Abstract

A procedure is presented for the use of the Kramers–Kronig dispersion relations in treatment of normal-incidence infrared reflection spectra to yield optical constants of crystals. The crux of the practical problem, treatment of unobserved wing regions in the integrations, is discussed in detail. Definitive methods of picking optimum upper and lower limits for integration over actual data and for fitting artificial wings are discussed.

The method is tested on a reflection spectrum generated by a model involving several damped harmonic oscillators. It is shown how a region involving a single band (but flanked by “unobserved” bands) can be treated, and how accurately the method reproduces the theoretically known optical indices. It appears feasible to obtain the indices of absorption and of refraction to ±0.005 and ±0.002, respectively, given accurate data on reflectance. Integrated band intensities are reproduced to within a few percent.

© 1965 Optical Society of America

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References

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  1. T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).
  2. F. C. Jahoda, Phys. Rev. 107, 1261 (1957).
    [Crossref]
  3. D. G. Thomas and J. Hopfield, Phys. Rev. 116, 573 (1959).
    [Crossref]
  4. M. Gottlieb, J. Opt. Soc. Am. 50, 343 (1960).
    [Crossref]
  5. G. R. Anderson and W. B. Person, J. Chem. Phys. 36, 62 (1962).
    [Crossref]
  6. H. J. Bowlden and J. K. Wilmshurst, J. Opt. Soc. Am. 53, 1073 (1963).
    [Crossref]
  7. P. N. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
    [Crossref]
  8. W. G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1324 (1961).
    [Crossref]
  9. P. W. Barnes and P. N. Schatz, J. Chem. Phys. 38, 2662 (1963).
    [Crossref]
  10. J. L. Hollenberg and D. A. Dows, Spectrochim. Acta 16, 1155 (1960).
    [Crossref]
  11. J. Vincent-Geisse, J. Phys. (Paris) 24, 259 (1963).
    [Crossref]
  12. K. S. Seshadin and R. N. Jones, Spectrochim. Acta 19, 1013 (1963).
    [Crossref]

1963 (5)

H. J. Bowlden and J. K. Wilmshurst, J. Opt. Soc. Am. 53, 1073 (1963).
[Crossref]

P. N. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

P. W. Barnes and P. N. Schatz, J. Chem. Phys. 38, 2662 (1963).
[Crossref]

J. Vincent-Geisse, J. Phys. (Paris) 24, 259 (1963).
[Crossref]

K. S. Seshadin and R. N. Jones, Spectrochim. Acta 19, 1013 (1963).
[Crossref]

1962 (1)

G. R. Anderson and W. B. Person, J. Chem. Phys. 36, 62 (1962).
[Crossref]

1961 (1)

W. G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1324 (1961).
[Crossref]

1960 (2)

J. L. Hollenberg and D. A. Dows, Spectrochim. Acta 16, 1155 (1960).
[Crossref]

M. Gottlieb, J. Opt. Soc. Am. 50, 343 (1960).
[Crossref]

1959 (1)

D. G. Thomas and J. Hopfield, Phys. Rev. 116, 573 (1959).
[Crossref]

1957 (1)

F. C. Jahoda, Phys. Rev. 107, 1261 (1957).
[Crossref]

1952 (1)

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Anderson, G. R.

G. R. Anderson and W. B. Person, J. Chem. Phys. 36, 62 (1962).
[Crossref]

Barnes, P. W.

P. W. Barnes and P. N. Schatz, J. Chem. Phys. 38, 2662 (1963).
[Crossref]

Bowlden, H. J.

Dows, D. A.

J. L. Hollenberg and D. A. Dows, Spectrochim. Acta 16, 1155 (1960).
[Crossref]

Gottlieb, M.

Hollenberg, J. L.

J. L. Hollenberg and D. A. Dows, Spectrochim. Acta 16, 1155 (1960).
[Crossref]

Hopfield, J.

D. G. Thomas and J. Hopfield, Phys. Rev. 116, 573 (1959).
[Crossref]

Jahoda, F. C.

F. C. Jahoda, Phys. Rev. 107, 1261 (1957).
[Crossref]

Jones, R. N.

K. S. Seshadin and R. N. Jones, Spectrochim. Acta 19, 1013 (1963).
[Crossref]

Kleinman, D. A.

W. G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1324 (1961).
[Crossref]

Kozima, K.

P. N. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

Maeda, S.

P. N. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

Person, W. B.

G. R. Anderson and W. B. Person, J. Chem. Phys. 36, 62 (1962).
[Crossref]

Robinson, T. S.

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Schatz, P. N.

P. N. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

P. W. Barnes and P. N. Schatz, J. Chem. Phys. 38, 2662 (1963).
[Crossref]

Seshadin, K. S.

K. S. Seshadin and R. N. Jones, Spectrochim. Acta 19, 1013 (1963).
[Crossref]

Spitzer, W. G.

W. G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1324 (1961).
[Crossref]

Thomas, D. G.

D. G. Thomas and J. Hopfield, Phys. Rev. 116, 573 (1959).
[Crossref]

Vincent-Geisse, J.

J. Vincent-Geisse, J. Phys. (Paris) 24, 259 (1963).
[Crossref]

Wilmshurst, J. K.

