Abstract

Computed and observed extinction spectra of colloidal silver particles with various size and shape distributions are compared. Excellent agreement is achieved through consideration of (i) the effects of surface and internal contamination and particle size on the optical constants of silver, (ii) the appropriate number of terms as a function of size in the Mie equation for spheres, (iii) a formulation of the Gans equation for prolate spheroids, to include the degree of particle orientation that occurs when a swollen gelatin matrix dries, (iv) the method of combining spectral characteristics determined by the Mie (size) and Gans (shape) relations, and (v) stereometric corrections of the particle measurements for the effects of microtome-section thickness and swelling.

© 1973 Optical Society of America

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References

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  1. D. C. Skillman and C. R. Berry, J. Chem. Phys. 48, 3297 (1968).
    [Crossref]
  2. E. Klein and H. J. Metz, Photogr. Sci. Eng. 5, 5 (1961).
  3. D. C. Skillman, J. Opt. Soc. Am. 61, 1264 (1971).
    [Crossref]
  4. G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).
  5. U. Kreibig and C. v. Fragstein, Z. Phys. 224, 307 (1969).
    [Crossref]
  6. R. H. Doremus, J. Chem. Phys. 42, 414 (1965).
    [Crossref]
  7. W. T. Doyle and A. Agarwal, J. Opt. Soc. Am. 55, 305 (1965).
    [Crossref]
  8. M. Kerker and D. D. Cooke, Appl. Opt. 10, 2670 (1971).
    [Crossref] [PubMed]
  9. D. C. Skillman, J. Microsc. 98, Pt. 1 (May1973).
    [Crossref]
  10. L. G. Schulz, (a)Adv. Phys. 6, 102 (1957); (b)J. Opt. Soc. Am. 44, 357 (1954); J. Opt. Soc. Am. 44, 362 (1954).
  11. S. N. Latysheva, A. N. Latyshev, and L. L. Orekhova, Opt. Spektrosk. 30, 524 (1971) [Opt. Spectrosc. 30, 285 (1971)].
  12. Soon after this manuscript was completed, the question arose of whether or not the crystal-imperfection factor, N, should multiply the frequency-dependent part as well as the frequency-independent part of the free-electron collision frequency, ω0(Appendix II). In retrospect, it seems more logical to multiply both parts of ω0. We recalculated in this new way the curves of Fig. 6 with N= 11 (and the same values of surface contaminant and coating weight) and obtained an improvement of 10% in the difference of area between the calculated and observed curves, demonstrating again the value of the excellent experimental data in resolving theoretical questions. To recalculate the curves of Figs. 7 and 8 would be too expensive, but unproved correlation due to increased half-peak width of the short λ peaks relative to the long λ peaks is certain. Thus, the range of values of the crystal imperfection factor now used to multiply both parts of the free-electron collision frequency will be from about 3.5 to 12 (instead of 8 to 27) for our samples.
  13. U. Kreibig, Z. Phys. 234, 307 (1970).
    [Crossref]
  14. R. Clark Jones and George R. Bird, Photogr. Sci. Eng. 16, 16 (1972).

1973 (1)

D. C. Skillman, J. Microsc. 98, Pt. 1 (May1973).
[Crossref]

1972 (1)

R. Clark Jones and George R. Bird, Photogr. Sci. Eng. 16, 16 (1972).

1971 (4)

M. Kerker and D. D. Cooke, Appl. Opt. 10, 2670 (1971).
[Crossref] [PubMed]

S. N. Latysheva, A. N. Latyshev, and L. L. Orekhova, Opt. Spektrosk. 30, 524 (1971) [Opt. Spectrosc. 30, 285 (1971)].

D. C. Skillman, J. Opt. Soc. Am. 61, 1264 (1971).
[Crossref]

G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).

1970 (1)

U. Kreibig, Z. Phys. 234, 307 (1970).
[Crossref]

1969 (1)

U. Kreibig and C. v. Fragstein, Z. Phys. 224, 307 (1969).
[Crossref]

1968 (1)

D. C. Skillman and C. R. Berry, J. Chem. Phys. 48, 3297 (1968).
[Crossref]

1965 (2)

1961 (1)

E. Klein and H. J. Metz, Photogr. Sci. Eng. 5, 5 (1961).

Agarwal, A.

Bastian, P. E.

G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).

Berry, C. R.

D. C. Skillman and C. R. Berry, J. Chem. Phys. 48, 3297 (1968).
[Crossref]

Bird, G. R.

G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).

Bird, George R.

R. Clark Jones and George R. Bird, Photogr. Sci. Eng. 16, 16 (1972).

Clark Jones, R.

R. Clark Jones and George R. Bird, Photogr. Sci. Eng. 16, 16 (1972).

Cooke, D. D.

Doremus, R. H.

R. H. Doremus, J. Chem. Phys. 42, 414 (1965).
[Crossref]

Doyle, W. T.

Fragstein, C. v.

U. Kreibig and C. v. Fragstein, Z. Phys. 224, 307 (1969).
[Crossref]

Gray, W. E.

