Abstract

Optical and electron energy-loss data for aluminum between 0.04 and 72 eV have been critically analyzed and used to test the validity of models for (ω), the complex frequency-dependent dielectric constant. Experimental data and models for (ω) can be effectively compared by use of a nonlinear least-squares computer program and, at least in simple cases, the model parameters have physical significance. Though aluminum has been widely regarded as a relatively ideal free-electron metal, it has been found that a Drude model for (ω) does not adequately describe the observed data. Deviations from the Drude model for photon energies greater than about 1 eV have been interpreted in terms of the effects of interband-electronic transitions and a significant L-shell contribution to the real part of (ω) between 10 and 72 eV. From a fit of reflectance data between 0.2 and 12 eV it has been possible to derive parameters describing the interband-transition contribution (b)(ω) to (ω); the imaginary part of (b)(ω) does not differ significantly from calculations based on the aluminum band structure. Optical constants have been derived in the range of fit and agree closely with the measurements of Hass and Waylonis between 1.9 and 5.6 eV. For photon energies less than 0.2 eV, the reflectance data can be fitted by the empirical formulation of Roberts. The optical absorption for photon energies greater than 12 eV is monotonic and greater than that expected from the tails of the electronic transitions at 1.5–2 eV.

© 1970 Optical Society of America

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  1. See, for example, J. R. Beattie and G. K. T. Conn, Phil. Mag. 46, 989 (1955); L. G. Schulz, Advan. Phys. 6, 102 (1957); T. S. Moss, Optical Properties of Semiconductors (Academic Press Inc., New York, 1959), Ch. 2; H. Mendlowitz, Proc. Phys. Soc. (London) 75, 664 (1960); A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966); A. P. Lenham, 57, 473 (1967); J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
    [Crossref]
  2. W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).
    [Crossref]
  3. H. E. Bennett, M. Silver, and E. J. Ashley, J. Opt. Soc. Am. 53, 1089 (1963).
    [Crossref]
  4. W. R. Hunter, J. Opt. Soc. Am. 54, 208 (1964); J. Phys. (Paris) 25, 154 (1964).
    [Crossref]
  5. H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918 (1963).
    [Crossref]
  6. B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965) and references to earlier papers cited therein.
    [Crossref]
  7. J. C. Phillips, in Solid State Physics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1966), Vol. 18, p. 55.
    [Crossref]
  8. L. W. Beeferman and H. Ehrenreich, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).
  9. J. W. McCaffrey, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).
  10. A. J. Hughes, D. Jones, and A. H. Lettington, J. Phys. Chem. Ser. 2,  2, 102 (1969) [formerly Proc. Phys. Soc. (London)].
  11. R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.
  12. J. G. Collins [Appl. Sci. Res. Sec. B7, 1 (1958)] and K. L. Kliewer and R. Fuchs [Phys. Rev. 172, 607 (1968)] have shown that, in principle, two dielectric functions are needed to represent the optical behavior of a solid, one for describing p polarization and another for s polarization. The variation of the polarization of the output from a grating monochromator with wavelength and with grating coating has been demonstrated by K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965).
    [Crossref]
  13. T. T. Cole and F. Oppenheimer [Appl. Opt. 1, 709 (1962)] report values of n and k for Al between 10.2 and 40.8 eV although, as they remark, their specimen-preparation conditions were not rigorous enough for their values to represent the properties of pure, bulk Al.
    [Crossref]
  14. J. Feinleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).
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  18. C. J. Powell, J. Opt. Soc. Am. 59, 738 (1968).
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  19. H. Ehrenreich and H. R. Philipp, Phys. Rev. 128, 1622 (1962).
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  20. L. G. Schulz, Advan. Phys. 6, 102 (1957).
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  21. A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966).
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  22. See, for example, M. P. Givens, in Solid State Physics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1958), Vol. 6, p. 313.
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  23. R. N. Gurzhi, Sov. Phys.—JETP 8, 673 (1959); R. N. Gurzhi and M. I. Kaganov, 22, 654 (1966).
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    [Crossref] [PubMed]
  25. See, for example, H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
    [Crossref]
  26. H. Knof, Physica 38, 300 (1968).
    [Crossref]
  27. A. Lonke and A. Ron, Phys. Rev. 160, 577 (1967).
    [Crossref]
  28. S. J. Nettel, Phys. Rev. 150, 421 (1966).
    [Crossref]
  29. E. N. Foo and J. J. Hopfield, Phys. Rev. 173, 635 (1968).
    [Crossref]
  30. S. Roberts, Phys. Rev. 114, 104 (1959).
    [Crossref]
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  32. For example, see H. Stone, J. Opt. Soc. Am. 52, 998 (1962); J. A. Blackburn, Anal. Chem. 37, 1000 (1965); E. Rhodes, W. O’Neal, and J. J. Spijkerman, Natl. Bur. Std. (U. S.) Tech. Note 404, 1966, p. 108; M. A. Mariscotti, Nucl. Instr. Methods 50, 309 (1967); R. G. Helmer, R. L. Heath, M. Putnam, and D. H. Gipson, 57, 46 (1967); D. G. Luenberger and V. E. Dennis, Anal. Chem. 38, 715 (1966).
    [Crossref]
  33. D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963). The computer program titled “Least-Squares Estimation of Nonlinear Parameters” can be obtained from the IBM Share General Purpose Library, Distribution No. 1428, December1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August1966. The author is indebted to Dr. D. L. Ederer for bringing the earlier program to his attention and for modifying it for use on the NBS computer.
    [Crossref]
  34. F. Sauter [Z. Physik 203, 488 (1967)] and F. Forstmann [Z. Physik 203, 495 (1967)] have shown that the usual Fresnel equations do not give correct values of the reflectance in the vicinity of the plasma frequency. The correction, however, is 1% or less in the case of Na and this correction is insignificant compared to the usual measurement accuracy.
    [Crossref]
  35. See, for example, W. Steinmann, Phys. Status Solidi 28, 437 (1968).
    [Crossref]
  36. Techniques and problems of least-square analysis using models that are nonlinear functions of the parameters are discussed by N. R. Draper and H. Smith, Applied Regression Analysis (John Wiley & Sons, Inc., New York, 1966), Ch. 10.
  37. A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 752 (1966).
    [Crossref]
  38. J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955).
  39. J. R. Beattie, Phil. Mag. 46, 235 (1955); J. R. Beattie and G. K. T. Conn, p. 989.
  40. A. I. Golovashkin, G. P. Motulevich, and A. A. Shubin, Sov. Phys.—JETP 11, 38 (1960).
  41. I. N. Shklyarevskii and R. G. Yarovaya, Opt. Spectrosc. 14, 130 (1963); Opt. Spectrosc. 16, 45 (1964).
  42. L. G. Schulz, J. Opt. Soc. Am. 44, 357 (1954); L. G. Schulz and F. R. Tangherlini, 44, 362 (1954).
    [Crossref]
  43. G. Hass and J. E. Waylonis, J. Opt. Soc. Am. 51, 719 (1961).
    [Crossref]
  44. R. P. Madden, L. R. Canfield, and G. Hass, J. Opt. Soc. Am. 53, 620 (1963).
    [Crossref]
  45. R. C. Vehse, E. T. Arakawa, and J. L. Stanford, J. Opt. Soc. Am. 57, 551 (1967).
    [Crossref] [PubMed]
  46. R. E. LaVilla and H. Mendlowitz, J. Phys. (Paris) 25, 114 (1964).
    [Crossref]
  47. R. W. Ditchburn and G. H. C. Freeman, Proc. Roy. Soc. (London) A294, 20 (1966).
  48. V. A. Fomichev and A. P. Lukirskii, Sov. Phys.—Solid State 8, 1674 (1967).
  49. H. E. Bennett, J. M. Bennett, and E. J. Ashley, J. Opt. Soc. Am. 52, 1245 (1962).
    [Crossref]
  50. A. P. Lenham, D. M. Treherne, and A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (North-Holland Publ. Co., Amsterdam, 1966), p. 40. A. P. Lenham and D. M. Treherne [Proc. Phys. Soc. (London) 85, 167 (1965)] briefly describe optical measurements on a hand-polished Al specimen in the 0.6–4.6-eV energy region but give no numerical data.
    [Crossref]
  51. J. E. Davey and T. Pankey, J. Appl. Phys. 36, 2571 (1965); L. Marton, J. A. Simpson, and T. F. McCraw, Phys. Rev. 99, 495 (1955).
    [Crossref]
  52. M. W. Williams, E. T. Arakawa, and L. C. Emerson, Surface Sci. 6, 127 (1967).
    [Crossref]
  53. R. H. Ritchie [Surface Sci. 3, 497 (1965)] has suggested that optical absorption due to surface plasmon excitation can occur. This prediction has been confirmed by three groups who find that silver samples show variable absorption near the surface plasmon energy of 3.6 eV that can be correlated with surface roughness [J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly13, 989 (1968); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968)].
    [Crossref]
  54. N. Swanson and C. J. Powell, Phys. Rev. 167, 592 (1968).
    [Crossref]
  55. N. Swanson and C. J. Powell, Phys. Rev. 145, 195 (1966); P. Wienhold, Z. Physik 208, 313 (1968); R. E. Burge and D. L. Misell, Phil. Mag. 18, 261 (1968).
    [Crossref]
  56. See, for example, H. E. Bennett and J. M. Bennett, in Abelès, Ref. 50, p. 175and A. P. Lenham and D. M. Treherne (Ref. 1).
  57. H. Bethe, Ann. Physik 5, 325 (1930); A. H. Wilson, The Theory of Metals (Cambridge University Press, London, 1936), 1st. ed., Ch. 4; U. Fano, Phys. Rev. 103, 1202 (1956); U. Fano and J. Cooper, Rev. Mod. Phys. 40, 441 (1968).
    [Crossref]
  58. H. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Co., Inc., New York, 1945).
  59. N. Swanson, J. Opt. Soc. Am. 54, 1130 (1964).
    [Crossref]
  60. W. A. M. Hartl, Z. Physik 191, 487 (1966); J. Geiger and K. Wittmaack, 195, 44 (1966); K. H. Gaukler, 196, 85 (1966); C. von Festenberg, 207, 47 (1967).
    [Crossref]
  61. C. Kunz, Z. Physik 167, 53 (1962).
    [Crossref]
  62. R. A. Ferrell, Phys. Rev. 111, 1214 (1958).
    [Crossref]
  63. E. T. Arakawa, R. N. Hamm, W. P. Hanson, and T. M. Jelinek, in Abelès, in Ref. 50, p. 374.
  64. H. E. Bennett, private communication.
  65. R. P. Madden, private communication.
  66. R. H. Ritchie, Phys. Rev. 106, 874 (1957).
    [Crossref]
  67. C. J. Powell and J. B. Swan, Phys. Rev. 115, 869 (1959); C. J. Powell, 175, 972 (1968).
    [Crossref]
  68. The upper limit of Eqs. (9) and (10) in Ref. 18 was shown erroneously as infinity.
  69. H. R. Philipp and H. Ehrenreich, J. Appl. Phys. 35, 1416 (1964).
    [Crossref]
  70. D. W. Marquardt, R. G. Bennett, and E. J. Burrell, J. Mol. Spectrosc. 7, 269 (1961).
    [Crossref]
  71. H. Stone, J. Roy. Statist. Soc., Ser. B,  22, 84 (1960); E. M. L. Beale, p. 41.
  72. Y. Beers, Introduction to the Theory of Error (Addison–Wesley Publ. Co., Inc., Reading, Mass., 1958).

