Abstract

The damping of lattice vibrations in solid compounds is treated using kinetic theory analogous to damping in gases. It is based on the collision frequency of atoms, taking into consideration the atomic coordination due to the crystalline structure, the cross section of collision, the radius ratio of the component atoms (atomic size factor), as well as an anharmonic factor which is an expression for the anharmonicity of lattice vibrations. A semiempirical formulation is derived without need for constants fitted to experimental data. This formulation of damping is shown valid for more than eighty solids, mostly binary compounds, also some ternary compounds and elements. They may have either ionic or covalent or metallic binding. They cover ten different structures and valencies from one through four. In addition, a close relationship is shown between damping and thermal expansion as a function of temperatures. Based on this relationship, the temperature dependence is empirically expressed by an exponential function of the coefficient of thermal expansion. This function agrees with the variation of ir energy absorption vs temperatures. The complete damping formulation is shown valid for the entire temperature range of solids, from absolute zero to the melting point, for a variety of solids for which all pertinent data were on hand.

© 1971 Optical Society of America

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  1. J. R. Jasperse, A. Kahan, J. N. Plendl, S. S. Mitra, Phys. Rev. 146, 526 (1966).
    [CrossRef]
  2. A. A. Maradudin, R. F. Wallis, Phys. Rev. 125, 4 (1962).
  3. J. N. Plendl, “Characteristic Energy Absorption Spectra of Solids,” AFCRL Special Rep. 353 (1969).
  4. J. N. Plendl, Appl. Opt. 9, 2768 (1970).
    [PubMed]
  5. W. Hume-Rothery, G. W. Mabbott, K. M. Channel-Evans, Phil. Trans. Roy. Soc. A233, 1 (1934).
  6. J. N. Plendl, Phys. Rev. 123, 1172 (1961).
    [CrossRef]
  7. J. N. Plendl, “New Concepts in the Physics of Solids,” Monograph, AFCRL Spec. Rep. 52 (1966).
  8. J. N. Plendl, “New Spectral and Atomistic Relations in Physics and Chemistry of Solids,” in Optical Properties of Solids (Plenum Press, New York, 1969).
    [CrossRef]
  9. R. K. Chang, B. Lacina, P. S. Pershan, Phys. Rev. Lett. 17, 755 (1966).
    [CrossRef]
  10. G. D. Wignall, AEREM, 1911 (1967).
  11. A. P. Roy, C. L. Thaper, P. K. Iyengar, Physica 34, 384 (1967).
    [CrossRef]
  12. J. N. Plendl, P. J. Gielisse, J. Phys. Chem. Solids 31, 874 (1970).
    [CrossRef]
  13. S. S. Mitra, R. S. Singh, Phys. Rev. Lett. 16, 694 (1966).
    [CrossRef]
  14. D. E. Schuele, C. S. Smith, J. Phys. Chem. Solids 25, 801 (1964).
    [CrossRef]
  15. H. F. Schaake, Martin Marietta Corp. Rep. on thermal expansion measurements on AFCRL Contr. No. AF19(628)-5663 (1965/69).
  16. J. E. Mooij, Phys. Lett. 24A, 249 (1967).
  17. A. Eucken, W. Dannoehl, Z. Elektrochem. 40, 814 (1934).
  18. R. A. Cowley, Advan. Phys. 12, 42 (1963).
    [CrossRef]
  19. G. Heilmann, Z. Phys. 152, 368 (1958).
    [CrossRef]
  20. R. J. Elliot, W. Hayes, G. D. Jones, H. F. MacDonald, C. T. Sennett, Proc. Roy. Soc. (London) 289, 1 (1965).
    [CrossRef]
  21. Critical Tables (Nat. Acad. Sci., Wash., D.C., 1928), Vol. 3, p. 43.
  22. S. S. Ballard, K. A. McCarthy, W. L. Wolfe, U. Michigan, Willow Run Lab. Rep. 2389-11-S (1959).

