Periodic HartreeâFock linear combination of crystalline orbitals ...

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7214 C A Perottoni et alTable 3. Elastic constants of titanium diboride (GPa).Reference c 11 c 12 c 13 c 33 c 44ExperimentalGilman and Roberts [25] 690 410 320 440 250Spoor et al [26] 660 48 93 432 260Manghnani et al a 588 72 84 503 238Wright b 672 40 125 224 232Wright c 711 17 118 349 240TheoreticalThis work 786 127 87 583 271Van Camp and van Doren d 777 e 83 568Van Camp and van Doren f 730 78 572a TiB 2 polycrystalline elastic stiffnesses [27].b Voigt estimates of the single-crystal elastic constants calculated from the reported values ofManghnani, Fisher, Li and Grady [27] after correction for texture effects [28].c The same as above, but using the Reuss approximation.d DFT with ab initio norm-conserving pseudopotentials in the local density approximation (LDA)for the exchange–correlation contribution to the energy [13].e c 11 + c 12 .f The same as in note d, but in the generalized gradient approximation (GGA).c 44 of TiB 2 compare well with the values measured at room conditions by Spoor et al [26]. Theagreement is worse for c 11 , c 33 and c 12 . However, one should note the great variance among theexperimental results, mainly for c 12 . The DFT value for c 11 + c 12 is closer to the experimentalresults than the corresponding HF estimate of this work. However, both HF and DFT (in theLDA and GGA approximation) approaches lead to a value for c 33 somewhat greater than theexperimental one. This tendency of HF calculations to overestimate some elastic constantswas already pointed out by several authors (see, e.g., references [30–32]).From the general relationship between elastic compliances (s ij ) and bulk compressibility[33]1B = s 11 + s 22 + s 33 +2(s 12 + s 23 + s 31 ) (4)one obtains the bulk modulus at zero pressure for a single crystal with hexagonal symmetry(without constraints on the c/a dependence on lattice strain) as given byB 0 = c 33(c 11 + c 12 ) − 2c 2 13c 11 + c 12 − 4c 13 +2c 33. (5)With the elastic constants from table 3, the above equation yields B 0 = 299 GPa, only 2.4%higher than the value obtained from the fitting of the Murnaghan equation of state to the pointsin figure 2(a). This reveals that the whole set of calculated elastic constants is consistent withthe bulk modulus obtained through the analysis of the volume dependence of the primitive-cellenergy, as done in section 3.The elastic anisotropy of a TiB 2 single crystal can be best appreciated by plotting atridimensional representation of the directional dependence of Young and linear bulk moduli[33] (figure 5). The surface of revolution representing the directional dependence of the TiB 2bulk modulus, calculated from the set of elastic constants given in table 3, intercepts the a-axis at 1043 GPa and the c-axis at 700 GPa, in good agreement with the B a0 - and B c0 -valuesdetermined from the pressure dependence of the lattice parameters (section 3).The Young modulus surface of revolution intercepts the c-axis at 566 GPa and the a-axisat 756 GPa. In other words, the Young modulus parallel to the basal plane is about 30% higher
Periodic HF-LCCO calculations for titanium diboride 7215Figure 5. The directional dependency of the bulk (above) and Young (below) moduli of titaniumdiboride. The axis scales are given in GPa.than the one parallel to the c-axis. This plot can be directly compared to the experimentalresults given by Spoor et al [26].The effect of the anisotropic linear compressibility on the overall compressibility oftitanium diboride can also be evaluated by calculating the isotropic bulk modulus (B iso ), i.e.,the bulk modulus obtained under the assumption that the c/a ratio remains unchanged whenthe lattice is subjected to an isotropic stress. B iso can be written in terms of the elastic
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Periodic HartreeâFock linear combination of crystalline orbitals ...