Abstract

Earlier results for coherent propagation of light in correlated random distributions of dielectric particles of radius a (with minimum separation b ≥ 2a small compared with wavelength λ = 2π/k) are generalized to obtain the refractive and absorptive terms to order (ka)2. The present results include the earlier multiple scattering by electric dipoles as well as scattering and multipole coupling by magnetic dipoles and electric quadrupoles. The correlation aspects are determined by the statistical-mechanics radial distribution function f(R) for impenetrable particles of diameter b. The new terms for slab scatterers and spheres involve the integral of fR (first moment) or of f ln R for cylinders. The new packing factor is evaluated exactly for slabs as a simple algebraic function of the volume fraction w, and it is shown that the bulk index of refraction reduces to that of one particle in the limit w = 1. A similar result is achieved for spheres in terms of the Percus–Yevick approximation and the unrealizable limit w = 1.

© 1983 Optical Society of America

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  1. V. Twersky, "Transparency of pair-correlated, random distributions of small scatterers with applications to the cornea," J. Opt. Soc. Am. 65, 524–530 (1975); "Propagation in pair-correlated distributions of small-spaced lossy scatterers," J. Opt. Soc. Am. 69, 1567–1572 (1979). The text cites the equations of the second paper.
    [CrossRef] [PubMed]
  2. V. Twersky, "Birefringence and dichroism," J. Opt. Soc. Am. 71, 1243–1249 (1981).
    [CrossRef]
  3. V. Twersky, "Coherent scalar field in pair-correlated random distributions of aligned scatterers," J. Math. Phys. 18, 2468–2486 (1977).
    [CrossRef]
  4. V. Twersky, "Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers," J. Math. Phys. 19, 215–230 (1978).
    [CrossRef]
  5. H. L. Frisch and J. L. Lebowitz, The Equilibrium Theory of Fluids (Benjamin, New York, 1964); R. L. Baxter, "Distribution functions," in Physical Chemistry, H. Eyring, D. Henderson, and W. Jost, eds. (Academic, New York, 1971), Vol. VIII A, Chap. 4, pp. 267–334.
  6. H. Reiss, H. L. Frisch, and J. L. Lebowitz, "Statistical mechanics of rigid spheres," J. Chem. Phys. 31, 369–380 (1959); E. Helfand, H. L. Frisch, and J. L. Lebowitz, "The theory of the two- and one-dimensional rigid sphere fluids," J. Chem. Phys. 34, 1037–1042 (1961).
    [CrossRef]
  7. M. S. Wertheim, "Exact solution of the Percus—Yevick integral equation for hard spheres," Phys. Rev. Lett. 10,321–323 (1963); E. Thiele, "Equation of state for hard spheres," J. Chem. Phys. 39, 474–479 (1963).
    [CrossRef]
  8. H. D. Jones, "Method for finding the equation of state of liquid metals," J. Chem. Phys. 55, 2640–2642 (1971).
    [CrossRef]
  9. G. J. Throop and R. J. Bearman, "Numerical solution of the Percus—Yevick equation for the hard-sphere potential," J. Chem. Phys. 42, 2408–2411 (1965); F. Mandell, R. J. Bearman, and M. Y. Bearman, "Numerical solution of the Percus—Yevick equation for the Lennard—Jones (6–12) and hard sphere potentials," J. Chem. Phys. 52, 3315–3323 (1970); D. Levesque, J. J. Weis, and J. P. Hansen, "Simulation of classial fluids," in Monte Carlo Methods in Statistical Physics, K. Binder, ed. (Springer-Verlag, New York, 1979), pp. 121–144.
    [CrossRef]
  10. Y. Uehara, T. Ree, and F. H. Ree, "Radial distribution function for hard disks from the B GY2 theory," J. Chem. Phys. 70, 1876–1883 (1979); J. Woodhead-Galloway and P. A. Machin, "X-ray scattering from a gas of uniform hard-disks using the Percus—Yevick approximation," Mol. Phys. 32, 41–48 (1976); F. Lado, "Equation of state for the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092–3096 (1968).
    [CrossRef]
  11. G. Placzek, B. R. A. Nijboer, and L. Van Hove, "Effects of short wavelength interference on neutron scattering by dense systems of heavy nucli," Phys. Rev. 82, 392–403 (1951); B. R. A. Nijboer and L. Van Hove, "Radial distribution function of a gas of hard spheres and the superposition approximation," Phys. Rev. 85, 777–783 (1952).
    [CrossRef]
  12. B. Larsen, J. C. Rasaiah, and G. Stell, "Thermodynamic perturbation theory for multipolar and ionic liquids," Mol. Phys. 33, 987–1027 (1977); G. Stell and K. C. Wu, "Pade approximant for the internal energy of a system of charged particles," J. Chem. Phys. 63, 491–498 (1975).
    [CrossRef]
  13. F. Zernike and J. A. Prins, "Die Beuging von Rontgenstrahlen in Flussigkeiten als Effekt der Molekulanordnung," Z. Phys. 41, 184–194 (1927).
    [CrossRef]
  14. V. Twersky, "Scattering by quasi-periodic and quasi-random distributions," IRE Trans. AP-7, S307–S319 (1959); "Multiple scattering of waves by planar random distributions of cylinders and bosses," Rep. No. EM58 (New York U. Press, New York, 1953).
  15. V. Twersky, "Multiple scattering of sound by correlated monolayers," J. Acoust. Soc. Am. 73, 68–84 (1983).
    [CrossRef]
  16. L. Tonks, "The complete equation of state of one, two and three-dimensional gases of hard elastic spheres," Phys. Rev. 50, 955–963 (1936).
    [CrossRef]
  17. V. Twersky, "Acoustic bulk parameters in distributions of paircorrelated scatterers," J. Acoust. Soc. Am. 64, 1710–1719 (1978); S. W. Hawley, T. H. Kays, and V. Twersky, "Comparison of distribution functions from scattering data on different sets of spheres," IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
    [CrossRef]
  18. V. Twersky, "Propagation in correlated distributions of largespaced scatterers," J. Opt. Soc. Am. 73, 313–320 (1983).
    [CrossRef]
  19. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  20. Lord Rayleigh, "On the transmission of light through the atmosphere containing small particles in suspension, and on the origin of the color of the sky," Philos. Mag. 47, 375–383 (1899).
  21. J. C. Maxwell, A Treatise on Electricity and Magnetism (Cambridge, 1873; Dover, New York, 1954); spheres are considered in Sec. 314 and slabs in Sec. 321.
  22. Lord Rayleigh, "On the influence of obstacles arranged in rectangular order upon the properties of a medium," Philos. Mag. 34,481–501 (1892).
  23. F. Reiche, "Zur Theorie der Dispersion in Gasen and Dampfen," Ann. Phys. 50, 1–121 (1916).
    [CrossRef]
  24. L. L. Foldy, "The multiple scattering of waves," Phys. Rev. 67, 107–119 (1945).
    [CrossRef]
  25. M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287–310 (1951); "The effective field in dense systems," Phys. Rev. 88, 621–629 (1952).
    [CrossRef]

