A Package for calculating elastic tensors of cubic - WIEN 2k

Background theory for calculating elasticconstants of hexagonal phases used in theHex-elasticPackage 1Morteza Jamal15 th Dec 2012Ghods City-Tehran-IranElastic constants are defined by means of a Taylor expansion of thetotal energy E(v,e) for the system, with respect to a small strain “e” of thelattice (v is the volume). In this Package we consider the hcp crystalstructure, which is spanned by three vectors ( √3a/2, -a/2, 0), ( 0, a, 0), and( 0, 0, c ). The Bravais lattice vectors are normally written in a matrix form,i.e.,√3a/2 -a/2 0R= 0 a 00 0 c

E(v,e) = E(v 0 ,0) +v 0 [(∆ 1+ ∆ 2 ) e + ( C 11 + C 12 ) e 2 + o(e 3 )]The second type of distortion is a volume conserved distortion and lead toOrthorhombic symmetry and written as1+e 1 0 0 1+e 1 =(1+x/1-x) 1/20 1+e 2 0 1+e 2 =(1-x/1+x) 1/20 0 1and the energy for this distortion can be obtained asE(v,e) = E(v 0 ,0) +v 0 [ ( C 11 - C 12 ) x 2 + o(x 3 )]The third strain we have used is given by1 0 00 1 00 0 1+eThis strain changes C lattice parameter and keep the symmetry of thestrained lattice hexagonal and the energy for this distortion can be obtainedbyE(v,e) = E(v 0 ,0) +v 0 [(∆ 3 ) e + ( C 33 ) e 2 /2+ o(e 3 )]The fourth elastic constant, C55, is determined by means of a deformation ofthe lattice, which produces an object with low symmetry. The deformation iswritten as1 0 e0 1 0e 0 1And it leads to triclinic symmetry and the energy for this deformation can bewritten asE(v,e) = E(v 0 ,0) +v 0 [(∆ 5 ) e + (2 C 55 ) e 2 + o(e 3 )]

Finally, the last strain we have used is volume conserved and keeps thesymmetry of the strained lattice hexagonal and can be written as1+e 1 0 0 1+e 1 =(1+x) - 1/30 1+e 2 0 1+e 2 =(1+x) - 1/30 0 1+ e 3 1+e 3 =(1+x) 2/3And the energy for this strain is given byE(v,e) = E(v 0 ,0) +v 0 [ ( C zz )/9 x 2 + o(x 3 )]AndC zz = C 11 + C 12 + 2C 33 - 4C 13In practice, to calculate elastic constants we fit E(v,e) to a polynomial ofdegree M and M changes from 2 till N-1 , N is number of data, and thenelastic constants is computed by using second order derivative ( E”(e) ) ofPolynomial fit ( E=E(v,e) ) of Energy vs. strains (e) at zero strain (e=0). Inthis situation we are able to check the sensitivity of our results (elasticconstants) to the order of fit however final results are for a polynomial ofdegree 2 ( M=2) because we are in the regime of Hooke’s law.[1] L. Fast, J. M. Wills, B. Johansson and O. Eriksson, Phys. Rev. B 51,(1995).[2] D.C. Wallace, Solid State Phys. 25, 301 (1970)