1970 (2)

J. N. Plendl, Appl. Opt. 9, 2768 (1970).
[PubMed]

J. N. Plendl, P. J. Gielisse, J. Phys. Chem. Solids 31, 874 (1970).
[CrossRef]

1967 (3)

J. E. Mooij, Phys. Lett. 24A, 249 (1967).

G. D. Wignall, AEREM, 1911 (1967).

A. P. Roy, C. L. Thaper, P. K. Iyengar, Physica 34, 384 (1967).
[CrossRef]

1966 (3)

R. K. Chang, B. Lacina, P. S. Pershan, Phys. Rev. Lett. 17, 755 (1966).
[CrossRef]

J. R. Jasperse, A. Kahan, J. N. Plendl, S. S. Mitra, Phys. Rev. 146, 526 (1966).
[CrossRef]

S. S. Mitra, R. S. Singh, Phys. Rev. Lett. 16, 694 (1966).
[CrossRef]

1965 (1)

R. J. Elliot, W. Hayes, G. D. Jones, H. F. MacDonald, C. T. Sennett, Proc. Roy. Soc. (London) 289, 1 (1965).
[CrossRef]

1964 (1)

D. E. Schuele, C. S. Smith, J. Phys. Chem. Solids 25, 801 (1964).
[CrossRef]

1963 (1)

R. A. Cowley, Advan. Phys. 12, 42 (1963).
[CrossRef]

1962 (1)

A. A. Maradudin, R. F. Wallis, Phys. Rev. 125, 4 (1962).

1961 (1)

J. N. Plendl, Phys. Rev. 123, 1172 (1961).
[CrossRef]

1958 (1)

G. Heilmann, Z. Phys. 152, 368 (1958).
[CrossRef]

1934 (2)

A. Eucken, W. Dannoehl, Z. Elektrochem. 40, 814 (1934).

W. Hume-Rothery, G. W. Mabbott, K. M. Channel-Evans, Phil. Trans. Roy. Soc. A233, 1 (1934).

Ballard, S. S.

S. S. Ballard, K. A. McCarthy, W. L. Wolfe, U. Michigan, Willow Run Lab. Rep. 2389-11-S (1959).

Chang, R. K.

R. K. Chang, B. Lacina, P. S. Pershan, Phys. Rev. Lett. 17, 755 (1966).
[CrossRef]

Channel-Evans, K. M.

W. Hume-Rothery, G. W. Mabbott, K. M. Channel-Evans, Phil. Trans. Roy. Soc. A233, 1 (1934).

Cowley, R. A.

R. A. Cowley, Advan. Phys. 12, 42 (1963).
[CrossRef]

Dannoehl, W.

A. Eucken, W. Dannoehl, Z. Elektrochem. 40, 814 (1934).

Elliot, R. J.

R. J. Elliot, W. Hayes, G. D. Jones, H. F. MacDonald, C. T. Sennett, Proc. Roy. Soc. (London) 289, 1 (1965).
[CrossRef]

Eucken, A.

A. Eucken, W. Dannoehl, Z. Elektrochem. 40, 814 (1934).

Gielisse, P. J.

J. N. Plendl, P. J. Gielisse, J. Phys. Chem. Solids 31, 874 (1970).
[CrossRef]

Hayes, W.

R. J. Elliot, W. Hayes, G. D. Jones, H. F. MacDonald, C. T. Sennett, Proc. Roy. Soc. (London) 289, 1 (1965).
[CrossRef]

Heilmann, G.

G. Heilmann, Z. Phys. 152, 368 (1958).
[CrossRef]

Hume-Rothery, W.

W. Hume-Rothery, G. W. Mabbott, K. M. Channel-Evans, Phil. Trans. Roy. Soc. A233, 1 (1934).

Iyengar, P. K.

A. P. Roy, C. L. Thaper, P. K. Iyengar, Physica 34, 384 (1967).
[CrossRef]

Jasperse, J. R.

J. R. Jasperse, A. Kahan, J. N. Plendl, S. S. Mitra, Phys. Rev. 146, 526 (1966).
[CrossRef]

Jones, G. D.

R. J. Elliot, W. Hayes, G. D. Jones, H. F. MacDonald, C. T. Sennett, Proc. Roy. Soc. (London) 289, 1 (1965).
[CrossRef]

Kahan, A.

J. R. Jasperse, A. Kahan, J. N. Plendl, S. S. Mitra, Phys. Rev. 146, 526 (1966).
[CrossRef]

Lacina, B.

