Abstract

Solid solution hardening has been reviewed with respect to existing experimental data. It is shown with the aid of new experimental data on two inorganic systems (NiO–CoO and CaF2–SrF2) that the large increase in hardness of intermediate compositions correlates qualitatively and can also be quantitatively justified by the use of selected values of the parameters in a previously developed atomistic hardness formula. The key parameters are the variations in the unit cell dimensions, the eigenfrequency, and anharmonic factor of the structure.

© 1971 Optical Society of America

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References

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  1. E. Friedrich, Fortschr. Chem. Phys. Physikal. Chem. 18, 1 (1926).
  2. V. M. Goldschmidt, Geochemische Verteilingsgesetze der Elemente (Oslo, 1927).
  3. A. S. Powarjonnych, Geologie, 93–99 (1959); also in Aspects of Theoretical Mineralogy in the USSR, M. H. Battey, S. I. Tomkieff, Eds. (Plenum Press, New York, 1964), pp.379 ff.
  4. J. N. Plendl, P. J. Gielisse, Phys. Rev. 125, 828 (1962).
    [CrossRef]
  5. J. N. Plendl, S. S. Mitra, P. J. Gielisse, Phys. Stat. Sol. 12, 376 (1965).
  6. J. N. Plendl, P. J. Gielisse, Z. Kristallogr. 118, 404 (1963).
    [CrossRef]
  7. G. Sachs, Praktische Metallkunde (Berlin, 1935), Vol. 3, p. 35.
  8. W. Kuntze, in G. Masing, Handbuch der Metallphysik (Akademie-Verlagsges., Leipzig, 1940), p. 276.
  9. H. G. F. Winkler, Struktur und Eigenschaften der Kristalle (Springer, Berlin, 1950), p. 151.
  10. W. Koester, W. Rauscher, Z. Metallkunde 39, 111 (1948).
  11. T. S. Liu, R. J. Stokes, C. H. Li, J. Amer. Ceram. Soc. 47, 276 (1964).
    [CrossRef]
  12. T. T. Kazandzhyan, V. D. Galstyan, R. S. Mkhetaryan, Nauk Trudy Erevan Univ. 44, Ser. Khim. 2, 95 (1954).
  13. A. A. Vorobev, E. K. Zavodoskaya, M. S. Ivankina, Izv. Vysshikh Ucheb Zavedenii Fiz. 6, 162 (1959); A. A. Vorobev. Zh. Tekh. Fiz. 26, 330 (1956). P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izv. Vysshikh Ucheb Zavedenii Fiz. 4, 86 (1958); G. A. Wolff, L. Toman, N. J. Field, J. C. Clark, Semiconductors and Phosphors, Proc. Intern. Colloq. Garmisch Partenkirchen 1956 (1958), pp. 463–469.
  14. P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izvest. Vysshikh Ucheb. Zavedenii Fiz. 4, 86 (1958).
  15. A. A. Vorobev, Zh. Tekh. Fiz. 26, 330 (1956) [Sov. Phys. Tech. Phys. 1, 324 (1956)].
  16. N. N. Sirota, V. V. Rozov., Akad. Nauk. BSSR Dokl. 7, 446 (1963).
  17. A. Smakula, Quarterly Progress Reports, AF 19(628)-395 and AF 19(628)-2418 (1962–1967).
  18. E. G. Chernevskaya, Sov. J. Opt. Technol. 33, 346 (1966).
  19. It should be noted here that the value of the structure constant of CaF2 (CN = 4; 8) is π/2. This value actually corresponds to the atomic coordination number CN = 8.
  20. J. N. Plendl, “Characteristic Energy Absorption Spectra of Solids,” in Far Infrared Properties of Solids (Plenum, New York, 1970). Also Appl. Opt. 9, 2768 (1970).
    [CrossRef] [PubMed]
  21. In previous publications the quantity Ψ(A) has been described as the anharmonicity of solids. The term anharmonic factor is felt to be more appropriate and is used to discriminate between its present use and anharmonicity as used to describe third- and fourth-order terms in lattice vibrational potential.
  22. P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
    [CrossRef]
  23. P. Tarte, U. of Liége (Belgium), Quarterly Reports on AFCRL contract #AF 61(052)917 (1966–1969).
  24. J. J. Stoker, Nonlinear Vibrations (Interscience, New York, 1950), p. 15.

