Defect Equilibria in Solids

  • George G. Libowitz
Part of the Treatise on Solid State Chemistry book series (TSSC, volume 1)


An ideal crystal consists of a perfectly ordered arrangement of atoms, ions, or molecules. However, in any real crystal, at temperatures above absolute zero, there are always imperfections or defects in the crystal lattice, as discussed in Chapter 5. This chapter will deal with defects whose distribution and concentration in the lattice are governed by the laws of thermodynamics.† In pure crystals such defects are called native defects. The existence of native defects in a lattice arises from a tendency of a crystal to increase its entropy or degree of disorder. As defects are introduced into a crystal, the entropy ΔS will increase. The number of defects will be limited, however, by the enthalpy necessary to form the defects, ΔH. The actual number of defects present at any temperature is that which gives a minimum in the free energy G of the crystal according to the relation
$$G = G^* + \Delta H - T\Delta S = G^* + N_D \Delta H_D - TN_D \Delta S_u - T\Delta S_c (N_D )$$
where G* is the free energy of the theoretically perfect crystal, N D is the number of defects, ΔH D is the enthalpy change per defect, ΔS V is the change in vibrational entropy per defect, and ΔS c (N D ) is the change in configurational entropy, which is a function of the number of defects.


Point Defect Defect Concentration Intrinsic Defect Native Defect Configurational Entropy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. J. Burton, Vacancy-formation entropy in cubic metals, Phys. Rev. B5, 2948–2957 (1972).Google Scholar
  2. 2.
    A. C. Damask and G. J. Dienes, Point Defects in Metals, Gordon and Breach, New York (1963).Google Scholar
  3. 3.
    R. A. Johnson and W. D. Wilson, in Interatomic Potentials and Simulation of Lattice Defects (P. C. Gehlen, J. R. Beeler, and R. I. Jaffee, eds.), pp. 301–319, Plenum Press, New York (1972).CrossRefGoogle Scholar
  4. 4.
    K. H. Bennemann, New methods for treating lattice defects in covalent crystals, Phys. Rev. 137A, 1497–1514 (1965).CrossRefGoogle Scholar
  5. 5.
    R. A. Swalin, Theoretical calculation of the enthalpies and entropies of diffusion and vacancy formation in semiconductors, J. Phys. Chem. Solids 18, 290–296 (1961).CrossRefGoogle Scholar
  6. 6.
    W. L. Korst and J. C. Warf, Rare earth hydrogen systems. I. Structural and thermo-dynamic properties, Inorg. Chem. 5, 1719–1726 (1966).CrossRefGoogle Scholar
  7. 7.
    G. G. Libowitz and J. B. Lightstone, in Proc. 6th Rare Earth Research Conf., Gatlinburg, Tenn., Air Force Report AFOSR 67-1214, pp. 132-144 (1967).Google Scholar
  8. 8.
    M. D. Banus and T. B. Reed, in The Chemistry of Extended Defects in Non-Metallic Solids (L. Eyring and M. O’Keeffe, eds.), pp. 488–522, North-Holland, Amsterdam (1970).Google Scholar
  9. 9.
    G. G. Libowitz and T. R. P. Gibb, High pressure dissociation studies of the uranium hydrogen system, J. Phys. Chem. 61, 793–795 (1957).CrossRefGoogle Scholar
  10. 10.
    J. P. Pemsler and E. J. Rapperport, Thermodynamic properties of solid Au-Zn alloys by atomic absorption spectroscopy, Met. Trans. 2, 79–84 (1971).CrossRefGoogle Scholar
  11. 11.
