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. 2017 Apr 26;7:1192. doi: 10.1038/s41598-017-01251-z

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© The Author(s) 2017

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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The speed of sound through a (6 × 6 × 10) unit cell of diamond as a function of vacancy concentration for both the longitudinal (top) and transverse (bottom) polarizations. The values and error bars plotted at each concentration correspond to the mean and standard deviation for the value of 50 randomly selected vacancy configurations. The inset shows a longitudinal wave propagating through the unit cell used for these calculations, and is meant to emphasize that potentials of this kind can couple short-range quantum mechanical information acquired from ab initio calculations with large, complex systems, facilitating the study of emergent behavior. The longitudinal speed of sound at zero vacancy concentration is in good agreement with experimental values⁴⁶.