Abstract

The thermal radiation from metals is calculated by including the Holstein processes of phononassisted and surface-assisted scattering in the free-electron model. This approach provides a fairly accurate description of the experimentally measured total emissivity of good conductors in terms of the dc resistivity and the radius of the free-electron sphere. These calculations do not support the Foote and Davisson-Weeks-type equations which have been used previously.

© 1978 Optical Society of America

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  1. E. Aschkinass, Ann. Phys. 17, 960 (1905).
  2. P. D. Foote, Bull. Natl. Bur. Stand. 11, 607 (1914–15).
  3. B. Donovan, Elementary Theory of Metals, (Pergamon, Oxford, 1967), p. 220.
  4. Thermophysical Properties of Matter, edited by Y. S. Touloukian, Vol. 7 (Plenum, New York, 1970), p. 24a.
  5. C. Davisson and J. R. Weeks, J. Opt. Soc. Am. 8, 581 (1924).
  6. W. J. Parker and G. C. Abbott, Symposium on Thermal Radiation of Solids, edited by S. Katzoff, (NASA SP-55, 1965), p. 11.
  7. M. Jakob, Heat Transfer, (Wiley, New York, 1949), p. 51.
  8. G. A. W. Rutgers, Handbuch der Physik, edited by S. Fligge (Springer-Verlag, Berlin 1958), Vol. 26, p. 154.
  9. K. Schocken, Radiative Transfer From Solid Materials, edited by H. Blau and H. Fischer (Macmillan, New York, 1962), p. 3.
  10. K. G. Ramanathan and S. H. Yen, J. Opt. Soc. Am. 67, 32 (1977).
  11. E. A. Estalote and K. G. Ramanathan, J. Opt. Soc. Am. 67, 39 (1977).
  12. K. G. Ramanathan, S. H. Yen, and E. A. Estalote, Appl. Opt. 16, 2810 (1977).
  13. H. E. Bennett and J. M. Bennett, Proceedings of the International Colloquium on Optical Properties, Paris, 1 965, edited by F. Abeles, (North Holland, Amsterdam, 1966), p. 175; H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
  14. J. A. McKay and J. A. Rayne, Phys. Rev. B 13, 673 (1976).
  15. N. W. Ashcroft and K. Sturm, Phys. Rev. B 3, 1898 (1971).
  16. B. Donovan, Elementary Theory of Metals, (Pergamon, Oxford, 1967), p. 215.
  17. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Massachusetts, 1960), p. 280.
  18. A. B. Pippard, Optical Properties and Electronic Structure of Metals and Alloys, edited by F. Abeles (North Holland, Amsterdam, 1966) p. 622.
  19. F. J. Blatt, Physics of Electronic Conduction in Solids (McGraw-Hill, New York, 1968), p. 194.
  20. N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys, (Dover, New York, 1936), p. 268.
  21. N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Holt, Rinehart and Winston, New York, 1976), p. 5, (see Appendix A for numerical relations between τ,ρµ,rs/a0,vF, and ωp).
  22. Ref. 20, Appendix II, p. 318.
  23. G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. A 195, 336 (1948).
  24. T. Holstein, Phys. Rev. 88, 1427 (1952).
  25. R. B. Dingle, Physica 19, 311 (1953).
  26. A. B. Pippard, in Advances in Electronics and Electron Physics, edited by L. Marton, (Academic, New York, 1954), p. 1.
  27. H. B. G. Casimir and J. Ubbink, Philips Tech. Rev. 28, 300 (1967).
  28. F. Wooten, Optical Properties of Solids, (Academic, New York, 1972), p. 93.
  29. The positions of the boundaries change slightly from metal to metal since they depend on relaxation time and the electron density. See Ref. 28 for a complete description of the boundary parameters.
  30. T. Holstein, Ann. Phys. (N.Y.) 29, 410 (1964); Phys. Rev. 96, 535 (1954).
  31. R. N. Gurzhi, Zh. Eksp. Teor. Fiz. 33, 660 (1957) [Sov. Phys. JETP 6, 506 (1958)1; 35, 965 (1958) [Sov. Phys. JETP 8, 673 (1959)].
  32. For Cu, Pt, and Ag, ρu(T) was taken from American Institute of Physics Handbook, 3rd Edition, McGraw Hill, New York) while for the rest of the elements ρu(T) was taken from Metals Reference Book Vol. III, edited by C. J. Smithells, (Butterworths, London, 1967). Values of rs were taken from Ref. (20) or (21).
  33. G. L. Abbott, in Measurement of Thermal Radiation Properties of Solids, edited by J. C. Richmond, (NASA SP-31, 1963), p. 293.
  34. R. Blickensderfer, D. K. Deardorff, and R. L. Lincoln, J. Less- Common Metals 51, 13 (1977).
  35. G. A. Zhorov, High Temp. 5 881 (1967) [English translation of Teplofiz, Uys. Temp., 5, 987 (1967)].
  36. H. T. Betz, O. H. Olson, B. O. Schurin, and J. C. Morris, Determination of Emissivity and Reflectivity Data on Aircraft Structural Materials, Pt. 2, AD 202493, 1958.
  37. D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), p. 322.
  38. R. Smalley and A. J. Sievers, J. Opt. Soc. Am. 68, 1516–1518 (1978).
  39. A. G. Mathewson and H. P. Myers, J. Phys. F. Metal Phys. 3, 623 (1973).
  40. R. T. Beach and R. W. Christy, Phys. Rev. B 16, 5277 (1977).
  41. R. N. Gurzhi, M. Y. Azbel, and H. P. Lin, Fiz. Tverd. Tela. 5, 759 (1963) [Sov. Phys. - Solid State 5, 554 (1963)].
  42. R. Suhrmann, H. Ober, and G. Wedler, Z. Phys. Chem. (Frankfurt) 29, 305 (1961).
  43. P. Wissmann, in Springer Tracts in Modern Physics 77, edited by G. Hbhler (Springer-Verlag, New York, 1975), p. 1.

