Hardness of solids


The study of the propeties of solids is based on ab-initio calculations of electron states of solids within the framework of density functional theory (DFT). Calculations should contribute to

a) understanding of the macroscopic properties of a solid based on its atomic arrangement and

b) explanation of experimental data.


InitiInitially, we interpreted the X-ray spectra of emission and absorption bands at our workplace and later at the sites in Halle, Warsaw, Munich, Paris and Vienna, which resulted in more than 100 publications. Recently, the most significant result has been to find the relationship between the atomic composition of a crystal and its hardness.

In 2006, we defined the hardness of the substance by means of the density of interatomic forces. It was the first time that all the variables needed to calculate the hardness could be obtained from the first principles. It is a completely new and fundamental feature in the theoretical approach to this very important property of materials. A new, very surprising result, which contradicted earlier opinions, is that the smaller number of closest atoms in the bond, i.e., structures with a smaller coordination number, leads to greater hardness of the substance than structures with a larger coordination number. This discovery was highlighted by Physics World Alerts web site as well as the Frontiers column of Physics World, Vol. 19, No. Page 4, April 2006 . The presented method clarifies the essence of hardness and makes it possible to predict the hardness of even purely hypothetical substances, which is crucial in the search for new superhard materials.

The dependence of hardness on the orientation of the measured sample was not taken into account in the 2006 and 2007 methods. The anisotropy of single crystal hardness, ie the effect of crystal orientation towards the external force direction, was described in an article from 2009. In contrast to previous ideas, it was found that not longitudinal but transverse interatomic bonds to the external force direction are decisive for crystal hardness. This result was also publicized by reputable journals Physical Review Focus Vol. 24, 4 September 2009 and in Research Highlights of Nature 461, 319 (17 September 2009).

Due to the great use of the above methods - around 500 citations - the approximation of interatomic forces in complex structures has been generalized and calculations have been made for the substances currently studied in recent publications (4) and (5). At present we optimize the approximations of interatomic forces.

The most important publications

1) Hardness of covalent and ionic crystals:First-principle calculations, A. Šimůnek and J. Vackář Phys.Rev.Lett. 96, 085501 (2006).

2) How to estimate hardness of crystals on a pocket calculator, A. Šimůnek, Phys.Rev. B75, 172108 (2007).

3) Anisotropy of hardness from first principles: The cases of ReB2 and OsB2, A. Šimůnek, Phys. Rev. B 80, 060103(R) (2009).

4) Generalized bond-strength model of Vickers hardness: Application to Cr4B, CrB, CrB2, CrB4, Mo2B, MoB2, OsB2, ReB2, WB2, WB3 a Ti1.87B50, A. Šimůnek and M. Dušek, Mechanics of Materials 112 (2017) 71-75.

5) Hardness of Re-, Ru-, Os-based borides and metal substituted aluminum diborides of MgB2 type Mo0.5Al0.5B2, A. Šimůnek and M. Dušek, Int.J.Refractory Metals & Hard Materials 82 (2019) 110-112. ​


Unit cell of ReB2. The bond Re-B are light, the bonds B-B are dark. On the left (a) vertical and horizontal direction runs along unit cell axes c and a, on the right (b) a perspective projection is shown along the axis c. Calculated hardness: along c 50.3, along a 41.8 GPa, perpendicular to a and c 40.0 GPa.


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