1969 (3)

L. W. Beeferman and H. Ehrenreich, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).

J. W. McCaffrey, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).

A. J. Hughes, D. Jones, and A. H. Lettington, J. Phys. Chem. Ser. 2,  2, 102 (1969) [formerly Proc. Phys. Soc. (London)].

1968 (8)

J. Feinleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).
[Crossref]

H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).
[Crossref]

C. J. Powell, J. Opt. Soc. Am. 59, 738 (1968).
[Crossref]

See, for example, H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

H. Knof, Physica 38, 300 (1968).
[Crossref]

E. N. Foo and J. J. Hopfield, Phys. Rev. 173, 635 (1968).
[Crossref]

See, for example, W. Steinmann, Phys. Status Solidi 28, 437 (1968).
[Crossref]

N. Swanson and C. J. Powell, Phys. Rev. 167, 592 (1968).
[Crossref]

1967 (6)

R. C. Vehse, E. T. Arakawa, and J. L. Stanford, J. Opt. Soc. Am. 57, 551 (1967).
[Crossref] [PubMed]

V. A. Fomichev and A. P. Lukirskii, Sov. Phys.—Solid State 8, 1674 (1967).

M. W. Williams, E. T. Arakawa, and L. C. Emerson, Surface Sci. 6, 127 (1967).
[Crossref]

F. Sauter [Z. Physik 203, 488 (1967)] and F. Forstmann [Z. Physik 203, 495 (1967)] have shown that the usual Fresnel equations do not give correct values of the reflectance in the vicinity of the plasma frequency. The correction, however, is 1% or less in the case of Na and this correction is insignificant compared to the usual measurement accuracy.
[Crossref]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 57, 476 (1967).
[Crossref] [PubMed]

A. Lonke and A. Ron, Phys. Rev. 160, 577 (1967).
[Crossref]

1966 (6)