R. K. Chang, B. Lacina, P. S. Pershan, Phys. Rev. Lett. 17, 755 (1966).
[CrossRef]

Mabbott, G. W.

W. Hume-Rothery, G. W. Mabbott, K. M. Channel-Evans, Phil. Trans. Roy. Soc. A233, 1 (1934).

MacDonald, H. F.

R. J. Elliot, W. Hayes, G. D. Jones, H. F. MacDonald, C. T. Sennett, Proc. Roy. Soc. (London) 289, 1 (1965).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, R. F. Wallis, Phys. Rev. 125, 4 (1962).

McCarthy, K. A.

S. S. Ballard, K. A. McCarthy, W. L. Wolfe, U. Michigan, Willow Run Lab. Rep. 2389-11-S (1959).

Michigan, U.

S. S. Ballard, K. A. McCarthy, W. L. Wolfe, U. Michigan, Willow Run Lab. Rep. 2389-11-S (1959).

Mitra, S. S.

J. R. Jasperse, A. Kahan, J. N. Plendl, S. S. Mitra, Phys. Rev. 146, 526 (1966).
[CrossRef]

S. S. Mitra, R. S. Singh, Phys. Rev. Lett. 16, 694 (1966).
[CrossRef]

Mooij, J. E.

J. E. Mooij, Phys. Lett. 24A, 249 (1967).

Pershan, P. S.

R. K. Chang, B. Lacina, P. S. Pershan, Phys. Rev. Lett. 17, 755 (1966).
[CrossRef]

Plendl, J. N.

J. N. Plendl, Appl. Opt. 9, 2768 (1970).
[PubMed]

J. N. Plendl, P. J. Gielisse, J. Phys. Chem. Solids 31, 874 (1970).
[CrossRef]

J. R. Jasperse, A. Kahan, J. N. Plendl, S. S. Mitra, Phys. Rev. 146, 526 (1966).
[CrossRef]

J. N. Plendl, Phys. Rev. 123, 1172 (1961).
[CrossRef]

J. N. Plendl, “New Concepts in the Physics of Solids,” Monograph, AFCRL Spec. Rep. 52 (1966).

J. N. Plendl, “New Spectral and Atomistic Relations in Physics and Chemistry of Solids,” in Optical Properties of Solids (Plenum Press, New York, 1969).
[CrossRef]

J. N. Plendl, “Characteristic Energy Absorption Spectra of Solids,” AFCRL Special Rep. 353 (1969).

Roy, A. P.

A. P. Roy, C. L. Thaper, P. K. Iyengar, Physica 34, 384 (1967).
[CrossRef]

Schaake, H. F.

H. F. Schaake, Martin Marietta Corp. Rep. on thermal expansion measurements on AFCRL Contr. No. AF19(628)-5663 (1965/69).

Schuele, D. E.

D. E. Schuele, C. S. Smith, J. Phys. Chem. Solids 25, 801 (1964).
[CrossRef]

Sennett, C. T.

R. J. Elliot, W. Hayes, G. D. Jones, H. F. MacDonald, C. T. Sennett, Proc. Roy. Soc. (London) 289, 1 (1965).
[CrossRef]

Singh, R. S.

S. S. Mitra, R. S. Singh, Phys. Rev. Lett. 16, 694 (1966).
[CrossRef]

Smith, C. S.

D. E. Schuele, C. S. Smith, J. Phys. Chem. Solids 25, 801 (1964).
[CrossRef]

Thaper, C. L.

A. P. Roy, C. L. Thaper, P. K. Iyengar, Physica 34, 384 (1967).
[CrossRef]

Wallis, R. F.

A. A. Maradudin, R. F. Wallis, Phys. Rev. 125, 4 (1962).

Wignall, G. D.

G. D. Wignall, AEREM, 1911 (1967).

Wolfe, W. L.

