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A Theory of Anharmonic Lattice Statics for Analysis of Defective Crystals

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Abstract

This paper develops a theory of anharmonic lattice statics for the analysis of defective complex lattices. This theory differs from the classical treatments of defects in lattice statics in that it does not rely on harmonic and homogenous force constants. Instead, it starts with an interatomic potential, possibly with infinite range as appropriate for situations with electrostatics, and calculates the equilibrium states of defects. In particular, the present theory accounts for the differences in the force constants near defects and in the bulk. The present formulation reduces the analysis of defective crystals to the solution of a system of nonlinear difference equations with appropriate boundary conditions. A harmonic problem is obtained by linearizing the nonlinear equations, and a method for obtaining analytical solutions is described in situations where one can exploit symmetry. It is then extended to the anharmonic problem using modified Newton–Raphson iteration. The method is demonstrated for model problems motivated by domain walls in ferroelectric materials.

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References

  1. Agarwal, R.P.: Difference Equations and Inequalities. Marcel Dekker, New York (2000)

    MATH  Google Scholar 

  2. Babŭska, I.: The Fourier transform in the theory of difference equations and its applications. Arch. Mech. Stosow. 11(4), 349–381 (1959)

    MATH  Google Scholar 

  3. Babŭska, I., Vitásek, E., Kroupa, F.: Some applications of the discrete Fourier transform to problems of crystal lattice deformation, I,II. Czechoslov. J. Phys. B. 10, 419–427, 488–504 (1960)

    Article  ADS  MATH  Google Scholar 

  4. Benedetto, J.J.: Harmonic Analysis and Applications. CRC, Boca Raton, Florida (1997)

    Google Scholar 

  5. Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford University Press, London, UK (1988)

    MATH  Google Scholar 

  6. Boyer, L.L., Hardy, J.R.: Lattice statics applied to screw dislocations in cubic metals. Philos. Mag. 24, 647–671 (1971)

    ADS  Google Scholar 

  7. Briggs, W.L., Hendon, V.E.: The DFT: An Owner’s Manual for the Discrete Fourier Transform. SIAM, Philadelphia, Pennsylvania (1995)

    MATH  Google Scholar 

  8. Bullough, R., Tewary, V.K.: Lattice theory of dislocations. In: Nabarro, F.R.N. (ed.) Dislocations in Solids, vol. 2. North Holland, Amsterdam, The Netherlands (1979)

  9. De Boor, C., Holling, K., Riemenschneider, S.: Fundamental solutions of multivariate difference equations. J. Am. Math. Soc. 111, 403–415 (1989)

    MATH  Google Scholar 

  10. Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, Pennsylvania (1996)

    MATH  Google Scholar 

  11. Elaydi, S.N.: An Introduction to Difference Equations. Springer, Berlin Heidelberg New York (1996)

    MATH  Google Scholar 

  12. Esterling, D.M.: Modified lattice-statics approach to dislocation calculations I. Formalism. J. Appl. Phys. 49(7), 3954–3959 (1978)

    Article  ADS  Google Scholar 

  13. Esterling, D.M., Moriarty, J.A.: Modified lattice-statics approach to dislocation calculations II. Application. J. Appl. Phys. 49(7), 3960–3966 (1978)

    Article  ADS  Google Scholar 

  14. Faux, I.D.: The polarization catastrophe in defect calculations in ionic crystals. J. Phys. C, Solid State Phys. 4, L211–L216 (1971)

    Article  ADS  Google Scholar 

  15. Flocken, J.W.: Modified lattice-statics approach to point defect calculations. Phys. Rev. B. 6(4), 1176–1181 (1972)

    Article  ADS  Google Scholar 

  16. Flocken, J.W.: Modified lattice-statics approach to surface calculations in a monatomic lattice. Phys. Rev. B. 15, 4132–4135 (1977)

    Article  ADS  Google Scholar 

  17. Flocken, J.W., Hardy, J.R.: Application of the method of lattice statics to interstitial Cu atoms in Cu. Phys. Rev. 175(3), 919–927 (1968)

    Article  ADS  Google Scholar 

  18. Flocken, J.W., Hardy, J.R.: Application of the method of lattice statics to vacancies in Na, K, Rb, and Cs. Phys. Rev. 117(3), 1054–1062 (1969)

    Article  ADS  Google Scholar 

  19. Flocken, J.W., Hardy, J.R.: The method of lattice statics. In: Eyring, H., Henderson, D., (eds.) Fundamental Aspects of Dislocation Theory, vol. 1 of J.A. Simmons and R. de Wit and R. Bullough, pp. 219–245 (1970)