S. J. Nettel, Phys. Rev. 150, 421 (1966).
[Crossref]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 752 (1966).
[Crossref]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966).
[Crossref]

R. W. Ditchburn and G. H. C. Freeman, Proc. Roy. Soc. (London) A294, 20 (1966).

W. A. M. Hartl, Z. Physik 191, 487 (1966); J. Geiger and K. Wittmaack, 195, 44 (1966); K. H. Gaukler, 196, 85 (1966); C. von Festenberg, 207, 47 (1967).
[Crossref]

N. Swanson and C. J. Powell, Phys. Rev. 145, 195 (1966); P. Wienhold, Z. Physik 208, 313 (1968); R. E. Burge and D. L. Misell, Phil. Mag. 18, 261 (1968).
[Crossref]

1965 (4)

J. E. Davey and T. Pankey, J. Appl. Phys. 36, 2571 (1965); L. Marton, J. A. Simpson, and T. F. McCraw, Phys. Rev. 99, 495 (1955).
[Crossref]

R. H. Ritchie [Surface Sci. 3, 497 (1965)] has suggested that optical absorption due to surface plasmon excitation can occur. This prediction has been confirmed by three groups who find that silver samples show variable absorption near the surface plasmon energy of 3.6 eV that can be correlated with surface roughness [J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly13, 989 (1968); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968)].
[Crossref]

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965) and references to earlier papers cited therein.
[Crossref]

W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).
[Crossref]

1964 (4)

W. R. Hunter, J. Opt. Soc. Am. 54, 208 (1964); J. Phys. (Paris) 25, 154 (1964).
[Crossref]

R. E. LaVilla and H. Mendlowitz, J. Phys. (Paris) 25, 114 (1964).
[Crossref]

N. Swanson, J. Opt. Soc. Am. 54, 1130 (1964).
[Crossref]

H. R. Philipp and H. Ehrenreich, J. Appl. Phys. 35, 1416 (1964).
[Crossref]

1963 (5)

I. N. Shklyarevskii and R. G. Yarovaya, Opt. Spectrosc. 14, 130 (1963); Opt. Spectrosc. 16, 45 (1964).

R. P. Madden, L. R. Canfield, and G. Hass, J. Opt. Soc. Am. 53, 620 (1963).
[Crossref]

H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918 (1963).
[Crossref]

H. E. Bennett, M. Silver, and E. J. Ashley, J. Opt. Soc. Am. 53, 1089 (1963).
[Crossref]

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963). The computer program titled “Least-Squares Estimation of Nonlinear Parameters” can be obtained from the IBM Share General Purpose Library, Distribution No. 1428, December1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August1966. The author is indebted to Dr. D. L. Ederer for bringing the earlier program to his attention and for modifying it for use on the NBS computer.
[Crossref]

1962 (5)

1961 (2)

G. Hass and J. E. Waylonis, J. Opt. Soc. Am. 51, 719 (1961).
[Crossref]

D. W. Marquardt, R. G. Bennett, and E. J. Burrell, J. Mol. Spectrosc. 7, 269 (1961).
[Crossref]

1960 (2)

H. Stone, J. Roy. Statist. Soc., Ser. B,  22, 84 (1960); E. M. L. Beale, p. 41.

A. I. Golovashkin, G. P. Motulevich, and A. A. Shubin, Sov. Phys.—JETP 11, 38 (1960).

1959 (3)

S. Roberts, Phys. Rev. 114, 104 (1959).
[Crossref]

R. N. Gurzhi, Sov. Phys.—JETP 8, 673 (1959); R. N. Gurzhi and M. I. Kaganov, 22, 654 (1966).

C. J. Powell and J. B. Swan, Phys. Rev. 115, 869 (1959); C. J. Powell, 175, 972 (1968).
[Crossref]

1958 (3)

R. A. Ferrell, Phys. Rev. 111, 1214 (1958).
[Crossref]

M. H. Cohen, Phil. Mag. 3, 762 (1958).
[Crossref]

J. G. Collins [Appl. Sci. Res. Sec. B7, 1 (1958)] and K. L. Kliewer and R. Fuchs [Phys. Rev. 172, 607 (1968)] have shown that, in principle, two dielectric functions are needed to represent the optical behavior of a solid, one for describing p polarization and another for s polarization. The variation of the polarization of the output from a grating monochromator with wavelength and with grating coating has been demonstrated by K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965).
[Crossref]

1957 (2)

L. G. Schulz, Advan. Phys. 6, 102 (1957).
[Crossref]

R. H. Ritchie, Phys. Rev. 106, 874 (1957).
[Crossref]

1955 (3)

See, for example, J. R. Beattie and G. K. T. Conn, Phil. Mag. 46, 989 (1955); L. G. Schulz, Advan. Phys. 6, 102 (1957); T. S. Moss, Optical Properties of Semiconductors (Academic Press Inc., New York, 1959), Ch. 2; H. Mendlowitz, Proc. Phys. Soc. (London) 75, 664 (1960); A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966); A. P. Lenham, 57, 473 (1967); J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955).

J. R. Beattie, Phil. Mag. 46, 235 (1955); J. R. Beattie and G. K. T. Conn, p. 989.

1954 (1)

1930 (1)

H. Bethe, Ann. Physik 5, 325 (1930); A. H. Wilson, The Theory of Metals (Cambridge University Press, London, 1936), 1st. ed., Ch. 4; U. Fano, Phys. Rev. 103, 1202 (1956); U. Fano and J. Cooper, Rev. Mod. Phys. 40, 441 (1968).
[Crossref]

Arakawa, E. T.

R. C. Vehse, E. T. Arakawa, and J. L. Stanford, J. Opt. Soc. Am. 57, 551 (1967).
[Crossref] [PubMed]

M. W. Williams, E. T. Arakawa, and L. C. Emerson, Surface Sci. 6, 127 (1967).
[Crossref]

E. T. Arakawa, R. N. Hamm, W. P. Hanson, and T. M. Jelinek, in Abelès, in Ref. 50, p. 374.

Ashley, E. J.

Beattie, J. R.

See, for example, J. R. Beattie and G. K. T. Conn, Phil. Mag. 46, 989 (1955); L. G. Schulz, Advan. Phys. 6, 102 (1957); T. S. Moss, Optical Properties of Semiconductors (Academic Press Inc., New York, 1959), Ch. 2; H. Mendlowitz, Proc. Phys. Soc. (London) 75, 664 (1960); A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966); A. P. Lenham, 57, 473 (1967); J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

J. R. Beattie, Phil. Mag. 46, 235 (1955); J. R. Beattie and G. K. T. Conn, p. 989.

Beeferman, L. W.

L. W. Beeferman and H. Ehrenreich, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).

Beers, Y.

Y. Beers, Introduction to the Theory of Error (Addison–Wesley Publ. Co., Inc., Reading, Mass., 1958).

Bennett, H. E.