1983 (2)

V. Twersky, "Multiple scattering of sound by correlated monolayers," J. Acoust. Soc. Am. 73, 68–84 (1983).
[CrossRef]

V. Twersky, "Propagation in correlated distributions of largespaced scatterers," J. Opt. Soc. Am. 73, 313–320 (1983).
[CrossRef]

1981 (1)

1979 (1)

Y. Uehara, T. Ree, and F. H. Ree, "Radial distribution function for hard disks from the B GY2 theory," J. Chem. Phys. 70, 1876–1883 (1979); J. Woodhead-Galloway and P. A. Machin, "X-ray scattering from a gas of uniform hard-disks using the Percus—Yevick approximation," Mol. Phys. 32, 41–48 (1976); F. Lado, "Equation of state for the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092–3096 (1968).
[CrossRef]

1978 (2)

V. Twersky, "Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers," J. Math. Phys. 19, 215–230 (1978).
[CrossRef]

V. Twersky, "Acoustic bulk parameters in distributions of paircorrelated scatterers," J. Acoust. Soc. Am. 64, 1710–1719 (1978); S. W. Hawley, T. H. Kays, and V. Twersky, "Comparison of distribution functions from scattering data on different sets of spheres," IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
[CrossRef]

1977 (2)

B. Larsen, J. C. Rasaiah, and G. Stell, "Thermodynamic perturbation theory for multipolar and ionic liquids," Mol. Phys. 33, 987–1027 (1977); G. Stell and K. C. Wu, "Pade approximant for the internal energy of a system of charged particles," J. Chem. Phys. 63, 491–498 (1975).
[CrossRef]