    G. G. Libowitz, Point defects and thermodynamic properties in CsCl-type intermetallic compounds, Met. Trans. 2, 85–93 (1971).CrossRefGoogle Scholar
  12. 12.
    B. Fisher and D. S. Tannhauser, Electrical properties of cobalt monoxide, J. Chem. Phys. 44, 1663–1672 (1966).CrossRefGoogle Scholar
  13. 13.
    T. C. Harman, B. Paris, S. E. Miller, and H. L. Goering, Preparation and some physical properties of Bi2Te3, Sb2Te3, and As2Te3, J. Phys. Chem. Solids 2, 181–190 (1957).CrossRefGoogle Scholar
  14. G. R. Miller and C. Li, Evidence for the existence of antistructure defects in bismuth telluride by density measurements, J. Phys. Chem. Solids 26, 173–177 (1965).CrossRefGoogle Scholar
  15. R. F. Brebrick, Homogeneity ranges and Te2-pressure along the three-phase curves for Bi2Te3(c) and a 55–58 at % Te Peritectic Phase, J. Phys. Chem. Solids 30, 719–731 (1969).CrossRefGoogle Scholar
  16. 14.
    R. W. Vest, N. M. Tallan, and W. C. Tripp, Electrical properties and defect structure of zirconia: I. Monoclinic phase, J. Am. Ceram. Soc. 47, 635–640 (1964).CrossRefGoogle Scholar
  17. 15.
    G. Brouwer, A general asymptotic solution of reaction equations common in solid state chemistry, Philips Res. Rep. 9, 366–376 (1954).Google Scholar
  18. 16.
    F. A. Kröger and H. J. Vink, in Solid State Physics, Advances in Research and Applications (F. Seitz and D. Turnbull, eds.), pp. 307–435, Academic Press, New York (1956).Google Scholar
  19. 17.
    F. A. Kröger, The Chemistry of Imperfect Crystals, North-Holland, Amsterdam (1964).Google Scholar
  20. 18.
    W. Van Gool, Principles of Defect Chemistry of Crystalline Solids, Academic Press, New York (1966).Google Scholar
  21. 19.
    F. W. G. Rose, On the mass action laws in degenerate semiconductors, Proc. Phys. Soc. London 71, 699–701 (1958).CrossRefGoogle Scholar
  22. A. J. Rosenberg, Activity coefficients of electrons and holes at high concentrations, J. Chem. Phys. 33, 665–667 (1960).CrossRefGoogle Scholar
  23. 20.
    E. A. Guggenheim, Mixtures, Oxford Univ. Press, London (1952).Google Scholar
  24. 21.
    C. E. Messer and G. W. Hung, Dissociation pressures in the system LaH2-LaH3, 250-350°C, J. Phys. Chem. 72, 3958–3962 (1968).CrossRefGoogle Scholar
  25. 22.
    J. B. Lightstone and G. G. Libowitz, Interaction between point defects in nonstoichiometric compounds, J. Phys. Chem. Solids 30, 1025–1036 (1969).CrossRefGoogle Scholar
  26. 23.
    J. S. Anderson, The conditions of equilibrium of nonstoichiometric chemical compounds, Proc. Roy. Soc. (London) A185, 69–89 (1946).Google Scholar
  27. 24.
    J. C. Ward, Interaction between cation vacancies in pyrrhotite, Solid State Commun. 9, 357–359 (1971).CrossRefGoogle Scholar
  28. 25.
    M. Hoch, Order-disorder reactions in α-Ti(O) and TiO, J. Phys. Chem. Solids 24, 157–159 (1963).CrossRefGoogle Scholar
  29. 26.
    G. G. Libowitz, Nonstoichiometry and lattice defects in transition metal hydrides, J. Appl. Phys. 33, 399–405 (1962).CrossRefGoogle Scholar
  30. G. G. Libowitz and J. G. Pack, The gadolinium-hydrogen system at elevated temperatures. Vacancy interactions in gadolinium dihydride, J. Phys. Chem. 73, 2352–2356 (1969).CrossRefGoogle Scholar
  31. 27.
    M. Hoch, in Phase Stability in Metals and Alloys (P. Rudman, J. Stringer, and R. I. Jaffee, eds.), pp. 419–429, McGraw-Hill, New York (1967).Google Scholar