See, for example, H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

H. E. Bennett, M. Silver, and E. J. Ashley, J. Opt. Soc. Am. 53, 1089 (1963).
[Crossref]

H. E. Bennett, J. M. Bennett, and E. J. Ashley, J. Opt. Soc. Am. 52, 1245 (1962).
[Crossref]

See, for example, H. E. Bennett and J. M. Bennett, in Abelès, Ref. 50, p. 175and A. P. Lenham and D. M. Treherne (Ref. 1).

H. E. Bennett, private communication.

Bennett, J. M.

See, for example, H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

H. E. Bennett, J. M. Bennett, and E. J. Ashley, J. Opt. Soc. Am. 52, 1245 (1962).
[Crossref]

See, for example, H. E. Bennett and J. M. Bennett, in Abelès, Ref. 50, p. 175and A. P. Lenham and D. M. Treherne (Ref. 1).

Bennett, R. G.

D. W. Marquardt, R. G. Bennett, and E. J. Burrell, J. Mol. Spectrosc. 7, 269 (1961).
[Crossref]

Bethe, H.

H. Bethe, Ann. Physik 5, 325 (1930); A. H. Wilson, The Theory of Metals (Cambridge University Press, London, 1936), 1st. ed., Ch. 4; U. Fano, Phys. Rev. 103, 1202 (1956); U. Fano and J. Cooper, Rev. Mod. Phys. 40, 441 (1968).
[Crossref]

Bode, H.

H. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Co., Inc., New York, 1945).

Burrell, E. J.

D. W. Marquardt, R. G. Bennett, and E. J. Burrell, J. Mol. Spectrosc. 7, 269 (1961).
[Crossref]

Canfield, L. R.

Cohen, M. H.

M. H. Cohen, Phil. Mag. 3, 762 (1958).
[Crossref]

Cole, T. T.

Collins, J. G.

J. G. Collins [Appl. Sci. Res. Sec. B7, 1 (1958)] and K. L. Kliewer and R. Fuchs [Phys. Rev. 172, 607 (1968)] have shown that, in principle, two dielectric functions are needed to represent the optical behavior of a solid, one for describing p polarization and another for s polarization. The variation of the polarization of the output from a grating monochromator with wavelength and with grating coating has been demonstrated by K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965).
[Crossref]

Conn, G. K. T.

See, for example, J. R. Beattie and G. K. T. Conn, Phil. Mag. 46, 989 (1955); L. G. Schulz, Advan. Phys. 6, 102 (1957); T. S. Moss, Optical Properties of Semiconductors (Academic Press Inc., New York, 1959), Ch. 2; H. Mendlowitz, Proc. Phys. Soc. (London) 75, 664 (1960); A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966); A. P. Lenham, 57, 473 (1967); J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

Cooper, B. R.

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965) and references to earlier papers cited therein.
[Crossref]

Davey, J. E.

J. E. Davey and T. Pankey, J. Appl. Phys. 36, 2571 (1965); L. Marton, J. A. Simpson, and T. F. McCraw, Phys. Rev. 99, 495 (1955).
[Crossref]

Ditchburn, R. W.

R. W. Ditchburn and G. H. C. Freeman, Proc. Roy. Soc. (London) A294, 20 (1966).

Draper, N. R.

Techniques and problems of least-square analysis using models that are nonlinear functions of the parameters are discussed by N. R. Draper and H. Smith, Applied Regression Analysis (John Wiley & Sons, Inc., New York, 1966), Ch. 10.

Ehrenreich, H.

L. W. Beeferman and H. Ehrenreich, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965) and references to earlier papers cited therein.
[Crossref]

H. R. Philipp and H. Ehrenreich, J. Appl. Phys. 35, 1416 (1964).
[Crossref]

H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918 (1963).
[Crossref]

H. Ehrenreich and H. R. Philipp, Phys. Rev. 128, 1622 (1962).
[Crossref]

Emerson, L. C.

M. W. Williams, E. T. Arakawa, and L. C. Emerson, Surface Sci. 6, 127 (1967).
[Crossref]

Feinleib, J.

J. Feinleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).
[Crossref]

Ferrell, R. A.

R. A. Ferrell, Phys. Rev. 111, 1214 (1958).
[Crossref]

Ferretti, A.

J. Feinleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).
[Crossref]

Fomichev, V. A.

V. A. Fomichev and A. P. Lukirskii, Sov. Phys.—Solid State 8, 1674 (1967).

Foo, E. N.

E. N. Foo and J. J. Hopfield, Phys. Rev. 173, 635 (1968).
[Crossref]

Freeman, G. H. C.

R. W. Ditchburn and G. H. C. Freeman, Proc. Roy. Soc. (London) A294, 20 (1966).

Givens, M. P.

See, for example, M. P. Givens, in Solid State Physics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1958), Vol. 6, p. 313.
[Crossref]

Golovashkin, A. I.

A. I. Golovashkin, G. P. Motulevich, and A. A. Shubin, Sov. Phys.—JETP 11, 38 (1960).

Gurzhi, R. N.

R. N. Gurzhi, Sov. Phys.—JETP 8, 673 (1959); R. N. Gurzhi and M. I. Kaganov, 22, 654 (1966).

Hamm, R. N.

E. T. Arakawa, R. N. Hamm, W. P. Hanson, and T. M. Jelinek, in Abelès, in Ref. 50, p. 374.

Hanson, W. P.

E. T. Arakawa, R. N. Hamm, W. P. Hanson, and T. M. Jelinek, in Abelès, in Ref. 50, p. 374.

Hartl, W. A. M.

W. A. M. Hartl, Z. Physik 191, 487 (1966); J. Geiger and K. Wittmaack, 195, 44 (1966); K. H. Gaukler, 196, 85 (1966); C. von Festenberg, 207, 47 (1967).
[Crossref]

Hass, G.

Hodgson, J. N.

J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955).

Hopfield, J. J.

E. N. Foo and J. J. Hopfield, Phys. Rev. 173, 635 (1968).
[Crossref]

Hughes, A. J.

A. J. Hughes, D. Jones, and A. H. Lettington, J. Phys. Chem. Ser. 2,  2, 102 (1969) [formerly Proc. Phys. Soc. (London)].

Hunter, W. R.

Jelinek, T. M.

E. T. Arakawa, R. N. Hamm, W. P. Hanson, and T. M. Jelinek, in Abelès, in Ref. 50, p. 374.

Jones, D.

A. J. Hughes, D. Jones, and A. H. Lettington, J. Phys. Chem. Ser. 2,  2, 102 (1969) [formerly Proc. Phys. Soc. (London)].

Knof, H.

H. Knof, Physica 38, 300 (1968).
[Crossref]

Kunz, C.