V. Twersky, "Coherent scalar field in pair-correlated random distributions of aligned scatterers," J. Math. Phys. 18, 2468–2486 (1977).
[CrossRef]

1971 (1)

H. D. Jones, "Method for finding the equation of state of liquid metals," J. Chem. Phys. 55, 2640–2642 (1971).
[CrossRef]

1963 (1)

M. S. Wertheim, "Exact solution of the Percus—Yevick integral equation for hard spheres," Phys. Rev. Lett. 10,321–323 (1963); E. Thiele, "Equation of state for hard spheres," J. Chem. Phys. 39, 474–479 (1963).
[CrossRef]

1959 (1)

H. Reiss, H. L. Frisch, and J. L. Lebowitz, "Statistical mechanics of rigid spheres," J. Chem. Phys. 31, 369–380 (1959); E. Helfand, H. L. Frisch, and J. L. Lebowitz, "The theory of the two- and one-dimensional rigid sphere fluids," J. Chem. Phys. 34, 1037–1042 (1961).
[CrossRef]

1951 (1)

G. Placzek, B. R. A. Nijboer, and L. Van Hove, "Effects of short wavelength interference on neutron scattering by dense systems of heavy nucli," Phys. Rev. 82, 392–403 (1951); B. R. A. Nijboer and L. Van Hove, "Radial distribution function of a gas of hard spheres and the superposition approximation," Phys. Rev. 85, 777–783 (1952).
[CrossRef]

1945 (1)

L. L. Foldy, "The multiple scattering of waves," Phys. Rev. 67, 107–119 (1945).
[CrossRef]

1936 (1)

L. Tonks, "The complete equation of state of one, two and three-dimensional gases of hard elastic spheres," Phys. Rev. 50, 955–963 (1936).
[CrossRef]

1927 (1)

F. Zernike and J. A. Prins, "Die Beuging von Rontgenstrahlen in Flussigkeiten als Effekt der Molekulanordnung," Z. Phys. 41, 184–194 (1927).
[CrossRef]

1916 (1)

F. Reiche, "Zur Theorie der Dispersion in Gasen and Dampfen," Ann. Phys. 50, 1–121 (1916).
[CrossRef]

1899 (1)

Lord Rayleigh, "On the transmission of light through the atmosphere containing small particles in suspension, and on the origin of the color of the sky," Philos. Mag. 47, 375–383 (1899).

1892 (1)

Lord Rayleigh, "On the influence of obstacles arranged in rectangular order upon the properties of a medium," Philos. Mag. 34,481–501 (1892).

Bearman, R. J.

G. J. Throop and R. J. Bearman, "Numerical solution of the Percus—Yevick equation for the hard-sphere potential," J. Chem. Phys. 42, 2408–2411 (1965); F. Mandell, R. J. Bearman, and M. Y. Bearman, "Numerical solution of the Percus—Yevick equation for the Lennard—Jones (6–12) and hard sphere potentials," J. Chem. Phys. 52, 3315–3323 (1970); D. Levesque, J. J. Weis, and J. P. Hansen, "Simulation of classial fluids," in Monte Carlo Methods in Statistical Physics, K. Binder, ed. (Springer-Verlag, New York, 1979), pp. 121–144.
[CrossRef]

Foldy, L. L.

L. L. Foldy, "The multiple scattering of waves," Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Frisch, H. L.

H. Reiss, H. L. Frisch, and J. L. Lebowitz, "Statistical mechanics of rigid spheres," J. Chem. Phys. 31, 369–380 (1959); E. Helfand, H. L. Frisch, and J. L. Lebowitz, "The theory of the two- and one-dimensional rigid sphere fluids," J. Chem. Phys. 34, 1037–1042 (1961).
[CrossRef]

H. L. Frisch and J. L. Lebowitz, The Equilibrium Theory of Fluids (Benjamin, New York, 1964); R. L. Baxter, "Distribution functions," in Physical Chemistry, H. Eyring, D. Henderson, and W. Jost, eds. (Academic, New York, 1971), Vol. VIII A, Chap. 4, pp. 267–334.

Jones, H. D.

