REDUCED ISOTROPIC CRYSTAL MODEL WITH RESPECT TO THE FOURTH-ORDER ELASTIC MODULI
Using a reduced isotropic crystal model the relationship between the fourth-order elastic moduli of an isotropic medium and the independent components of the fourth-order elastic moduli tensor of real crystals of various crystal systems is found. To calculate the coefficients of these relations, computer algebra systems Redberry and Mathematica for working with high order tensors in the symbolic and explicit form were used, in light of the overly complex computation. In an isotropic medium, there are four independent fourth order elastic moduli. This is due to the presence of four invariants for an eighth-rank tensor in the three-dimensional space, that has symmetries over the pairs of indices. As an example, the moduli of elasticity of an isotropic medium corresponding to certain crystals of cubic system are given (LiF, NaCl, MgO, CaF2). From the obtained results it can be seen that the reduced isotropic crystal model can be most effectively applied to high-symmetry crystal systems.
2. Landau L.D., Lifshitz E.M. Theory of Elasticity. − Oxford: Butterworth-Heinemann, 1986. − 187 p.
3. Fedorov F.I. Theory of Elastic Waves in Crystals. − New York: Plenum Press, 1968. – 375 p.
4. Akhiezer A.I., Aleksin V.F., Khodusov V.D. Gas dynamics of quasi-particles // Low Temp. Phys. – 1994. – Vol.20. – P.939.
5. Poluektov Yu.M. Self-Consistent Description Of Interacting Phonons In A Crystal Lattice // East Eur. J. Phys. − 2016. – Vol. 3. – No.3. − P. 35-46.
6. Krishnamurty T.S.G. Fourth-order Elastic Coefficients in Crystals // Acta Cryst. − 1963. − Vol. 16. − P. 839-840.
7. Destradea M., Ogdenb R.W. On the third- and fourth-order constants of incompressible isotropic elasticity // The Journal of the Acoustical Society of America. − 2010. − Vol. 128. − P. 3334-3343.
8. Garg V.K., Puri D.S., Verma M.P. Many-Body Effects on the Fourth-Order Elastic Constants of Ionic Solids // Phys. Stat. Sol. (b). – 1977. − Vol. 82. − P. 325-334.
9. Goyal S.C., Sharma A.K. Fourth-Order Elastic Constants of Fluorite Structure Solids // Phys. Stat. Sol. (b). – 1988. − Vol. 146. − P. 435-443.
10. Krishnamurty T.S.G., Gopalakrishnamurty P. Elastic coefficients in crystals // Acta Cryst. − 1968. − Vol. 24. − P. 563-565.
Nonlinear Effects in the Phonon System of Diamond Crystal
(2020) East European Journal of Physics
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).