C. Kunz, Z. Physik 167, 53 (1962).
[Crossref]

LaVilla, R. E.

R. E. LaVilla and H. Mendlowitz, J. Phys. (Paris) 25, 114 (1964).
[Crossref]

Lenham, A. P.

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 57, 476 (1967).
[Crossref] [PubMed]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 752 (1966).
[Crossref]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966).
[Crossref]

A. P. Lenham, D. M. Treherne, and A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (North-Holland Publ. Co., Amsterdam, 1966), p. 40. A. P. Lenham and D. M. Treherne [Proc. Phys. Soc. (London) 85, 167 (1965)] briefly describe optical measurements on a hand-polished Al specimen in the 0.6–4.6-eV energy region but give no numerical data.
[Crossref]

Lettington, A. H.

A. J. Hughes, D. Jones, and A. H. Lettington, J. Phys. Chem. Ser. 2,  2, 102 (1969) [formerly Proc. Phys. Soc. (London)].

Lonke, A.

A. Lonke and A. Ron, Phys. Rev. 160, 577 (1967).
[Crossref]

Lukirskii, A. P.

V. A. Fomichev and A. P. Lukirskii, Sov. Phys.—Solid State 8, 1674 (1967).

Madden, R. P.

R. P. Madden, L. R. Canfield, and G. Hass, J. Opt. Soc. Am. 53, 620 (1963).
[Crossref]

R. P. Madden, private communication.

R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.

Marquardt, D. W.

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963). The computer program titled “Least-Squares Estimation of Nonlinear Parameters” can be obtained from the IBM Share General Purpose Library, Distribution No. 1428, December1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August1966. The author is indebted to Dr. D. L. Ederer for bringing the earlier program to his attention and for modifying it for use on the NBS computer.
[Crossref]

D. W. Marquardt, R. G. Bennett, and E. J. Burrell, J. Mol. Spectrosc. 7, 269 (1961).
[Crossref]

McCaffrey, J. W.

J. W. McCaffrey, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).

Mendlowitz, H.

R. E. LaVilla and H. Mendlowitz, J. Phys. (Paris) 25, 114 (1964).
[Crossref]

Motulevich, G. P.

A. I. Golovashkin, G. P. Motulevich, and A. A. Shubin, Sov. Phys.—JETP 11, 38 (1960).

Motyka, R. J.

See, for example, H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

Nettel, S. J.

S. J. Nettel, Phys. Rev. 150, 421 (1966).
[Crossref]

Oppenheimer, F.

Pankey, T.

J. E. Davey and T. Pankey, J. Appl. Phys. 36, 2571 (1965); L. Marton, J. A. Simpson, and T. F. McCraw, Phys. Rev. 99, 495 (1955).
[Crossref]

Philipp, H. R.

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965) and references to earlier papers cited therein.
[Crossref]

H. R. Philipp and H. Ehrenreich, J. Appl. Phys. 35, 1416 (1964).
[Crossref]

H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918 (1963).
[Crossref]

H. Ehrenreich and H. R. Philipp, Phys. Rev. 128, 1622 (1962).
[Crossref]

Phillips, J. C.

J. C. Phillips, in Solid State Physics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1966), Vol. 18, p. 55.
[Crossref]

Powell, C. J.

C. J. Powell, J. Opt. Soc. Am. 59, 738 (1968).
[Crossref]

N. Swanson and C. J. Powell, Phys. Rev. 167, 592 (1968).
[Crossref]

N. Swanson and C. J. Powell, Phys. Rev. 145, 195 (1966); P. Wienhold, Z. Physik 208, 313 (1968); R. E. Burge and D. L. Misell, Phil. Mag. 18, 261 (1968).
[Crossref]

C. J. Powell and J. B. Swan, Phys. Rev. 115, 869 (1959); C. J. Powell, 175, 972 (1968).
[Crossref]

Ritchie, R. H.

R. H. Ritchie [Surface Sci. 3, 497 (1965)] has suggested that optical absorption due to surface plasmon excitation can occur. This prediction has been confirmed by three groups who find that silver samples show variable absorption near the surface plasmon energy of 3.6 eV that can be correlated with surface roughness [J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly13, 989 (1968); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968)].
[Crossref]

R. H. Ritchie, Phys. Rev. 106, 874 (1957).
[Crossref]

Roberts, S.

S. Roberts, Phys. Rev. 114, 104 (1959).
[Crossref]

Ron, A.

A. Lonke and A. Ron, Phys. Rev. 160, 577 (1967).
[Crossref]

Sauter, F.

F. Sauter [Z. Physik 203, 488 (1967)] and F. Forstmann [Z. Physik 203, 495 (1967)] have shown that the usual Fresnel equations do not give correct values of the reflectance in the vicinity of the plasma frequency. The correction, however, is 1% or less in the case of Na and this correction is insignificant compared to the usual measurement accuracy.
[Crossref]

Schulz, L. G.

Scouler, W. J.

J. Feinleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).
[Crossref]

Segall, B.

H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918 (1963).
[Crossref]

Shklyarevskii, I. N.

I. N. Shklyarevskii and R. G. Yarovaya, Opt. Spectrosc. 14, 130 (1963); Opt. Spectrosc. 16, 45 (1964).

Shubin, A. A.

A. I. Golovashkin, G. P. Motulevich, and A. A. Shubin, Sov. Phys.—JETP 11, 38 (1960).

Silver, M.

Slater, J. C.

For example, see J. C. Slater, Insulators, Semiconductors and Metals, Vol. 3 ofQuantum Theory of Molecules and Solids (McGraw–Hill Book Co., New York, 1967), Chs. 4 and 5.

Smith, H.

Techniques and problems of least-square analysis using models that are nonlinear functions of the parameters are discussed by N. R. Draper and H. Smith, Applied Regression Analysis (John Wiley & Sons, Inc., New York, 1966), Ch. 10.

Stanford, J. L.

Steinmann, W.

See, for example, W. Steinmann, Phys. Status Solidi 28, 437 (1968).
[Crossref]

Stone, H.

Swan, J. B.

C. J. Powell and J. B. Swan, Phys. Rev. 115, 869 (1959); C. J. Powell, 175, 972 (1968).
[Crossref]

Swanson, N.

N. Swanson and C. J. Powell, Phys. Rev. 167, 592 (1968).
[Crossref]

N. Swanson and C. J. Powell, Phys. Rev. 145, 195 (1966); P. Wienhold, Z. Physik 208, 313 (1968); R. E. Burge and D. L. Misell, Phil. Mag. 18, 261 (1968).
[Crossref]

N. Swanson, J. Opt. Soc. Am. 54, 1130 (1964).
[Crossref]

Treherne, D. M.