J. Chem. Phys. (3)

G. R. Anderson and W. B. Person, J. Chem. Phys. 36, 62 (1962).
[Crossref]

P. N. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

P. W. Barnes and P. N. Schatz, J. Chem. Phys. 38, 2662 (1963).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. (Paris) (1)

J. Vincent-Geisse, J. Phys. (Paris) 24, 259 (1963).
[Crossref]

Phys. Rev. (3)

F. C. Jahoda, Phys. Rev. 107, 1261 (1957).
[Crossref]

D. G. Thomas and J. Hopfield, Phys. Rev. 116, 573 (1959).
[Crossref]

W. G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1324 (1961).
[Crossref]

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

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Spectrochim. Acta (2)

K. S. Seshadin and R. N. Jones, Spectrochim. Acta 19, 1013 (1963).
[Crossref]

J. L. Hollenberg and D. A. Dows, Spectrochim. Acta 16, 1155 (1960).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Synthetic k spectrum.

Fig. 2
Fig. 2

Synthetic n spectrum.

Fig. 3
Fig. 3

Synthetic % R spectrum.

Fig. 4
Fig. 4

Correlation of δ(ωi) with nU at fixed nL. ○, 580 cm−1 data; ×, 620 cm−1 data.

Fig. 5
Fig. 5

Left: Distortion in phase angle θM using Eq. (1); curve I (– – –) illustrates θM; curve II (— – — – —) corresponds to ΔθLθU; curve III (——) demonstrates theoretical value of θ. Right: Distortion in phase angle θM using Eq. (2); curve I (– – –) illustrates θM; curve II (— – — – —) corresponds to ΔθLθU; curve III (——) demonstrates theoretical value of θ.

Fig. 6
Fig. 6

Left: Comparison of reflectance between flanked and isolated damped harmonic oscillators; curve I (——) corresponds to the single isolated oscillator R0; curve II (– – –) corresponds to the observed central, flanked oscillator Robs. Right: Use of undamped oscillator for testing influence of central, flanked band; curve I (——) corresponds to the ratio of R0/Ruho, where Ruho is the reflectance of the undamped oscillator; R0 is defined as before; curve II (– – –) corresponds to the ratio of Robs/Ruho.

Tables (6)

Tables Icon

Table I Parameters for the synthetic three-band system.

Tables Icon

Table II Absorption index values from forced match, for band B.

Tables Icon

Table III Refractive index values from forced match, for band B.

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Table IV Cauchy analysis. n*k* and nk′ data.

Tables Icon

Table V Combined Cauchy, Kramers–Kronig analysis results.

Tables Icon

Table VI Kramers–Kronig results based on k=0 requirement.

Equations (23)

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θ ( ω i ) = 2 ω i π 0 ln r ( ω ) ω 2 - ω i 2 d ω .
θ ( ω i ) = 2 ω i π 0 ln r ( ω ) - ln r ( ω i ) ω i 2 - ω 2 d ω .
n ˆ = n + i k
r ˆ = r exp ( i θ ) = ( n ˆ - 1 ) / ( n ˆ + 1 )
R = r ˆ 2 = [ ( n - 1 ) 2 + k 2 ] / [ ( n + 1 ) 2 + k 2 ]
n = ( 1 - r 2 ) / ( 1 - 2 r cos θ + r 2 )
k = ( 2 r sin θ ) / ( 1 - 2 r cos θ + r 2 ) .
θ ( ω i ) = Δ θ L ( ω i ) + θ M ( ω i ) + Δ θ U ( ω i ) ,
θ M ( ω i ) = 2 ω i π ω L ω U ln r ( ω ) - ln r ( ω i ) ω i 2 - ω 2 d ω ,
Δ θ L ( ω i ) = 2 ω i π 0 ω L ln r ( ω ) - ln r ( ω i ) ω i 2 - ω 2 d ω ,
Δ θ U ( ω i ) = 2 ω i π ω U ln r ( ω ) - ln r ( ω i ) ω i 2 - ω 2 d ω .
n 2 ( ω ) = n L 2 + A L / ( ω O , L 2 - ω 2 ) ,
A L = 1 p j = 1 p - 1 [ n 2 ( ω j ) - n L 2 ] [ ω O , L 2 - ω j 2 ] .
R ( ω j ) 1 2 = r ( ω j ) = [ n ( ω j ) - 1 ] / [ n ( ω j ) + 1 ] .
n 2 ( ω ) = n U 2 + A U / ( ω O , U 2 - ω 2 ) ,
Δ θ U ( ω i ) = 2 ω i π ω U ω S ln r ( ω ) - ln r ( ω i ) ω i 2 - ω 2 d ω + ( 1 / π ) [ ln r ( ω S ) - ln r ( ω i ) ] × { ln [ ( ω S - ω i ) / ( ω S + ω i ) ] } .
n 2 - k 2 = n 0 2 + j { B j ( ω j 2 - ω 2 ) / [ ( ω j 2 - ω 2 ) 2 + γ j 2 ω 2 ] } ,
2 n k = j { B j γ j ω / [ ( ω j 2 - ω 2 ) 2 + γ j 2 ω 2 ] } ,
n k = a / [ ( ω - ω 0 ) 2 + b 2 ] ,
R uho = ( n ˜ - 1 ) 2 / ( n ˜ + 1 ) 2 ,
n ˜ 2 = n ˜ 0 2 + B ˜ / ( ω 0 2 - ω 2 ) ,
n ˜ 0 = n * k * d ω / k * d ω .
B ˜ = 4 a ω 0 / b