Crystal structure of . (a) View perpendicular to the to show the oxygen stacking. (b) View parallel to the to show relative positions of Na(1) and Na(2).
GGA formation energies of various Na-vacancy ordered states in with some of the ground state structures shown. (a) The concentration range, where each of the three ordering patterns (ROW, LZZ, and DRO) are favored, are shaded in red, yellow, and green, respectively; ROW ordering occurs at (b) and (c) ; LZZ ordering occurs at (d) and (e) ; DRO ordering occurs at concentration , e.g., (f) , (g) , and (h) .
Na ordering at (0.5556). The pattern can be recognized as a combination of ordering at (0.50) ⟨‘1’2⟩ and (0.60) .
Average numbers of bonds per formula unit for bond length up to for various ordering patterns. (a) ROW ordering pattern and (b) LZZ ordering pattern for ; (c) comparison for ROW and LZZ orderings for ; (d) types of bonds within (each type of bond is labeled and corresponds to the bond length in the histogram), the Na(1)–Na(2) nearest neighbor is circled with a dashed line since it never appears in any calculations. (e) LZZ ordering and (f) DRO ordering of ; (g) comparison of the number of bonds per formula unit for band length up to in LZZ and DRO ordering for .
The stabilization energy of Na(1) cluster formation in the dilute vacancy limit. Decrease in energy per vacancy when Na in is replaced by vacancies in various environments where singlet, triplet, sextet, and decatet Na(1) droplets form. The concentration is at the dilute vacancy limit , where there are no interactions between motifs. The energy change is calculated based on the changes in bonds between Na ions per vacancy up to distance . The strength of the pair interactions is obtained with a cluster expansion. Details of the cluster expansion will be reported in a separate paper.
lattice constant as a function of Na concentration. The open symbols show available experimental data from both the single crystal (Refs. 50 and 57) and powder diffraction (Ref. 42) measurements. The solid symbols are from our DFT calculations. A parabolic curve is fitted to the calculated values.
Change in Na(1)/Na(2) occupation ratio with Na concentration. The symbols show neutron data (Refs. 10 and 42–44). The red line shows the results obtained from previous DFT studies (Ref. 8). The blue line shows that our DFT calculation results are in good agreement with experiments, although the Na(1)/Na(2) ratio is from ground state calculations.
Ordering patterns at (0.75) proposed by (a) Ref. 7, named as “diamond,” (b) Ref. 11, named as “zigzag,” (c) Ref. 9, named as “stripe,” (d) Ref. 10, named as “droplet.”
Summary of ground states for . (Values in parentheses are not used in Figs. 6 and 7 since their energies are not on the convex hull.)
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