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 57, 476 (1967).
[Crossref] [PubMed]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 752 (1966).
[Crossref]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966).
[Crossref]

A. P. Lenham, D. M. Treherne, and A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (North-Holland Publ. Co., Amsterdam, 1966), p. 40. A. P. Lenham and D. M. Treherne [Proc. Phys. Soc. (London) 85, 167 (1965)] briefly describe optical measurements on a hand-polished Al specimen in the 0.6–4.6-eV energy region but give no numerical data.
[Crossref]

Vehse, R. C.

Verleur, H. W.

Waylonis, J. E.

Williams, M. W.

M. W. Williams, E. T. Arakawa, and L. C. Emerson, Surface Sci. 6, 127 (1967).
[Crossref]

Woodall, A. J.

A. P. Lenham, D. M. Treherne, and A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (North-Holland Publ. Co., Amsterdam, 1966), p. 40. A. P. Lenham and D. M. Treherne [Proc. Phys. Soc. (London) 85, 167 (1965)] briefly describe optical measurements on a hand-polished Al specimen in the 0.6–4.6-eV energy region but give no numerical data.
[Crossref]

Yarovaya, R. G.

I. N. Shklyarevskii and R. G. Yarovaya, Opt. Spectrosc. 14, 130 (1963); Opt. Spectrosc. 16, 45 (1964).

Ziman, J. M.

J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, London, England, 1964), p. 241.

Advan. Phys. (1)

L. G. Schulz, Advan. Phys. 6, 102 (1957).
[Crossref]

Ann. Physik (1)

H. Bethe, Ann. Physik 5, 325 (1930); A. H. Wilson, The Theory of Metals (Cambridge University Press, London, 1936), 1st. ed., Ch. 4; U. Fano, Phys. Rev. 103, 1202 (1956); U. Fano and J. Cooper, Rev. Mod. Phys. 40, 441 (1968).
[Crossref]

Appl. Opt. (1)

Appl. Sci. Res. Sec. (1)

J. G. Collins [Appl. Sci. Res. Sec. B7, 1 (1958)] and K. L. Kliewer and R. Fuchs [Phys. Rev. 172, 607 (1968)] have shown that, in principle, two dielectric functions are needed to represent the optical behavior of a solid, one for describing p polarization and another for s polarization. The variation of the polarization of the output from a grating monochromator with wavelength and with grating coating has been demonstrated by K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965).
[Crossref]

Bull. Am. Phys. Soc., Ser. II (2)

L. W. Beeferman and H. Ehrenreich, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).

J. W. McCaffrey, Bull. Am. Phys. Soc., Ser. II,  14, 397 (1969).

J. Appl. Phys. (2)

J. E. Davey and T. Pankey, J. Appl. Phys. 36, 2571 (1965); L. Marton, J. A. Simpson, and T. F. McCraw, Phys. Rev. 99, 495 (1955).
[Crossref]

H. R. Philipp and H. Ehrenreich, J. Appl. Phys. 35, 1416 (1964).
[Crossref]

J. Mol. Spectrosc. (1)

D. W. Marquardt, R. G. Bennett, and E. J. Burrell, J. Mol. Spectrosc. 7, 269 (1961).
[Crossref]

J. Opt. Soc. Am. (15)

N. Swanson, J. Opt. Soc. Am. 54, 1130 (1964).
[Crossref]

L. G. Schulz, J. Opt. Soc. Am. 44, 357 (1954); L. G. Schulz and F. R. Tangherlini, 44, 362 (1954).
[Crossref]

G. Hass and J. E. Waylonis, J. Opt. Soc. Am. 51, 719 (1961).
[Crossref]

R. P. Madden, L. R. Canfield, and G. Hass, J. Opt. Soc. Am. 53, 620 (1963).
[Crossref]

R. C. Vehse, E. T. Arakawa, and J. L. Stanford, J. Opt. Soc. Am. 57, 551 (1967).
[Crossref] [PubMed]

H. E. Bennett, J. M. Bennett, and E. J. Ashley, J. Opt. Soc. Am. 52, 1245 (1962).
[Crossref]

W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).
[Crossref]

H. E. Bennett, M. Silver, and E. J. Ashley, J. Opt. Soc. Am. 53, 1089 (1963).
[Crossref]

W. R. Hunter, J. Opt. Soc. Am. 54, 208 (1964); J. Phys. (Paris) 25, 154 (1964).
[Crossref]

H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).
[Crossref]

C. J. Powell, J. Opt. Soc. Am. 59, 738 (1968).
[Crossref]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966).
[Crossref]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 57, 476 (1967).
[Crossref] [PubMed]

For example, see H. Stone, J. Opt. Soc. Am. 52, 998 (1962); J. A. Blackburn, Anal. Chem. 37, 1000 (1965); E. Rhodes, W. O’Neal, and J. J. Spijkerman, Natl. Bur. Std. (U. S.) Tech. Note 404, 1966, p. 108; M. A. Mariscotti, Nucl. Instr. Methods 50, 309 (1967); R. G. Helmer, R. L. Heath, M. Putnam, and D. H. Gipson, 57, 46 (1967); D. G. Luenberger and V. E. Dennis, Anal. Chem. 38, 715 (1966).
[Crossref]

A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 752 (1966).
[Crossref]

J. Phys. (Paris) (1)

R. E. LaVilla and H. Mendlowitz, J. Phys. (Paris) 25, 114 (1964).
[Crossref]

J. Phys. Chem. Ser. 2 (1)

A. J. Hughes, D. Jones, and A. H. Lettington, J. Phys. Chem. Ser. 2,  2, 102 (1969) [formerly Proc. Phys. Soc. (London)].

J. Roy. Statist. Soc., Ser. B (1)

H. Stone, J. Roy. Statist. Soc., Ser. B,  22, 84 (1960); E. M. L. Beale, p. 41.

J. Soc. Indust. Appl. Math. (1)

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963). The computer program titled “Least-Squares Estimation of Nonlinear Parameters” can be obtained from the IBM Share General Purpose Library, Distribution No. 1428, December1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August1966. The author is indebted to Dr. D. L. Ederer for bringing the earlier program to his attention and for modifying it for use on the NBS computer.
[Crossref]

Opt. Spectrosc. (1)

I. N. Shklyarevskii and R. G. Yarovaya, Opt. Spectrosc. 14, 130 (1963); Opt. Spectrosc. 16, 45 (1964).

Phil. Mag. (3)

J. R. Beattie, Phil. Mag. 46, 235 (1955); J. R. Beattie and G. K. T. Conn, p. 989.

M. H. Cohen, Phil. Mag. 3, 762 (1958).
[Crossref]

See, for example, J. R. Beattie and G. K. T. Conn, Phil. Mag. 46, 989 (1955); L. G. Schulz, Advan. Phys. 6, 102 (1957); T. S. Moss, Optical Properties of Semiconductors (Academic Press Inc., New York, 1959), Ch. 2; H. Mendlowitz, Proc. Phys. Soc. (London) 75, 664 (1960); A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 1076 (1966); A. P. Lenham, 57, 473 (1967); J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