Abbott, G. C.

W. J. Parker and G. C. Abbott, Symposium on Thermal Radiation of Solids, edited by S. Katzoff, (NASA SP-55, 1965), p. 11.

Abbott, G. L.

G. L. Abbott, in Measurement of Thermal Radiation Properties of Solids, edited by J. C. Richmond, (NASA SP-31, 1963), p. 293.

Aschkinass, E.

E. Aschkinass, Ann. Phys. 17, 960 (1905).

Ashcroft, N. W.

N. W. Ashcroft and K. Sturm, Phys. Rev. B 3, 1898 (1971).

N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Holt, Rinehart and Winston, New York, 1976), p. 5, (see Appendix A for numerical relations between τ,ρµ,rs/a0,vF, and ωp).

Azbel, M. Y.

R. N. Gurzhi, M. Y. Azbel, and H. P. Lin, Fiz. Tverd. Tela. 5, 759 (1963) [Sov. Phys. - Solid State 5, 554 (1963)].

Beach, R. T.

R. T. Beach and R. W. Christy, Phys. Rev. B 16, 5277 (1977).

Bennett, H. E.

H. E. Bennett and J. M. Bennett, Proceedings of the International Colloquium on Optical Properties, Paris, 1 965, edited by F. Abeles, (North Holland, Amsterdam, 1966), p. 175; H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).

Bennett, J. M.

H. E. Bennett and J. M. Bennett, Proceedings of the International Colloquium on Optical Properties, Paris, 1 965, edited by F. Abeles, (North Holland, Amsterdam, 1966), p. 175; H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).

Betz, H. T.

H. T. Betz, O. H. Olson, B. O. Schurin, and J. C. Morris, Determination of Emissivity and Reflectivity Data on Aircraft Structural Materials, Pt. 2, AD 202493, 1958.

Blatt, F. J.

F. J. Blatt, Physics of Electronic Conduction in Solids (McGraw-Hill, New York, 1968), p. 194.

Blickensderfer, R.

R. Blickensderfer, D. K. Deardorff, and R. L. Lincoln, J. Less- Common Metals 51, 13 (1977).

Casimir, H. B. G.

H. B. G. Casimir and J. Ubbink, Philips Tech. Rev. 28, 300 (1967).

Christy, R. W.