1966 (1)

E. G. Chernevskaya, Sov. J. Opt. Technol. 33, 346 (1966).

1965 (2)

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

J. N. Plendl, S. S. Mitra, P. J. Gielisse, Phys. Stat. Sol. 12, 376 (1965).

1964 (1)

T. S. Liu, R. J. Stokes, C. H. Li, J. Amer. Ceram. Soc. 47, 276 (1964).
[CrossRef]

1963 (2)

J. N. Plendl, P. J. Gielisse, Z. Kristallogr. 118, 404 (1963).
[CrossRef]

N. N. Sirota, V. V. Rozov., Akad. Nauk. BSSR Dokl. 7, 446 (1963).

1962 (1)

J. N. Plendl, P. J. Gielisse, Phys. Rev. 125, 828 (1962).
[CrossRef]

1959 (2)

A. S. Powarjonnych, Geologie, 93–99 (1959); also in Aspects of Theoretical Mineralogy in the USSR, M. H. Battey, S. I. Tomkieff, Eds. (Plenum Press, New York, 1964), pp.379 ff.

A. A. Vorobev, E. K. Zavodoskaya, M. S. Ivankina, Izv. Vysshikh Ucheb Zavedenii Fiz. 6, 162 (1959); A. A. Vorobev. Zh. Tekh. Fiz. 26, 330 (1956). P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izv. Vysshikh Ucheb Zavedenii Fiz. 4, 86 (1958); G. A. Wolff, L. Toman, N. J. Field, J. C. Clark, Semiconductors and Phosphors, Proc. Intern. Colloq. Garmisch Partenkirchen 1956 (1958), pp. 463–469.

1958 (1)

P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izvest. Vysshikh Ucheb. Zavedenii Fiz. 4, 86 (1958).

1956 (1)

A. A. Vorobev, Zh. Tekh. Fiz. 26, 330 (1956) [Sov. Phys. Tech. Phys. 1, 324 (1956)].

1948 (1)

W. Koester, W. Rauscher, Z. Metallkunde 39, 111 (1948).

1926 (1)

E. Friedrich, Fortschr. Chem. Phys. Physikal. Chem. 18, 1 (1926).

Chernevskaya, E. G.

E. G. Chernevskaya, Sov. J. Opt. Technol. 33, 346 (1966).

Friedrich, E.

E. Friedrich, Fortschr. Chem. Phys. Physikal. Chem. 18, 1 (1926).

Galstyan, V. D.

T. T. Kazandzhyan, V. D. Galstyan, R. S. Mkhetaryan, Nauk Trudy Erevan Univ. 44, Ser. Khim. 2, 95 (1954).

Gielisse, P. J.

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

J. N. Plendl, S. S. Mitra, P. J. Gielisse, Phys. Stat. Sol. 12, 376 (1965).

J. N. Plendl, P. J. Gielisse, Z. Kristallogr. 118, 404 (1963).
[CrossRef]

J. N. Plendl, P. J. Gielisse, Phys. Rev. 125, 828 (1962).
[CrossRef]

Goldschmidt, V. M.

V. M. Goldschmidt, Geochemische Verteilingsgesetze der Elemente (Oslo, 1927).

Ivankina, M. S.

A. A. Vorobev, E. K. Zavodoskaya, M. S. Ivankina, Izv. Vysshikh Ucheb Zavedenii Fiz. 6, 162 (1959); A. A. Vorobev. Zh. Tekh. Fiz. 26, 330 (1956). P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izv. Vysshikh Ucheb Zavedenii Fiz. 4, 86 (1958); G. A. Wolff, L. Toman, N. J. Field, J. C. Clark, Semiconductors and Phosphors, Proc. Intern. Colloq. Garmisch Partenkirchen 1956 (1958), pp. 463–469.

Kazandzhyan, T. T.