Phys. Rev. (14)

J. Feinleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).
[Crossref]

H. Ehrenreich and H. R. Philipp, Phys. Rev. 128, 1622 (1962).
[Crossref]

H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918 (1963).
[Crossref]

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965) and references to earlier papers cited therein.
[Crossref]

A. Lonke and A. Ron, Phys. Rev. 160, 577 (1967).
[Crossref]

S. J. Nettel, Phys. Rev. 150, 421 (1966).
[Crossref]

E. N. Foo and J. J. Hopfield, Phys. Rev. 173, 635 (1968).
[Crossref]

S. Roberts, Phys. Rev. 114, 104 (1959).
[Crossref]

See, for example, H. E. Bennett, J. M. Bennett, E. J. Ashley, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

N. Swanson and C. J. Powell, Phys. Rev. 167, 592 (1968).
[Crossref]

N. Swanson and C. J. Powell, Phys. Rev. 145, 195 (1966); P. Wienhold, Z. Physik 208, 313 (1968); R. E. Burge and D. L. Misell, Phil. Mag. 18, 261 (1968).
[Crossref]

R. H. Ritchie, Phys. Rev. 106, 874 (1957).
[Crossref]

C. J. Powell and J. B. Swan, Phys. Rev. 115, 869 (1959); C. J. Powell, 175, 972 (1968).
[Crossref]

R. A. Ferrell, Phys. Rev. 111, 1214 (1958).
[Crossref]

Phys. Status Solidi (1)

See, for example, W. Steinmann, Phys. Status Solidi 28, 437 (1968).
[Crossref]

Physica (1)

H. Knof, Physica 38, 300 (1968).
[Crossref]

Proc. Phys. Soc. (London) (1)

J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955).

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

R. W. Ditchburn and G. H. C. Freeman, Proc. Roy. Soc. (London) A294, 20 (1966).

Sov. Phys.—JETP (2)

A. I. Golovashkin, G. P. Motulevich, and A. A. Shubin, Sov. Phys.—JETP 11, 38 (1960).

R. N. Gurzhi, Sov. Phys.—JETP 8, 673 (1959); R. N. Gurzhi and M. I. Kaganov, 22, 654 (1966).

Sov. Phys.—Solid State (1)

V. A. Fomichev and A. P. Lukirskii, Sov. Phys.—Solid State 8, 1674 (1967).

Surface Sci. (2)

M. W. Williams, E. T. Arakawa, and L. C. Emerson, Surface Sci. 6, 127 (1967).
[Crossref]

R. H. Ritchie [Surface Sci. 3, 497 (1965)] has suggested that optical absorption due to surface plasmon excitation can occur. This prediction has been confirmed by three groups who find that silver samples show variable absorption near the surface plasmon energy of 3.6 eV that can be correlated with surface roughness [J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly13, 989 (1968); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968)].
[Crossref]

Z. Physik (3)

W. A. M. Hartl, Z. Physik 191, 487 (1966); J. Geiger and K. Wittmaack, 195, 44 (1966); K. H. Gaukler, 196, 85 (1966); C. von Festenberg, 207, 47 (1967).
[Crossref]

C. Kunz, Z. Physik 167, 53 (1962).
[Crossref]

F. Sauter [Z. Physik 203, 488 (1967)] and F. Forstmann [Z. Physik 203, 495 (1967)] have shown that the usual Fresnel equations do not give correct values of the reflectance in the vicinity of the plasma frequency. The correction, however, is 1% or less in the case of Na and this correction is insignificant compared to the usual measurement accuracy.
[Crossref]

Other (14)

Techniques and problems of least-square analysis using models that are nonlinear functions of the parameters are discussed by N. R. Draper and H. Smith, Applied Regression Analysis (John Wiley & Sons, Inc., New York, 1966), Ch. 10.

See, for example, M. P. Givens, in Solid State Physics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1958), Vol. 6, p. 313.
[Crossref]

J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, London, England, 1964), p. 241.

J. C. Phillips, in Solid State Physics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1966), Vol. 18, p. 55.
[Crossref]

R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.

For example, see J. C. Slater, Insulators, Semiconductors and Metals, Vol. 3 ofQuantum Theory of Molecules and Solids (McGraw–Hill Book Co., New York, 1967), Chs. 4 and 5.

H. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Co., Inc., New York, 1945).

See, for example, H. E. Bennett and J. M. Bennett, in Abelès, Ref. 50, p. 175and A. P. Lenham and D. M. Treherne (Ref. 1).

A. P. Lenham, D. M. Treherne, and A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (North-Holland Publ. Co., Amsterdam, 1966), p. 40. A. P. Lenham and D. M. Treherne [Proc. Phys. Soc. (London) 85, 167 (1965)] briefly describe optical measurements on a hand-polished Al specimen in the 0.6–4.6-eV energy region but give no numerical data.
[Crossref]

E. T. Arakawa, R. N. Hamm, W. P. Hanson, and T. M. Jelinek, in Abelès, in Ref. 50, p. 374.

H. E. Bennett, private communication.

R. P. Madden, private communication.

Y. Beers, Introduction to the Theory of Error (Addison–Wesley Publ. Co., Inc., Reading, Mass., 1958).

The upper limit of Eqs. (9) and (10) in Ref. 18 was shown erroneously as infinity.

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Figures (14)

Fig. 1
Fig. 1

Measured and computed reflectance of Al in the 0–1-eV photon-energy region, using the sources of data listed in Table I.

Fig. 2
Fig. 2

Measured and computed reflectance of Al in the 0.5–12.5-eV photon-energy region, using the sources of data listed in Table I.

Fig. 3
Fig. 3

Measured and computed reflectance of Al in the 9–20-eV photon-energy region, using the sources of data listed in Table I.

Fig. 4
Fig. 4

Plot of (1−1(f)) vs wavelength squared to test the validity of Eq. (8) using the data sources listed in Table I. The scales in the various plots have been adjusted by factors of 10 so that the slopes in the different regions are comparable. The photon-energy range covered is 0.13–64 eV.

Fig. 5
Fig. 5

Plot of f0 derived from a solution of Eqs. (8) and (9) vs photon energy, using the data sources listed in Table I. As discussed in the text, estimated values of 1(L) have been subtracted from the experimental values of 1. Some of the computed points have had to be omitted for clarity. The inset shows the low-energy region in more detail.