S. S. Ballard, K. A. McCarthy, W. L. Wolfe, U. Michigan, Willow Run Lab. Rep. 2389-11-S (1959).

Advan. Phys. (1)

R. A. Cowley, Advan. Phys. 12, 42 (1963).
[CrossRef]

AEREM (1)

G. D. Wignall, AEREM, 1911 (1967).

Appl. Opt. (1)

J. Phys. Chem. Solids (2)

J. N. Plendl, P. J. Gielisse, J. Phys. Chem. Solids 31, 874 (1970).
[CrossRef]

D. E. Schuele, C. S. Smith, J. Phys. Chem. Solids 25, 801 (1964).
[CrossRef]

Phil. Trans. Roy. Soc. (1)

W. Hume-Rothery, G. W. Mabbott, K. M. Channel-Evans, Phil. Trans. Roy. Soc. A233, 1 (1934).

Phys. Lett. (1)

J. E. Mooij, Phys. Lett. 24A, 249 (1967).

Phys. Rev. (3)

J. N. Plendl, Phys. Rev. 123, 1172 (1961).
[CrossRef]

J. R. Jasperse, A. Kahan, J. N. Plendl, S. S. Mitra, Phys. Rev. 146, 526 (1966).
[CrossRef]

A. A. Maradudin, R. F. Wallis, Phys. Rev. 125, 4 (1962).

Phys. Rev. Lett. (2)

S. S. Mitra, R. S. Singh, Phys. Rev. Lett. 16, 694 (1966).
[CrossRef]

R. K. Chang, B. Lacina, P. S. Pershan, Phys. Rev. Lett. 17, 755 (1966).
[CrossRef]

Physica (1)

A. P. Roy, C. L. Thaper, P. K. Iyengar, Physica 34, 384 (1967).
[CrossRef]

Proc. Roy. Soc. (London) (1)

R. J. Elliot, W. Hayes, G. D. Jones, H. F. MacDonald, C. T. Sennett, Proc. Roy. Soc. (London) 289, 1 (1965).
[CrossRef]

Z. Elektrochem. (1)

A. Eucken, W. Dannoehl, Z. Elektrochem. 40, 814 (1934).

Z. Phys. (1)

G. Heilmann, Z. Phys. 152, 368 (1958).
[CrossRef]

Other (6)

H. F. Schaake, Martin Marietta Corp. Rep. on thermal expansion measurements on AFCRL Contr. No. AF19(628)-5663 (1965/69).

J. N. Plendl, “Characteristic Energy Absorption Spectra of Solids,” AFCRL Special Rep. 353 (1969).

J. N. Plendl, “New Concepts in the Physics of Solids,” Monograph, AFCRL Spec. Rep. 52 (1966).

J. N. Plendl, “New Spectral and Atomistic Relations in Physics and Chemistry of Solids,” in Optical Properties of Solids (Plenum Press, New York, 1969).
[CrossRef]

Critical Tables (Nat. Acad. Sci., Wash., D.C., 1928), Vol. 3, p. 43.

S. S. Ballard, K. A. McCarthy, W. L. Wolfe, U. Michigan, Willow Run Lab. Rep. 2389-11-S (1959).

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Figures (18)

Fig. 1
Fig. 1

Characteristic energy absorption spectra of selected sodium and fluorite salts and of MgO.3

Fig. 2
Fig. 2

Laser Raman line and corresponding energy absorption line for α-quartz.3

Fig. 3
Fig. 3

Cold neutron spectrum of NaI with calculated data from elastic constants3,4 (arrows).

Fig. 4
Fig. 4

Characteristic energy absorption spectra of Al2O3 and Cr2O3.3,4

Fig. 5
Fig. 5

Characteristic energy absorption spectrum of MgF2.3,4

Fig. 6
Fig. 6

Equality between damping data of thirty solids obtained from ir energy absorption spectra and calculated from ionic radii.

Fig. 7
Fig. 7

Equality between damping data of seventeen solids obtained from laser Raman lines and calculated from covalent radii.

Fig. 8
Fig. 8

Cold neutron spectrum of vanadium (partial)10 with data calculated from elastic constants3 (arrow).

Fig. 9
Fig. 9

Cold neutron spectrum of aluminum at 923 K, with data calculated from elastic constants3 (arrows).

