Limits of the stability of hexagonal phases upon uniaxial loading

Abstract

In modern studies, experimental methods for estimating nonlinear elastic moduli are often replaced with calculations of these quantities in mathematical simulation. However, different reliable models and experiments give different values of nonlinear elastic moduli. This work proposes a poßsible solution to the problem of which sets of elastic moduli should be used to predict the properties of substances. Two sets of the second-({c_αβ,γδ}), third-({c_αβ,γδ}), and fourth-order elastic moduli ({c_{αβ,γδ,μη,τρ}}) of hexagonal Gd crystal proposed in the literature are considered as examples. The elastic moduli are defined as partial derivatives of the nonequilibrium Landau potential Φ_L {u_αβ{ with respect to components of the tensor of homogeneous deformation of the crystal (Ü). Necessary information about the nonequilibrium Landau potential as a function of {u_αβ{ is given in the second section. An analytical way of deriving relationships between generally independent values of nonlinear elastic moduli caused by symmetry is proposed in Sections 3 and 4. The approach is based on using the integral rational basis of invariants (IRBI), which have the form of polynomials {u_aß{. Aspects of the theory of phase transitions based on IRBI are discussed in Section 3. The set of polynomials of the second, third, and fourth order included in the list of basic invariants {J _i ^(P)}, and the form of Landau potential Φ({J _i ^(P)}, are clearly defined. The forms of the chosen dependences {J _i ^(P)} and (Φ_L{J _i ^(P)}) are defined in Section 4 to compare the results from different works. Once the forms of Landau potential Φ_L{J _i ^(P)} and Φ_L{u_αβ} are defined, they are compared. The comparison results allow derivation of the nontrivial relationships between the components of the third-C ^III and fourth-rank elastic moduli tensors C ^IV due Gd hexagonal symmetry. Two different sets of calculated elastic moduli of Gd crystals, found in two different works, are given in Section 5. The criteria for selecting the most suitable set of numerical values of elastic moduli are described in Sections 6 and 7. One criterion is a comparison of load limits calculated from isothermal Gd elastic moduli and experimentally determined numerical values of the limits of stability of a certain Gd phase. In the last section, we show how the criterion based on comparing the phase stability limits allows us to dertermine which sets of third-rank elastic moduli should be used in, e.g., predicting Raman spectra.