Fig. 6
Fig. 6

Plot of ħg0 derived from a solution of Eqs. (8) and (9) vs photon energy using the data sources listed in Table I. Estimated values of 1(L) have been subtracted from the experimental values of 1, as for f0 in Fig. 5. The scatter in the points derived from the data of LaVilla and Mendlowitz46 is not significant. The horizontal dashed line represents the value of ħg0 computed using Eq. (4), assuming σ0 to be the bulk dc conductivity and computing ne with three electrons per atom. Some points have had to be omitted for clarity. The inset shows the low-energy region in more detail.

Fig. 7
Fig. 7

Computed and measured reflectance of Al vs photon energy in the ir. The dots represent the data of Bennett et al.3 and the three curves are computed assuming a Drude model for (ω). For curve (a) f0 = 0.65, ħg0 = 0.085 eV; curve (b) f0 = 1.5, ħg0 = 0.13 eV; curve (c) f0=6, ħg0 = 0.33 eV. Curve (c) is the final least-squares fit to the data.

Fig. 8
Fig. 8

Variations of the normal-incidence reflectance (solid line) for Al computed using the parameters in Table II for photon energies less than 0.3 eV. The points show reflectance values measured by Bennett et al.3 The dashed line shows a fit to the Bennett et al. reflectance data between 0.039 and 0.18 eV using a model with two sets of Drude parameters, as described in Sec. IV. The values of the parameters are: f01 = 0.70±0.84, ħg0 = 0.13±0.05 eV, f02 = 0.14±0.02, ħg02≈0±0.3 eV, where the uncertainties represent the computed standard error of each parameter. In this fit, the input data were inadequate to define g02 and the fitting program attempted to make this parameter negative; data at lower energies would be required to obtain a more accurate determination of g02. The dot–dashed line represents computed values of reflectance using the values of n and k calculated for Al by Lonke and Ron.27

Fig. 9
Fig. 9

The dashed lines show the variation of n and k for Al between 0.039 and 0.18 eV based on the four-parameter fit to the reflectance shown in Fig. 8. The computed standard error of n was 25 at 0.04 eV, a maximum value of 32 at 0.06 eV, and decreased to 16 at 0.18 eV. The computed standard error of k was between 25 and 33 for all photon energies between 0.04 and 0.18 eV. The dot–dashed lines show the predicted variation of n and k as calculated by Lonke and Ron.27 The points indicated by ⦵ were deduced by Bennett and Bennett56 who reported that their reflectance data3,56 was consistent with a Drude model. The other symbols refer to the data of the authors listed in Table I.

Fig. 10
Fig. 10

Results of a fit (solid line) to the measured Al reflectance of Bennett et al.3 and of Madden et al.44 The parameters describing the solid curve are listed in Table II.

Fig. 11
Fig. 11

(a) The solid line represents 2(b) deduced from Al experimental data by Ehrenreich et al.5 while the dashed lines represent calculations of 2(b) by the same authors for transitions at ≈1.4 eV and beginning at ≈2 eV. (b) The solid line represents 2(b) for Al calculated using the parameters determined from the fit shown in Fig. 10. The dashed lines represent the separate contribution to 2(b) of the transitions labeled with subscripts 1 and 2 in the list of parameters of Table II. The vertical bars indicate the limits of the computed standard error for several representative energies on each curve.

Fig. 12
Fig. 12

n and k as a function of photon energy for Al computed using the parameters given in Table II. The standard error in the calculated n varied from about 10% at the lower energies to about 6% for energies between 1.4 and 12 eV. The standard error in the calculated k varied between 2% and 3% for energies between 0.2 and 7 eV anti increased to 8% for an energy of 12 eV. The points indicated by josa-60-1-78-i001 were derived by Ehrenerich et al.5 from their analysis of Al reflectance data. The other symbols refer to the data of the authors listed in Table I.

Fig. 13
Fig. 13

Variation of 2 (solid line) computed using the parameters of Table II for photon energies less than 0.25 eV; the standard error in 2 was between 4% and 8% for energies between 0.04 and 0.2 eV. The dashed line represents the variation of 2 based on the fit (Fig. 8) to the Al reflectance data using two pairs of Drude parameters; the standard error in these values of 2 increased from 44% at 0.04 eV to 130% at 0.2 eV. The points refer to the data sources listed in Table I.

Fig. 14
Fig. 14

Variation of 2 (solid line) computed using the parameters of Table II for photon energies between 10 and 70 eV. The standard error in 2 was computed to be between 9% and 12% over the range shown. The points refer to the data sources shown in Table I.

Tables (3)

Tables Icon

Table I Summary of recent measurements of optical data for Al. The third column indicates the type of data presented in each paper and the fourth column indicates the symbols used in the figures to present the results of the corresponding authors.

Tables Icon

Table II Parameters of (ω) [Eqs. (2) and (3)] derived from the fit to the Al reflectance data shown in Fig. 10. The three uncertainty estimates are defined in the Appendix and are based on a 95% confidence level. The oscillator strengths are based on a plasma energy ħωp of 15.78 eV.

Tables Icon

Table III Values of n and k for Al computed using the parameters given in Table II. The quantities σn and σk are the calculated standard errors of n and k, respectively (see Appendix). [The reflectance of Al for photon energies between about 9 and 11 eV appears to be a function of surface roughness (Sec. III) and there may therefore be a significant systematic error in the derived optical constants in this energy range.]

Equations (15)

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( ω ) = ( f ) ( ω ) + ( b ) ( ω ) ,
( f ) ( ω ) = 1 - f 0 ω p 2 / ( ω 2 + i g 0 ω ) ,
( b ) ( ω ) = - j = 1 r f j ω p 2 ω 2 - ω j 2 + i g j ω .
τ 0 = m σ 0 / n e e 2 ,
n = [ ( 1 2 + 2 2 ) 1 2 + 1 ] 1 2 / 2 ,
k = [ ( 1 2 + 2 2 ) 1 2 - 1 ] 1 2 / 2 ,
R = [ ( n - 1 ) 2 + k 2 ] / [ ( n + 1 ) 2 + k 2 ] ,
1 ( f ) ( ω ) = 1 - f 0 ω p 2 / ( ω 2 + g 0 2 )
2 ( f ) ( ω ) = f 0 ω p 2 g 0 / ( ω 2 + g 0 2 ) .
1 ( ω ) 1 - ω p 2 ( f 0 + Σ j f j ) / ω 2 = 1 - ( f * ω p 2 / ω 2 ) ,
2 ( ω ) ( f 0 ω p 2 / ω 3 ) [ g 0 + ( Σ j f j g j / f 0 ) ] = f 0 ω p 2 g * / ω 3 .
0 ω m ω 2 d ω = π S o ω p 2 / 2
- 0 ω m ω Im - 1 d ω = π S e ω p 2 / 2 ,
φ c = φ { 1 + [ K / ( N - K ) ] F 1 - α ( K , N - K ) } .
s F 2 = Σ j , j ( F / j ) ( F / j ) s j s j ρ j j ,