Fig. 10
Fig. 10

Equality between experimental and calculated damping data for metals.3

Fig. 11
Fig. 11

Damping data, as a function of radius ratio, for a variety of solids (fifty-seven) with ionic binding, for the three structures: ZnS (CN = 4) (lower curve), NaCl (CN = 6) (middle curve), CsCl (CN = 8) (upper curve). Crosses = experimental data, circles = calculated data.12

Fig. 12
Fig. 12

Variation of energy absorption F[AC(ν)], absorption frequencies ν (cm−1), and damping Δν/νmax of RbI for temperatures below 300 K.3

Fig. 13
Fig. 13

Variation of energy absorption F[AC(ν)], absorption frequencies ν (cm−1), and damping Δν/νmax of CuI for temperatures below 300 K.3

Fig. 14
Fig. 14

Damping data of RbI (5–300 K) derived from ir energy absorption spectra (Fig. 14) and calculated by Eq. (8) (Table VI). Also data of linear thermal expansions.14 Squares = experimental damping data, crosses = calculated damping values, circles = experimental thermal expansion data.

Fig. 15
Fig. 15

Damping data of CuI (5–300 K) derived from ir energy absorption spectra (Fig. 15) and calculated by Eq. (7) (Table VI). Also data of linear thermal expansion.15 Squares = experimental damping data, crosses = calculated damping values, circles = experimental thermal expansion data.

Fig. 16
Fig. 16

Damping data of KBr (300–800 K) derived from ir emission spectra14 and calculated by Eq. (8) (Table V). Also data of linear thermal expansion.17,18 Crosses = calculated damping values, circles = experimental data of damping also of thermal expansion.

Fig. 17
Fig. 17

Damping data of LiF (300–900 K) derived from ir reflection spectra17 and calculated by Eq. (8) (Table VI). Also data of linear thermal expansion.17 Crosses = calculated damping values, circles = experimental data of thermal expansion and of damping, the latter modified as πγ/ω0 to equal, at room temperature, Δν/ν0 of ir energy absorption spectrum.

Fig. 18
Fig. 18

Damping data of CaF2:H (20–300 K) derived from absorption lines of localized modes20 and calculated by Eq. (8) (Table VII). Also data of linear thermal expansion of CaF2.21,22 Crosses = calculated damping values, circles = experimental data of damping (second harmonic) also of thermal expansion.

Tables (7)

Tables Icon

Table I Characteristic Constant for Various Crystalline Structures.

Tables Icon

Table II Damping Data of the Main TO Modes of Thirty Solids as Determined from Characteristic Energy Absorption Spectra3,4 and Calculated for Ionic Binding

Tables Icon

Table III Damping Data of Seventeen Solids as Determined from Laser Raman Spectra3,4 and Calculated for Covalent Binding

Tables Icon

Table IV Damping Data for TO and LO Modes for Selected Dielectrics

Tables Icon

Table V Damping Data for the Mean T Mode of Five Metallic Elements, as Experimentally Determined from Cold Neutron Spectra3,4 and Calculated for Metallic Binding in Elements

Tables Icon

Table VI Temperature Dependence of the Damping Data of Five Solids as Determined from ir Spectral Data3,4 and Calculated for Ionic Binding

Tables Icon

Table VII Damping Data of the Main TO Modes of Some III–V Compounds, Calculated for Ionic Bonding with CN = 4 and Cstr = π

Equations (9)

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M / ρ N = C str r 0 3 ( cm 3 ) ,
π [ ( r a + r b ) / 2 ] 2 = ( π / 4 ) r 0 2 ( cm 2 ) ,
V TO = 2 r 0 ν TO ( cm / sec ) .
Z = ( 1 2 ) ( 2 C N - 1 ) C str r 0 3 ( π 4 r 0 2 ) ( 2 r 0 ν TO ) = π 4 2 C N - 1 C str ν TO ( 1 / sec ) ,
γ = Z ( r b / r a ) [ π Ψ ( A ) ] - 2 .
Δ ν / ν TO = ( 1 8 π 2 ) [ ( 2 C N - 1 ) / C str ] ( r b / r a ) Ψ ( A ) - 2 .
( Δ ν / ν 0 ) T = ( Δ ν / ν 0 ) R T · exp [ b ( α T - α R T ) / α R T ] ,
b = 2 = constant for θ < T < T m , and b = 2 - ( θ - T / θ ) for 0 < T < Θ .
( Δ ν / ν 0 ) T = 1 8 π 2 2 C N - 1 C str r b r a Ψ ( A ) - 2 · exp [ b ( α T - α R T ) / α R T ] .

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