R. T. Beach and R. W. Christy, Phys. Rev. B 16, 5277 (1977).

Davisson, C.

C. Davisson and J. R. Weeks, J. Opt. Soc. Am. 8, 581 (1924).

Deardorff, D. K.

R. Blickensderfer, D. K. Deardorff, and R. L. Lincoln, J. Less- Common Metals 51, 13 (1977).

Dingle, R. B.

R. B. Dingle, Physica 19, 311 (1953).

Donovan, B.

B. Donovan, Elementary Theory of Metals, (Pergamon, Oxford, 1967), p. 215.

B. Donovan, Elementary Theory of Metals, (Pergamon, Oxford, 1967), p. 220.

Estalote, E. A.

E. A. Estalote and K. G. Ramanathan, J. Opt. Soc. Am. 67, 39 (1977).

K. G. Ramanathan, S. H. Yen, and E. A. Estalote, Appl. Opt. 16, 2810 (1977).

Foote, P. D.

P. D. Foote, Bull. Natl. Bur. Stand. 11, 607 (1914–15).

Gurzhi, R. N.

R. N. Gurzhi, Zh. Eksp. Teor. Fiz. 33, 660 (1957) [Sov. Phys. JETP 6, 506 (1958)1; 35, 965 (1958) [Sov. Phys. JETP 8, 673 (1959)].

R. N. Gurzhi, M. Y. Azbel, and H. P. Lin, Fiz. Tverd. Tela. 5, 759 (1963) [Sov. Phys. - Solid State 5, 554 (1963)].

Holstein, T.

T. Holstein, Phys. Rev. 88, 1427 (1952).

T. Holstein, Ann. Phys. (N.Y.) 29, 410 (1964); Phys. Rev. 96, 535 (1954).

Jakob, M.

M. Jakob, Heat Transfer, (Wiley, New York, 1949), p. 51.

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), p. 322.

Jones, H.

N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys, (Dover, New York, 1936), p. 268.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Massachusetts, 1960), p. 280.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Massachusetts, 1960), p. 280.

Lin, H. P.

R. N. Gurzhi, M. Y. Azbel, and H. P. Lin, Fiz. Tverd. Tela. 5, 759 (1963) [Sov. Phys. - Solid State 5, 554 (1963)].

Lincoln, R. L.

R. Blickensderfer, D. K. Deardorff, and R. L. Lincoln, J. Less- Common Metals 51, 13 (1977).

Mathewson, A. G.

A. G. Mathewson and H. P. Myers, J. Phys. F. Metal Phys. 3, 623 (1973).

McKay, J. A.

J. A. McKay and J. A. Rayne, Phys. Rev. B 13, 673 (1976).

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Holt, Rinehart and Winston, New York, 1976), p. 5, (see Appendix A for numerical relations between τ,ρµ,rs/a0,vF, and ωp).

Morris, J. C.

H. T. Betz, O. H. Olson, B. O. Schurin, and J. C. Morris, Determination of Emissivity and Reflectivity Data on Aircraft Structural Materials, Pt. 2, AD 202493, 1958.

Mott, N. F.

N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys, (Dover, New York, 1936), p. 268.

Myers, H. P.

A. G. Mathewson and H. P. Myers, J. Phys. F. Metal Phys. 3, 623 (1973).

Ober, H.

R. Suhrmann, H. Ober, and G. Wedler, Z. Phys. Chem. (Frankfurt) 29, 305 (1961).

Olson, O. H.

H. T. Betz, O. H. Olson, B. O. Schurin, and J. C. Morris, Determination of Emissivity and Reflectivity Data on Aircraft Structural Materials, Pt. 2, AD 202493, 1958.

Parker, W. J.

W. J. Parker and G. C. Abbott, Symposium on Thermal Radiation of Solids, edited by S. Katzoff, (NASA SP-55, 1965), p. 11.

Pippard, A. B.

A. B. Pippard, Optical Properties and Electronic Structure of Metals and Alloys, edited by F. Abeles (North Holland, Amsterdam, 1966) p. 622.