T. T. Kazandzhyan, V. D. Galstyan, R. S. Mkhetaryan, Nauk Trudy Erevan Univ. 44, Ser. Khim. 2, 95 (1954).

Koester, W.

W. Koester, W. Rauscher, Z. Metallkunde 39, 111 (1948).

Kuntze, W.

W. Kuntze, in G. Masing, Handbuch der Metallphysik (Akademie-Verlagsges., Leipzig, 1940), p. 276.

Li, C. H.

T. S. Liu, R. J. Stokes, C. H. Li, J. Amer. Ceram. Soc. 47, 276 (1964).
[CrossRef]

Liu, T. S.

T. S. Liu, R. J. Stokes, C. H. Li, J. Amer. Ceram. Soc. 47, 276 (1964).
[CrossRef]

Mansur, L. C.

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

Marshall, R.

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

Mitra, S. S.

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

J. N. Plendl, S. S. Mitra, P. J. Gielisse, Phys. Stat. Sol. 12, 376 (1965).

Mkhetaryan, R. S.

T. T. Kazandzhyan, V. D. Galstyan, R. S. Mkhetaryan, Nauk Trudy Erevan Univ. 44, Ser. Khim. 2, 95 (1954).

Mykolajewycz, R.

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

Naumov, A. F.

P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izvest. Vysshikh Ucheb. Zavedenii Fiz. 4, 86 (1958).

Plendl, J. N.

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

J. N. Plendl, S. S. Mitra, P. J. Gielisse, Phys. Stat. Sol. 12, 376 (1965).

J. N. Plendl, P. J. Gielisse, Z. Kristallogr. 118, 404 (1963).
[CrossRef]

J. N. Plendl, P. J. Gielisse, Phys. Rev. 125, 828 (1962).
[CrossRef]

J. N. Plendl, “Characteristic Energy Absorption Spectra of Solids,” in Far Infrared Properties of Solids (Plenum, New York, 1970). Also Appl. Opt. 9, 2768 (1970).
[CrossRef] [PubMed]

Powarjonnych, A. S.

A. S. Powarjonnych, Geologie, 93–99 (1959); also in Aspects of Theoretical Mineralogy in the USSR, M. H. Battey, S. I. Tomkieff, Eds. (Plenum Press, New York, 1964), pp.379 ff.

Rauscher, W.

W. Koester, W. Rauscher, Z. Metallkunde 39, 111 (1948).

Rozov, V. V.

N. N. Sirota, V. V. Rozov., Akad. Nauk. BSSR Dokl. 7, 446 (1963).

Sachs, G.

G. Sachs, Praktische Metallkunde (Berlin, 1935), Vol. 3, p. 35.

Savintsev, P. A.

P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izvest. Vysshikh Ucheb. Zavedenii Fiz. 4, 86 (1958).

Sirota, N. N.

N. N. Sirota, V. V. Rozov., Akad. Nauk. BSSR Dokl. 7, 446 (1963).

Smakula, A.

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

A. Smakula, Quarterly Progress Reports, AF 19(628)-395 and AF 19(628)-2418 (1962–1967).

Stoker, J. J.

J. J. Stoker, Nonlinear Vibrations (Interscience, New York, 1950), p. 15.

Stokes, R. J.

T. S. Liu, R. J. Stokes, C. H. Li, J. Amer. Ceram. Soc. 47, 276 (1964).
[CrossRef]

Tarte, P.

P. Tarte, U. of Liége (Belgium), Quarterly Reports on AFCRL contract #AF 61(052)917 (1966–1969).

Vorobev, A. A.

A. A. Vorobev, E. K. Zavodoskaya, M. S. Ivankina, Izv. Vysshikh Ucheb Zavedenii Fiz. 6, 162 (1959); A. A. Vorobev. Zh. Tekh. Fiz. 26, 330 (1956). P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izv. Vysshikh Ucheb Zavedenii Fiz. 4, 86 (1958); G. A. Wolff, L. Toman, N. J. Field, J. C. Clark, Semiconductors and Phosphors, Proc. Intern. Colloq. Garmisch Partenkirchen 1956 (1958), pp. 463–469.