H. D. Jones, "Method for finding the equation of state of liquid metals," J. Chem. Phys. 55, 2640–2642 (1971).
[CrossRef]

Larsen, B.

B. Larsen, J. C. Rasaiah, and G. Stell, "Thermodynamic perturbation theory for multipolar and ionic liquids," Mol. Phys. 33, 987–1027 (1977); G. Stell and K. C. Wu, "Pade approximant for the internal energy of a system of charged particles," J. Chem. Phys. 63, 491–498 (1975).
[CrossRef]

Lax, M.

M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287–310 (1951); "The effective field in dense systems," Phys. Rev. 88, 621–629 (1952).
[CrossRef]

Lebowitz, J. L.

H. Reiss, H. L. Frisch, and J. L. Lebowitz, "Statistical mechanics of rigid spheres," J. Chem. Phys. 31, 369–380 (1959); E. Helfand, H. L. Frisch, and J. L. Lebowitz, "The theory of the two- and one-dimensional rigid sphere fluids," J. Chem. Phys. 34, 1037–1042 (1961).
[CrossRef]

H. L. Frisch and J. L. Lebowitz, The Equilibrium Theory of Fluids (Benjamin, New York, 1964); R. L. Baxter, "Distribution functions," in Physical Chemistry, H. Eyring, D. Henderson, and W. Jost, eds. (Academic, New York, 1971), Vol. VIII A, Chap. 4, pp. 267–334.

Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Cambridge, 1873; Dover, New York, 1954); spheres are considered in Sec. 314 and slabs in Sec. 321.

Nijboer, B. R. A.

G. Placzek, B. R. A. Nijboer, and L. Van Hove, "Effects of short wavelength interference on neutron scattering by dense systems of heavy nucli," Phys. Rev. 82, 392–403 (1951); B. R. A. Nijboer and L. Van Hove, "Radial distribution function of a gas of hard spheres and the superposition approximation," Phys. Rev. 85, 777–783 (1952).
[CrossRef]

Placzek, G.

G. Placzek, B. R. A. Nijboer, and L. Van Hove, "Effects of short wavelength interference on neutron scattering by dense systems of heavy nucli," Phys. Rev. 82, 392–403 (1951); B. R. A. Nijboer and L. Van Hove, "Radial distribution function of a gas of hard spheres and the superposition approximation," Phys. Rev. 85, 777–783 (1952).
[CrossRef]

Prins, J. A.

F. Zernike and J. A. Prins, "Die Beuging von Rontgenstrahlen in Flussigkeiten als Effekt der Molekulanordnung," Z. Phys. 41, 184–194 (1927).
[CrossRef]

Rasaiah, J. C.

B. Larsen, J. C. Rasaiah, and G. Stell, "Thermodynamic perturbation theory for multipolar and ionic liquids," Mol. Phys. 33, 987–1027 (1977); G. Stell and K. C. Wu, "Pade approximant for the internal energy of a system of charged particles," J. Chem. Phys. 63, 491–498 (1975).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, "On the transmission of light through the atmosphere containing small particles in suspension, and on the origin of the color of the sky," Philos. Mag. 47, 375–383 (1899).

Lord Rayleigh, "On the influence of obstacles arranged in rectangular order upon the properties of a medium," Philos. Mag. 34,481–501 (1892).

Ree, F. H.

Y. Uehara, T. Ree, and F. H. Ree, "Radial distribution function for hard disks from the B GY2 theory," J. Chem. Phys. 70, 1876–1883 (1979); J. Woodhead-Galloway and P. A. Machin, "X-ray scattering from a gas of uniform hard-disks using the Percus—Yevick approximation," Mol. Phys. 32, 41–48 (1976); F. Lado, "Equation of state for the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092–3096 (1968).
[CrossRef]

Ree, T.

Y. Uehara, T. Ree, and F. H. Ree, "Radial distribution function for hard disks from the B GY2 theory," J. Chem. Phys. 70, 1876–1883 (1979); J. Woodhead-Galloway and P. A. Machin, "X-ray scattering from a gas of uniform hard-disks using the Percus—Yevick approximation," Mol. Phys. 32, 41–48 (1976); F. Lado, "Equation of state for the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092–3096 (1968).
[CrossRef]

Reiche, F.

F. Reiche, "Zur Theorie der Dispersion in Gasen and Dampfen," Ann. Phys. 50, 1–121 (1916).
[CrossRef]

Reiss, H.