A. B. Pippard, in Advances in Electronics and Electron Physics, edited by L. Marton, (Academic, New York, 1954), p. 1.

Ramanathan, K. G.

E. A. Estalote and K. G. Ramanathan, J. Opt. Soc. Am. 67, 39 (1977).

K. G. Ramanathan and S. H. Yen, J. Opt. Soc. Am. 67, 32 (1977).

K. G. Ramanathan, S. H. Yen, and E. A. Estalote, Appl. Opt. 16, 2810 (1977).

Rayne, J. A.

J. A. McKay and J. A. Rayne, Phys. Rev. B 13, 673 (1976).

Reuter, G. E. H.

G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. A 195, 336 (1948).

Rutgers, G. A. W.

G. A. W. Rutgers, Handbuch der Physik, edited by S. Fligge (Springer-Verlag, Berlin 1958), Vol. 26, p. 154.

Schocken, K.

K. Schocken, Radiative Transfer From Solid Materials, edited by H. Blau and H. Fischer (Macmillan, New York, 1962), p. 3.

Schurin, B. O.

H. T. Betz, O. H. Olson, B. O. Schurin, and J. C. Morris, Determination of Emissivity and Reflectivity Data on Aircraft Structural Materials, Pt. 2, AD 202493, 1958.

Sievers, A. J.

R. Smalley and A. J. Sievers, J. Opt. Soc. Am. 68, 1516–1518 (1978).

Smalley, R.

R. Smalley and A. J. Sievers, J. Opt. Soc. Am. 68, 1516–1518 (1978).

Sondheimer, E. H.

G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. A 195, 336 (1948).

Sturm, K.

N. W. Ashcroft and K. Sturm, Phys. Rev. B 3, 1898 (1971).

Suhrmann, R.

R. Suhrmann, H. Ober, and G. Wedler, Z. Phys. Chem. (Frankfurt) 29, 305 (1961).

Ubbink, J.

H. B. G. Casimir and J. Ubbink, Philips Tech. Rev. 28, 300 (1967).

Wedler, G.

R. Suhrmann, H. Ober, and G. Wedler, Z. Phys. Chem. (Frankfurt) 29, 305 (1961).

Weeks, J. R.

C. Davisson and J. R. Weeks, J. Opt. Soc. Am. 8, 581 (1924).

Wissmann, P.

P. Wissmann, in Springer Tracts in Modern Physics 77, edited by G. Hbhler (Springer-Verlag, New York, 1975), p. 1.

Wooten, F.

F. Wooten, Optical Properties of Solids, (Academic, New York, 1972), p. 93.

Yen, S. H.

K. G. Ramanathan, S. H. Yen, and E. A. Estalote, Appl. Opt. 16, 2810 (1977).

K. G. Ramanathan and S. H. Yen, J. Opt. Soc. Am. 67, 32 (1977).

Zhorov, G. A.

G. A. Zhorov, High Temp. 5 881 (1967) [English translation of Teplofiz, Uys. Temp., 5, 987 (1967)].

Other (43)

E. Aschkinass, Ann. Phys. 17, 960 (1905).

P. D. Foote, Bull. Natl. Bur. Stand. 11, 607 (1914–15).

B. Donovan, Elementary Theory of Metals, (Pergamon, Oxford, 1967), p. 220.

Thermophysical Properties of Matter, edited by Y. S. Touloukian, Vol. 7 (Plenum, New York, 1970), p. 24a.

C. Davisson and J. R. Weeks, J. Opt. Soc. Am. 8, 581 (1924).

W. J. Parker and G. C. Abbott, Symposium on Thermal Radiation of Solids, edited by S. Katzoff, (NASA SP-55, 1965), p. 11.

M. Jakob, Heat Transfer, (Wiley, New York, 1949), p. 51.

G. A. W. Rutgers, Handbuch der Physik, edited by S. Fligge (Springer-Verlag, Berlin 1958), Vol. 26, p. 154.

K. Schocken, Radiative Transfer From Solid Materials, edited by H. Blau and H. Fischer (Macmillan, New York, 1962), p. 3.