A. A. Vorobev, Zh. Tekh. Fiz. 26, 330 (1956) [Sov. Phys. Tech. Phys. 1, 324 (1956)].

Winkler, H. G. F.

H. G. F. Winkler, Struktur und Eigenschaften der Kristalle (Springer, Berlin, 1950), p. 151.

Zavodoskaya, E. K.

A. A. Vorobev, E. K. Zavodoskaya, M. S. Ivankina, Izv. Vysshikh Ucheb Zavedenii Fiz. 6, 162 (1959); A. A. Vorobev. Zh. Tekh. Fiz. 26, 330 (1956). P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izv. Vysshikh Ucheb Zavedenii Fiz. 4, 86 (1958); G. A. Wolff, L. Toman, N. J. Field, J. C. Clark, Semiconductors and Phosphors, Proc. Intern. Colloq. Garmisch Partenkirchen 1956 (1958), pp. 463–469.

Zlenko, V. Ya.

P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izvest. Vysshikh Ucheb. Zavedenii Fiz. 4, 86 (1958).

Akad. Nauk. BSSR Dokl. (1)

N. N. Sirota, V. V. Rozov., Akad. Nauk. BSSR Dokl. 7, 446 (1963).

Fortschr. Chem. Phys. Physikal. Chem. (1)

E. Friedrich, Fortschr. Chem. Phys. Physikal. Chem. 18, 1 (1926).

Geologie (1)

A. S. Powarjonnych, Geologie, 93–99 (1959); also in Aspects of Theoretical Mineralogy in the USSR, M. H. Battey, S. I. Tomkieff, Eds. (Plenum Press, New York, 1964), pp.379 ff.

Izv. Vysshikh Ucheb Zavedenii Fiz. (1)

A. A. Vorobev, E. K. Zavodoskaya, M. S. Ivankina, Izv. Vysshikh Ucheb Zavedenii Fiz. 6, 162 (1959); A. A. Vorobev. Zh. Tekh. Fiz. 26, 330 (1956). P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izv. Vysshikh Ucheb Zavedenii Fiz. 4, 86 (1958); G. A. Wolff, L. Toman, N. J. Field, J. C. Clark, Semiconductors and Phosphors, Proc. Intern. Colloq. Garmisch Partenkirchen 1956 (1958), pp. 463–469.

Izvest. Vysshikh Ucheb. Zavedenii Fiz. (1)

P. A. Savintsev, V. Ya. Zlenko, A. F. Naumov, Izvest. Vysshikh Ucheb. Zavedenii Fiz. 4, 86 (1958).

J. Amer. Ceram. Soc. (1)

T. S. Liu, R. J. Stokes, C. H. Li, J. Amer. Ceram. Soc. 47, 276 (1964).
[CrossRef]

J. Appl. Phys. (1)

P. J. Gielisse, J. N. Plendl, L. C. Mansur, R. Marshall, S. S. Mitra, R. Mykolajewycz, A. Smakula, J. Appl. Phys. 36, 2246 (1965).
[CrossRef]

Nauk Trudy Erevan Univ. (1)

T. T. Kazandzhyan, V. D. Galstyan, R. S. Mkhetaryan, Nauk Trudy Erevan Univ. 44, Ser. Khim. 2, 95 (1954).

Phys. Rev. (1)

J. N. Plendl, P. J. Gielisse, Phys. Rev. 125, 828 (1962).
[CrossRef]

Phys. Stat. Sol. (1)

J. N. Plendl, S. S. Mitra, P. J. Gielisse, Phys. Stat. Sol. 12, 376 (1965).

Sov. J. Opt. Technol. (1)

E. G. Chernevskaya, Sov. J. Opt. Technol. 33, 346 (1966).

Z. Kristallogr. (1)

J. N. Plendl, P. J. Gielisse, Z. Kristallogr. 118, 404 (1963).
[CrossRef]

Z. Metallkunde (1)

W. Koester, W. Rauscher, Z. Metallkunde 39, 111 (1948).

Zh. Tekh. Fiz. (1)

A. A. Vorobev, Zh. Tekh. Fiz. 26, 330 (1956) [Sov. Phys. Tech. Phys. 1, 324 (1956)].

Other (10)

A. Smakula, Quarterly Progress Reports, AF 19(628)-395 and AF 19(628)-2418 (1962–1967).

It should be noted here that the value of the structure constant of CaF2 (CN = 4; 8) is π/2. This value actually corresponds to the atomic coordination number CN = 8.