G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).

Johnson, J.

G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).

Kerker, M.

Klein, E.

E. Klein and H. J. Metz, Photogr. Sci. Eng. 5, 5 (1961).

Kreibig, U.

U. Kreibig, Z. Phys. 234, 307 (1970).
[Crossref]

U. Kreibig and C. v. Fragstein, Z. Phys. 224, 307 (1969).
[Crossref]

Latyshev, A. N.

S. N. Latysheva, A. N. Latyshev, and L. L. Orekhova, Opt. Spektrosk. 30, 524 (1971) [Opt. Spectrosc. 30, 285 (1971)].

Latysheva, S. N.

S. N. Latysheva, A. N. Latyshev, and L. L. Orekhova, Opt. Spektrosk. 30, 524 (1971) [Opt. Spectrosc. 30, 285 (1971)].

Metz, H. J.

E. Klein and H. J. Metz, Photogr. Sci. Eng. 5, 5 (1961).

Morse, M.

G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).

Orekhova, L. L.

S. N. Latysheva, A. N. Latyshev, and L. L. Orekhova, Opt. Spektrosk. 30, 524 (1971) [Opt. Spectrosc. 30, 285 (1971)].

Rodriguez, H.

G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).

Schulz, L. G.

L. G. Schulz, (a)Adv. Phys. 6, 102 (1957); (b)J. Opt. Soc. Am. 44, 357 (1954); J. Opt. Soc. Am. 44, 362 (1954).

Skillman, D. C.

D. C. Skillman, J. Microsc. 98, Pt. 1 (May1973).
[Crossref]

D. C. Skillman, J. Opt. Soc. Am. 61, 1264 (1971).
[Crossref]

D. C. Skillman and C. R. Berry, J. Chem. Phys. 48, 3297 (1968).
[Crossref]

Appl. Opt. (1)

J. Chem. Phys. (2)

R. H. Doremus, J. Chem. Phys. 42, 414 (1965).
[Crossref]

D. C. Skillman and C. R. Berry, J. Chem. Phys. 48, 3297 (1968).
[Crossref]

J. Microsc. (1)

D. C. Skillman, J. Microsc. 98, Pt. 1 (May1973).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Spektrosk. (1)

S. N. Latysheva, A. N. Latyshev, and L. L. Orekhova, Opt. Spektrosk. 30, 524 (1971) [Opt. Spectrosc. 30, 285 (1971)].

Photogr. Sci. Eng. (3)

R. Clark Jones and George R. Bird, Photogr. Sci. Eng. 16, 16 (1972).

G. R. Bird, M. Morse, H. Rodriguez, P. E. Bastian, J. Johnson, and W. E. Gray, Photogr. Sci. Eng. 15, 356 (1971).

E. Klein and H. J. Metz, Photogr. Sci. Eng. 5, 5 (1961).

Z. Phys. (2)

U. Kreibig and C. v. Fragstein, Z. Phys. 224, 307 (1969).
[Crossref]

U. Kreibig, Z. Phys. 234, 307 (1970).
[Crossref]

Other (2)

Soon after this manuscript was completed, the question arose of whether or not the crystal-imperfection factor, N, should multiply the frequency-dependent part as well as the frequency-independent part of the free-electron collision frequency, ω0(Appendix II). In retrospect, it seems more logical to multiply both parts of ω0. We recalculated in this new way the curves of Fig. 6 with N= 11 (and the same values of surface contaminant and coating weight) and obtained an improvement of 10% in the difference of area between the calculated and observed curves, demonstrating again the value of the excellent experimental data in resolving theoretical questions. To recalculate the curves of Figs. 7 and 8 would be too expensive, but unproved correlation due to increased half-peak width of the short λ peaks relative to the long λ peaks is certain. Thus, the range of values of the crystal imperfection factor now used to multiply both parts of the free-electron collision frequency will be from about 3.5 to 12 (instead of 8 to 27) for our samples.

L. G. Schulz, (a)Adv. Phys. 6, 102 (1957); (b)J. Opt. Soc. Am. 44, 357 (1954); J. Opt. Soc. Am. 44, 362 (1954).

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Figures (8)

Fig. 1
Fig. 1

Calculated extinction coefficients for spheres of indicated diameter using first electric partial wave. Dotted curves are second electric partial wave. Optical constants have X = 0.030 and N = 30.

Fig. 2
Fig. 2

Extinction-maximum position, relative extinction, and relative half-peak width vs sphere diameter.

Fig. 3
Fig. 3

Calculated extinction coefficients for prolate spheroids of indicated axial ratio using the Gans equation with a1 = 0.49 a1 + 0.51a1. Dotted curve is the envelope of long-λ maxima. Optical constants have X = 0.030, N = 30, and D = ∞ (bulk values).

Fig. 4
Fig. 4

Extinction-maximum position, maximum extinction, and half-peak width vs axial ratio for prolate spheroids contracted 94% into a plane. Curves beginning in the lower right are extensions.