K. G. Ramanathan and S. H. Yen, J. Opt. Soc. Am. 67, 32 (1977).

E. A. Estalote and K. G. Ramanathan, J. Opt. Soc. Am. 67, 39 (1977).

K. G. Ramanathan, S. H. Yen, and E. A. Estalote, Appl. Opt. 16, 2810 (1977).

H. E. Bennett and J. M. Bennett, Proceedings of the International Colloquium on Optical Properties, Paris, 1 965, edited by F. Abeles, (North Holland, Amsterdam, 1966), p. 175; H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).

J. A. McKay and J. A. Rayne, Phys. Rev. B 13, 673 (1976).

N. W. Ashcroft and K. Sturm, Phys. Rev. B 3, 1898 (1971).

B. Donovan, Elementary Theory of Metals, (Pergamon, Oxford, 1967), p. 215.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Massachusetts, 1960), p. 280.

A. B. Pippard, Optical Properties and Electronic Structure of Metals and Alloys, edited by F. Abeles (North Holland, Amsterdam, 1966) p. 622.

F. J. Blatt, Physics of Electronic Conduction in Solids (McGraw-Hill, New York, 1968), p. 194.

N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys, (Dover, New York, 1936), p. 268.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Holt, Rinehart and Winston, New York, 1976), p. 5, (see Appendix A for numerical relations between τ,ρµ,rs/a0,vF, and ωp).

Ref. 20, Appendix II, p. 318.

G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. A 195, 336 (1948).

T. Holstein, Phys. Rev. 88, 1427 (1952).

R. B. Dingle, Physica 19, 311 (1953).

A. B. Pippard, in Advances in Electronics and Electron Physics, edited by L. Marton, (Academic, New York, 1954), p. 1.

H. B. G. Casimir and J. Ubbink, Philips Tech. Rev. 28, 300 (1967).

F. Wooten, Optical Properties of Solids, (Academic, New York, 1972), p. 93.

The positions of the boundaries change slightly from metal to metal since they depend on relaxation time and the electron density. See Ref. 28 for a complete description of the boundary parameters.

T. Holstein, Ann. Phys. (N.Y.) 29, 410 (1964); Phys. Rev. 96, 535 (1954).

R. N. Gurzhi, Zh. Eksp. Teor. Fiz. 33, 660 (1957) [Sov. Phys. JETP 6, 506 (1958)1; 35, 965 (1958) [Sov. Phys. JETP 8, 673 (1959)].

For Cu, Pt, and Ag, ρu(T) was taken from American Institute of Physics Handbook, 3rd Edition, McGraw Hill, New York) while for the rest of the elements ρu(T) was taken from Metals Reference Book Vol. III, edited by C. J. Smithells, (Butterworths, London, 1967). Values of rs were taken from Ref. (20) or (21).

G. L. Abbott, in Measurement of Thermal Radiation Properties of Solids, edited by J. C. Richmond, (NASA SP-31, 1963), p. 293.

R. Blickensderfer, D. K. Deardorff, and R. L. Lincoln, J. Less- Common Metals 51, 13 (1977).

G. A. Zhorov, High Temp. 5 881 (1967) [English translation of Teplofiz, Uys. Temp., 5, 987 (1967)].

H. T. Betz, O. H. Olson, B. O. Schurin, and J. C. Morris, Determination of Emissivity and Reflectivity Data on Aircraft Structural Materials, Pt. 2, AD 202493, 1958.

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), p. 322.

R. Smalley and A. J. Sievers, J. Opt. Soc. Am. 68, 1516–1518 (1978).

A. G. Mathewson and H. P. Myers, J. Phys. F. Metal Phys. 3, 623 (1973).

R. T. Beach and R. W. Christy, Phys. Rev. B 16, 5277 (1977).

R. N. Gurzhi, M. Y. Azbel, and H. P. Lin, Fiz. Tverd. Tela. 5, 759 (1963) [Sov. Phys. - Solid State 5, 554 (1963)].

R. Suhrmann, H. Ober, and G. Wedler, Z. Phys. Chem. (Frankfurt) 29, 305 (1961).

P. Wissmann, in Springer Tracts in Modern Physics 77, edited by G. Hbhler (Springer-Verlag, New York, 1975), p. 1.

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