J. N. Plendl, “Characteristic Energy Absorption Spectra of Solids,” in Far Infrared Properties of Solids (Plenum, New York, 1970). Also Appl. Opt. 9, 2768 (1970).
[CrossRef] [PubMed]

In previous publications the quantity Ψ(A) has been described as the anharmonicity of solids. The term anharmonic factor is felt to be more appropriate and is used to discriminate between its present use and anharmonicity as used to describe third- and fourth-order terms in lattice vibrational potential.

G. Sachs, Praktische Metallkunde (Berlin, 1935), Vol. 3, p. 35.

W. Kuntze, in G. Masing, Handbuch der Metallphysik (Akademie-Verlagsges., Leipzig, 1940), p. 276.

H. G. F. Winkler, Struktur und Eigenschaften der Kristalle (Springer, Berlin, 1950), p. 151.

V. M. Goldschmidt, Geochemische Verteilingsgesetze der Elemente (Oslo, 1927).

P. Tarte, U. of Liége (Belgium), Quarterly Reports on AFCRL contract #AF 61(052)917 (1966–1969).

J. J. Stoker, Nonlinear Vibrations (Interscience, New York, 1950), p. 15.

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Figures (12)

Fig. 1
Fig. 1

Variation of hardness as a function of composition for several solid solution systems. For references, see text.

Fig. 2
Fig. 2

Schematic presentation of energy absorption curves for solids with a hard (a/b > 1) or a soft (a/b < 1) anharmonic factor. A fully symmetric energy absorption curve would be obtained for the linear case, where the anharmonic factor is unity.

Fig. 3
Fig. 3

Definition of soft and hard force characteristics as deviations from linear behavior. The two type behavior is classically characterized by different trends of the force characteristic as a function of displacement (see, e.g., Stoker24). – – –, Soft force characteristic; ——, harmonic force characteristic; — - —, hard force characteristic.

Fig. 4
Fig. 4

Infrared reflectivity spectra in the area of the fundamental lattice vibrations for five members of the NiO–CoO solid solution system.22

Fig. 5
Fig. 5

Infrared transmission spectra for three compositions of the NiO–CoO solid solution system.

Fig. 6
Fig. 6

Infrared transmission spectra for five compositions of the CaF2–SrF2 solid solution system. Composition (percent):

Fig. 7
Fig. 7

Infrared reflectivity spectra for five members of the CaF2–SrF2 solid solution system.

Fig. 8
Fig. 8

Characteristic energy absorption spectra for three compositions in the solid solution system NiO–CoO.20

Fig. 9
Fig. 9

Characteristic energy absorption spectra for five compositions in the solid solution system CaF2–SrF2.20

Fig. 10
Fig. 10

Comparison between experimental hardness values (curves a and b) and those calculated with the proposed method (curve c) for the solid solution system NiO–CoO. The variation of the spectral parameter (ν0/ψA)2 is also indicated (see SP).

Fig. 11
Fig. 11

Comparison between experimental hardness values (curve a) and those calculated with the proposed method (curve c) for the solid solution system CaF2–SrF2. The lower curve indicates the variation of the spectral parameter in this system (SP).

Fig. 12
Fig. 12

Identation hardness vs volumetric lattice energy (physical hardness) for eight pertinent solids.

Tables (2)

Tables Icon

Table I Data Used in the Evaluation of the Systems NiO–CoO and CaF2–SrF2

Tables Icon

Table II Mechanical Properties of Solids Used to Establish Proportionality Between Physical and Measured Hardness Data

Equations (1)

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H = 1 2 Z m r V ( ω 0 r 0 Ψ ( A ) ) 2 .

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