H. Reiss, H. L. Frisch, and J. L. Lebowitz, "Statistical mechanics of rigid spheres," J. Chem. Phys. 31, 369–380 (1959); E. Helfand, H. L. Frisch, and J. L. Lebowitz, "The theory of the two- and one-dimensional rigid sphere fluids," J. Chem. Phys. 34, 1037–1042 (1961).
[CrossRef]

Stell, G.

B. Larsen, J. C. Rasaiah, and G. Stell, "Thermodynamic perturbation theory for multipolar and ionic liquids," Mol. Phys. 33, 987–1027 (1977); G. Stell and K. C. Wu, "Pade approximant for the internal energy of a system of charged particles," J. Chem. Phys. 63, 491–498 (1975).
[CrossRef]

Throop, G. J.

G. J. Throop and R. J. Bearman, "Numerical solution of the Percus—Yevick equation for the hard-sphere potential," J. Chem. Phys. 42, 2408–2411 (1965); F. Mandell, R. J. Bearman, and M. Y. Bearman, "Numerical solution of the Percus—Yevick equation for the Lennard—Jones (6–12) and hard sphere potentials," J. Chem. Phys. 52, 3315–3323 (1970); D. Levesque, J. J. Weis, and J. P. Hansen, "Simulation of classial fluids," in Monte Carlo Methods in Statistical Physics, K. Binder, ed. (Springer-Verlag, New York, 1979), pp. 121–144.
[CrossRef]

Tonks, L.

L. Tonks, "The complete equation of state of one, two and three-dimensional gases of hard elastic spheres," Phys. Rev. 50, 955–963 (1936).
[CrossRef]

Twersky, V.

V. Twersky, "Propagation in correlated distributions of largespaced scatterers," J. Opt. Soc. Am. 73, 313–320 (1983).
[CrossRef]

V. Twersky, "Multiple scattering of sound by correlated monolayers," J. Acoust. Soc. Am. 73, 68–84 (1983).
[CrossRef]

V. Twersky, "Birefringence and dichroism," J. Opt. Soc. Am. 71, 1243–1249 (1981).
[CrossRef]

V. Twersky, "Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers," J. Math. Phys. 19, 215–230 (1978).
[CrossRef]

V. Twersky, "Acoustic bulk parameters in distributions of paircorrelated scatterers," J. Acoust. Soc. Am. 64, 1710–1719 (1978); S. W. Hawley, T. H. Kays, and V. Twersky, "Comparison of distribution functions from scattering data on different sets of spheres," IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
[CrossRef]

V. Twersky, "Coherent scalar field in pair-correlated random distributions of aligned scatterers," J. Math. Phys. 18, 2468–2486 (1977).
[CrossRef]

V. Twersky, "Transparency of pair-correlated, random distributions of small scatterers with applications to the cornea," J. Opt. Soc. Am. 65, 524–530 (1975); "Propagation in pair-correlated distributions of small-spaced lossy scatterers," J. Opt. Soc. Am. 69, 1567–1572 (1979). The text cites the equations of the second paper.
[CrossRef] [PubMed]

V. Twersky, "Scattering by quasi-periodic and quasi-random distributions," IRE Trans. AP-7, S307–S319 (1959); "Multiple scattering of waves by planar random distributions of cylinders and bosses," Rep. No. EM58 (New York U. Press, New York, 1953).

Uehara, Y.

Y. Uehara, T. Ree, and F. H. Ree, "Radial distribution function for hard disks from the B GY2 theory," J. Chem. Phys. 70, 1876–1883 (1979); J. Woodhead-Galloway and P. A. Machin, "X-ray scattering from a gas of uniform hard-disks using the Percus—Yevick approximation," Mol. Phys. 32, 41–48 (1976); F. Lado, "Equation of state for the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092–3096 (1968).
[CrossRef]

Van Hove, L.

G. Placzek, B. R. A. Nijboer, and L. Van Hove, "Effects of short wavelength interference on neutron scattering by dense systems of heavy nucli," Phys. Rev. 82, 392–403 (1951); B. R. A. Nijboer and L. Van Hove, "Radial distribution function of a gas of hard spheres and the superposition approximation," Phys. Rev. 85, 777–783 (1952).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wertheim, M. S.

