota$ phases as a function of pressure using either periodic HF (solid lines) or AMOEBA (dotted lines) for the many-body treatment.