  20. Gallego, R., Ortiz, M.: A harmonic/anharmonic energy partition method for lattice statics computations. Model. Simul. Mater. Sci. Eng. 1, 417–436 (1993)

    Article  ADS  Google Scholar 

  21. Gregor, J.: The Cauchy problem for partial difference equations. Acta Appl. Math. 53, 247–263 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hsieh, C., Thomson, J.: Lattice theory of fracture and crack creep. J. Appl. Phys. 44, 2051–2063 (1973)

    Article  ADS  Google Scholar 

  23. Kanazaki, H.: Point defects in face-centered cubic lattice-I distortion around defects. J. Phys. Chem. Solids 2, 24–36 (1957)

    Article  ADS  Google Scholar 

  24. Keating, P.N.: Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure. Phys. Rev. 145(2), 637–645 (1966)

    Article  ADS  Google Scholar 

  25. King, K.C., Mura, T.: The eigenstrain method for small defects in a lattice. J. Phys. Chem. Solids 52(8), 1019–1030 (1991)

    Article  ADS  Google Scholar 

  26. Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications. Academic, New York (1988)

    MATH  Google Scholar 

  27. Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial difference equations by tensor product methods. Numer. Math. 6, 185–199 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  28. Maradudin, A.A.: Screw dislocations and discrete elastic theory. J. Phys. Chem. Solids 9, 1–20 (1958)

    Article  MathSciNet  ADS  Google Scholar 

  29. Maradudin, A.A., Montroll, E.W., Weiss, G.H.: Theory of Lattice Dynamics in The Harmonic Approximation. Academic, New York (1971)

    Google Scholar 

  30. Matsubara, T.J.: Theory of diffuse scattering of x-rays by local lattice distortions. J. Phys. Soc. Japan 7, 270–274 (1952)

    Article  ADS  Google Scholar 

  31. Meyer, B., Vanderbilt, D.: Ab initio study of ferroelectric domain walls in \(\textrm{PbTiO}_{3}\). Phys. Rev. B. 65, 1–11 (2001)

    Google Scholar 

  32. Mickens, R.E.: Difference Equations: Theory and Applications. Chapman & Hall, London, UK (1990)

    MATH  Google Scholar 

  33. Mura, T.: Micromechanics of Defects in Solids. Martinus Nijhoff, Boston, Massachussetts (1982)

    Google Scholar 

  34. Ortiz, M., Phillips, R.: Nanomechanics of defects in solids. Adv. Appl. Mech. 59(1), 1217–1233 (1999)

    MathSciNet  Google Scholar 

  35. Shenoy, V.B., Ortiz, M., Phillips, R.: The atomistic structure and energy of nascent dislocation loops. Model. Simul. Mater. Sci. Eng. 7, 603–619 (1999)

    Article  ADS  Google Scholar 

  36. Tewary, V.K.: Green-function method for lattice statics. Adv. Phys. 22, 757–810 (1973)

    Article  ADS  Google Scholar 

  37. Tewary, V.K.: Lattice-statics model for edge dislocations in crystals. Philos. Mag. A. 80(6), 1445–1452 (2000)

    Article  ADS  Google Scholar 

  38. Thomson, R., Zhou, S.J., Carlsson, A.E., Tewary, V.K.: Lattice imperfections studied by use of lattice Green’s functions. Phys. Rev., B. 46(17), 613–622 (1992)

    Article  Google Scholar 

  39. Tosi, M.P., Doyama, M.: Ion-model theory of polar molecules. Phys. Rev. 160(3), 716–718 (1967)

    Article  ADS  Google Scholar 

  40. Veit, J.: Fundamental solutions of partial difference equations. ZAMM 83(1), 51–59 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vitásek, E.: The n-dimensional fourier transform in the theory of difference equations. Arch. Mech. Stosow. 12(2), 185–202, 488–504 (1959)

    Google Scholar 

  42. Yavari, A.: Atomic structure of ferroelectric domain walls, free surfaces and steps. PhD thesis, California Institute of Technology (2004)

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Arash Yavari

  2. Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA, 91125, USA

    Michael Ortiz & Kaushik Bhattacharya

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  1. Arash Yavari
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  2. Michael Ortiz
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  3. Kaushik Bhattacharya
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Correspondence to Arash Yavari.

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Yavari, A., Ortiz, M. & Bhattacharya, K. A Theory of Anharmonic Lattice Statics for Analysis of Defective Crystals. J Elasticity 86, 41–83 (2007). https://doi.org/10.1007/s10659-006-9079-8

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  • DOI: https://doi.org/10.1007/s10659-006-9079-8

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