  32. 28.
    G. G. Libowitz, Nonstoichiometry in metal hydrides, Advances in Chem. Series No. 39, pp. 74–86 (1963).CrossRefGoogle Scholar
  33. 29.
    A. L. G. Rees, Statistical mechanics of two-component interstitial solid solutions, Trans. Faraday Soc. 50, 335–342 (1954).CrossRefGoogle Scholar
  34. 30.
    W. A. Oates, J. A. Lambert, and P. T. Gallagher, Monte Carlo calculations of configurational entropies in interstitial solid solutions, Trans. Met. Soc. AIME 245, 47–54 (1969).Google Scholar
  35. 31.
    A. B. Lidiard, Vacancy pairs in ionic crystals, Phys. Rev. 112, 54–55 (1958).CrossRefGoogle Scholar
  36. 32.
    K. P. Chik, D. Schumacher, and A. Seeger, in Phase Stabilities in Metals and Alloys (P. Rudman, J. Stringer, and R. I. Jaffee, eds.), pp. 449–467, McGraw-Hill, New York (1967).Google Scholar
  37. 33.
    H. Reiss and C. S. Fuller, Influence of holes and electrons on the solubility of lithium in boron-doped silicon, Trans. Met. Soc. AIME 206, 276–282 (1956).Google Scholar
  38. 34.
    R. Parker and M. S. Smith, The solubility of nickel in nickel ferrite, J. Phys. Chem. Solids 21, 76–80 (1961).CrossRefGoogle Scholar
  39. 35.
    R. Adams, P. Russo, R. Arnott and A. Wold, Preparation and properties of the systems CuFeS2.00−x and Cu1.00−xFe1.00+xS2.00−y, Met. Res. Bull. 7, 93–100 (1972).CrossRefGoogle Scholar
  40. 36.
    Y. D. Tretyakov and R. A. Rapp, Nonstoichiometric and defect structures in pure nickel oxide and lithium ferrite, Trans. Met. Soc. AIME 245, 1235–1241 (1969).Google Scholar
  41. 37.
    H. Schmalzried, Point defects in ternary ionic crystals, in Progress in Solid State Chemistry, Vol. 2, pp. 265–303, Pergamon Press (1965).CrossRefGoogle Scholar
  42. 38.
    W. L. George and R. E. Grace, Formation of point defects in calcium titanate, J. Phys. Chem. Solids 30, 881–887 (1969).CrossRefGoogle Scholar
  43. 39.
    L. Eyring and M. O’Keeffe (eds.), The Chemistry of Extended Defects in Non-Metallic Solids, North-Holland, Amsterdam (1970).Google Scholar
  44. 40.
    B. T. M. Willis, Positions of the oxygen atoms in UO2.13, Nature 197, 755–756 (1963).CrossRefGoogle Scholar
  45. 41.
    W. L. Roth, Defects in the crystal and magnetic structure of ferrous oxide, Acta Cryst. 13, 140–149 (1960).CrossRefGoogle Scholar
  46. 42.
    P. Kofstad and A.Z. Hed, Defect structure model for wustite, J. Electrochem. Soc. 115, 102–105 (1968).CrossRefGoogle Scholar
  47. 43.
    G. G. Libowitz, in Mass Transport in Oxides (J. B. Wachtman and A. D. Franklin, eds.), pp. 109-118, Natl. Bur. Std. Special Publ. 296 (1968).Google Scholar
  48. 44.
    F. Koch and J. B. Cohen, The defect structure of Fe1−x O, Acta Cryst. B25, 275–287 (1969).Google Scholar
  49. 45.
    R. A. Huggins and M. L. Huggins, Structural defect equilibria in vitreous silica and dilute silicates, J. Solid State Chem. 2, 385–395 (1970).CrossRefGoogle Scholar
  50. 46.
    J. S. Anderson, in Problems of Nonstoichiometry (A. Rabenau, ed.), pp. 1–76, North-Holland, Amsterdam (1970).Google Scholar

Copyright information

© Bell Telephone Laboratories, Incorporated 1921

Authors and Affiliations

  • George G. Libowitz
    • 1
  1. 1.Materials Research CenterAllied Chemical CorporationMorristownUSA

Personalised recommendations