Fig. 4

The density plot of the distances from lattice sites of mobile ions (Li⁺, Na⁺, Mg²⁺, and Al³⁺) to the surfaces of their nearest framework ions in 12,448 coordination environments. The density estimate plots of the minimal distances under different coordination environments (distinguished by coordination number) for Li⁺, Na⁺, Mg²⁺, and Al³⁺ are shown in S3 (a), (b), (c), and (d) of the Supplementary Information, respectively. Except that the maxima at the higher distances correspond to the next coordination shell, most of the minimum distances are distributed within a well-defined range. The double peeks of Al³⁺ are caused by inconsistent data distribution. The density estimate plot of Al³⁺ is obtained from the minimum distances calculated under CN = 4 (1,478 samples), CN = 5 (66 samples), and CN = 6 coordination (1,283 samples), but the minimum distances corresponding to CN = 4, CN = 5, and CN = 6 are sparsely distributed.