Fig. 5
Fig. 5

Transmittance in collimated light, τ, measured by Klein and Metz2 for spheres of indicated diameter in gelatin. Coating weight 0.08 g Ag/m2.

Fig. 6
Fig. 6

Calculated extinction for spheres of indicated diameter, using five terms in the Mie equation and optical constants for X = 0.034 and N = 24. Coating weight 0.11 g Ag/m2.

Fig. 7
Fig. 7

Calculated (solid) and observed (dotted) extinction spectra.

Fig. 8
Fig. 8

Calculated (solid) and observed (dotted) extinction spectra.

Equations (9)

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long λ = 411.3 + Δ λ ( shape ) + Δ λ ( sphere of size a ) , short λ = 411.3 Δ λ ( shape ) + Δ λ ( sphere of size b ) ,
K c = ( K · D s ) ( HPW ) c 2 ( 2 Δ λ ) 2 + ( HPW ) c 2 ,
CP = D / A = 0.4343 K / ρ = 0.04136 K ( cm 2 / g ) ,
K ext = 6 π m 0 λ ( Im [ U 1 m 2 V 1 m 2 + 2 W 1 + U 1 1 V 1 1 + 2 W 1 + α 2 12 U 2 m 2 V 2 m 2 + 3 2 W 2 + α 2 12 U 2 1 V 2 1 + 3 2 W 2 + 2 α 4 675 U 3 m 2 V 3 m 2 + 4 3 W 3 ) ] , K sca = 4 π m 0 λ α 3 [ | U 1 m 2 V 1 m 3 + 2 W 1 | 2 + | U 1 1 V 1 1 + 2 W 1 | 2 + 3 5 | α 2 12 U 2 m 2 V 2 m 2 + 3 2 W 2 | 2 + 3 5 | α 2 12 U 2 1 V 2 1 + 3 2 W 2 | 2 + 3 7 | 2 α 4 675 U 3 m 2 V 3 m 2 + 4 3 W 3 | 2 ] ,
U 1 = e i α 1 + i α α 2 ( 1 α 2 5 + 3 α 4 280 ) , V 1 = 1 α 2 / 10 + α 4 / 280 1 α 2 / 5 + 3 α 4 / 280 · 1 α 2 m 2 / 5 + 3 α 4 m 4 / 280 1 α 2 m 2 / 10 + α 4 m 4 / 280 , W 1 = 1 + i α 1 + i α α 2 · 1 α 2 m 2 / 5 + 3 α 4 m 4 / 280 1 α 2 m 2 / 10 + α 4 m 4 / 280 , U 2 = e i α 1 + i α α 2 / 2 i α 3 / 6 ( 1 5 α 2 42 + α 4 216 α 6 11088 ) , V 2 = 1 α 2 / 14 + α 4 / 504 1 5 α 2 / 42 + α 4 / 216 · 1 5 α 2 m 2 / 42 + α 4 m 4 / 216 1 α 2 m 2 / 14 + α 4 m 4 / 504 , W 2 = 1 + i α α 2 / 3 1 + i α ( 1 α 2 / 6 ) α 2 / 2 · 1 5 α 2 m 2 / 42 + α 4 m 4 / 216 1 α 2 m 2 / 14 + α 4 m 4 / 504 , U 3 = e i α 1 + i α ( 7 / 15 ) α 2 ( 2 / 15 ) i α 3 + α 4 / 45 ( 1 α 2 12 + α 4 396 ) , V 3 = 1 α 2 / 18 + α 4 / 792 1 α 2 / 12 + α 4 / 396 · 1 α 2 m 2 / 12 + α 4 m 4 / 396 1 α 2 m 2 / 18 + α 4 m 4 / 792 , W 3 = 1 2 5 α 2 + i α ( 1 α 2 / 15 ) 1 ( 7 / 15 ) α 2 + i α ( 2 / 15 ) i α 3 + α 4 / 45 · 1 α 2 m 2 / 12 + α 4 m 4 / 396 1 α 2 m 2 / 18 + α 4 m 4 / 792 .
1 = 1 186.258 ω 2 + ω 0 2 + 9.48 ( 40.25 ω 2 ) ( 40.25 ω 2 ) 2 + ( 0.408 ω ) 2 + ( 0.1047 ω 4 ) ( 51.69 ω 2 ) ( 51.69 ω 2 ) 2 + ( 2.65 ω ) 2 , 2 = 186.258 ω 2 + ω 0 2 · ω 0 ω + 9.48 ( 0.408 ω ) ( 40.26 ω 2 ) 2 + ( 0.408 ω ) 2 ,
a 1 = ( m 2 1 ) / [ 3 + ( 3 P / 4 π ) ( m 2 1 ) ]
a 1 = ( m 2 1 ) / [ 3 + ( 3 P / 4 π ) ( m 2 1 ) ] ,
P = 4 π [ ( 1 e 2 ) / e 2 ] { ( 1 / 2 e ) ln [ ( 1 + e ) / ( 1 e ) ] 1 } ,