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. 2018 Mar 5;8:3988. doi: 10.1038/s41598-018-22279-9

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© The Author(s) 2018

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Linear thermal expansion coefficient (a) along a-axis (α_a), (b) along c-axis (α_c), and (c) α_c/α_a against a-axis lattice constant (a). The broken lines in Fig. 4(a,b and c) represent α_a, α_c, and α_c/α_a against a calculated by the hard-sphere model with minimum Madelung energy ( $α_{a}^{calc}$ , $α_{c}^{c a l c}$ , and $α_{c}^{c a l c}$ / $α_{a}^{calc}$ ).