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 ε(ω), the complex frequency-dependent dielectric constant. Experimental data and models for ε(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 ε(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 <i>L</i>-shell contribution to the real part of ε(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 ε<sup>(b)</sup> (ω) to ε(ω); the imaginary part of ε<sup>(b)</sup>(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.

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Arakawa, E. T

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

Arakawa, E. T.

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

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

Ashley, E. J.

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

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

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

Beattie, J . R.

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

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).

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.

H. E. Bennett, private communication.

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

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

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

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

Bennett, J. M.

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

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

See, for example, H. E. Bennett and J. M. Bennett, in Abelès, Ref. 50, p. 175 and 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).

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).

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).

Canfield, L. R.

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

Cohen, M. H.

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

Cole, T. T.

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.

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).

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).

Cooper, B. R.

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

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).

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.

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

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

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

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

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

Emerson, L. C.

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

Feinleib, J.

J.F einleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).

Ferrell, R. A

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

Ferretti, A.

J.F einleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).

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).

Forstmann, 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.

Freeman, G. H. C.

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

Fuchs, R.

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).

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.

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).

Hass, G.

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

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

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).

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.

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

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

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)].

Kliewer, K. L.

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).

Knof, H.

H. Knof, Physica 38, 300 (1968).

Kunz, C.

C. Kunz, Z. Physik 167, 53 (1962).

LaVilla, R. E.

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

Lenham, A. P.

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.

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

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

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

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).

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).

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, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966. 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.

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

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).

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).

Nettel, S. J.

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

Oppenheimer, F.

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.

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).

Philipp, H. R.

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

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

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

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

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.

Powell, C. J.

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

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

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).

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

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. Schnatterly 13, 989 (1968); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968)].

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

Roberts, S.

S. Roberts, Phys. Rev. 114, 104 (1959).

Ron, A.

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

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.

Schulz, L. G.

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

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

Scouler, W. J.

J.F einleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).

Segall, B.

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

Shklyarevskii, I. N.

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

Shubin, A. A.

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

Silver, M.

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

Slater, J. C.

For example, see J. C. Slater, Insulalors, Semiconductors and Metals, Vol. 3 of Quantum 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.

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

Steinmann, W.

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

Stone, H.

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).

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

Swan, J. B.

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

Swanson, N.

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

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).

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

Treherne, D. M.

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.

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

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

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

Vehse, R. C.

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

Verleur, H. W.

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

Waylonis, J. E.

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

Williams, M. W.

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

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.

Yarovaya, R. G.

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

Ziman, J. M.

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

Other (72)

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).

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

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

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

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

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

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

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)].

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.

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).

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.

J.F einleib, W. J. Scouler, and A. Ferretti, Phys. Rev. 165, 765 (1968).

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

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

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

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

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

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

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

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.

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

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

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

H. Knof, Physica 38, 300 (1968).

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

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

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

S. Roberts, Phys. Rev. 114, 104 (1959).

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

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).

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, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966. 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.

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.

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

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.

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

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.

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

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

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

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

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

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

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

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

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

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

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.

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).

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

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. Schnatterly 13, 989 (1968); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968)].

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

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).

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

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

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

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).

C. Kunz, Z. Physik 167, 53 (1962).

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

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.

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

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

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

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

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

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

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

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