M. S. Wertheim, "Exact solution of the Percus—Yevick integral equation for hard spheres," Phys. Rev. Lett. 10,321–323 (1963); E. Thiele, "Equation of state for hard spheres," J. Chem. Phys. 39, 474–479 (1963).
[CrossRef]

Zernike, F.

F. Zernike and J. A. Prins, "Die Beuging von Rontgenstrahlen in Flussigkeiten als Effekt der Molekulanordnung," Z. Phys. 41, 184–194 (1927).
[CrossRef]

Ann. Phys. (1)

F. Reiche, "Zur Theorie der Dispersion in Gasen and Dampfen," Ann. Phys. 50, 1–121 (1916).
[CrossRef]

J. Acoust. Soc. Am. (2)

V. Twersky, "Multiple scattering of sound by correlated monolayers," J. Acoust. Soc. Am. 73, 68–84 (1983).
[CrossRef]

V. Twersky, "Acoustic bulk parameters in distributions of paircorrelated scatterers," J. Acoust. Soc. Am. 64, 1710–1719 (1978); S. W. Hawley, T. H. Kays, and V. Twersky, "Comparison of distribution functions from scattering data on different sets of spheres," IEEE Trans. Antennas Propag. AP-15, 118–135 (1967).
[CrossRef]

J. Chem. Phys. (3)

Y. Uehara, T. Ree, and F. H. Ree, "Radial distribution function for hard disks from the B GY2 theory," J. Chem. Phys. 70, 1876–1883 (1979); J. Woodhead-Galloway and P. A. Machin, "X-ray scattering from a gas of uniform hard-disks using the Percus—Yevick approximation," Mol. Phys. 32, 41–48 (1976); F. Lado, "Equation of state for the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092–3096 (1968).
[CrossRef]

H. Reiss, H. L. Frisch, and J. L. Lebowitz, "Statistical mechanics of rigid spheres," J. Chem. Phys. 31, 369–380 (1959); E. Helfand, H. L. Frisch, and J. L. Lebowitz, "The theory of the two- and one-dimensional rigid sphere fluids," J. Chem. Phys. 34, 1037–1042 (1961).
[CrossRef]

H. D. Jones, "Method for finding the equation of state of liquid metals," J. Chem. Phys. 55, 2640–2642 (1971).
[CrossRef]

J. Math. Phys. (2)

V. Twersky, "Coherent scalar field in pair-correlated random distributions of aligned scatterers," J. Math. Phys. 18, 2468–2486 (1977).
[CrossRef]

V. Twersky, "Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers," J. Math. Phys. 19, 215–230 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

Mol. Phys. (1)

B. Larsen, J. C. Rasaiah, and G. Stell, "Thermodynamic perturbation theory for multipolar and ionic liquids," Mol. Phys. 33, 987–1027 (1977); G. Stell and K. C. Wu, "Pade approximant for the internal energy of a system of charged particles," J. Chem. Phys. 63, 491–498 (1975).
[CrossRef]

Philos. Mag. (2)

Lord Rayleigh, "On the transmission of light through the atmosphere containing small particles in suspension, and on the origin of the color of the sky," Philos. Mag. 47, 375–383 (1899).

Lord Rayleigh, "On the influence of obstacles arranged in rectangular order upon the properties of a medium," Philos. Mag. 34,481–501 (1892).

Phys. Rev. (3)

L. L. Foldy, "The multiple scattering of waves," Phys. Rev. 67, 107–119 (1945).
[CrossRef]

G. Placzek, B. R. A. Nijboer, and L. Van Hove, "Effects of short wavelength interference on neutron scattering by dense systems of heavy nucli," Phys. Rev. 82, 392–403 (1951); B. R. A. Nijboer and L. Van Hove, "Radial distribution function of a gas of hard spheres and the superposition approximation," Phys. Rev. 85, 777–783 (1952).
[CrossRef]

L. Tonks, "The complete equation of state of one, two and three-dimensional gases of hard elastic spheres," Phys. Rev. 50, 955–963 (1936).
[CrossRef]

Phys. Rev. Lett. (1)

M. S. Wertheim, "Exact solution of the Percus—Yevick integral equation for hard spheres," Phys. Rev. Lett. 10,321–323 (1963); E. Thiele, "Equation of state for hard spheres," J. Chem. Phys. 39, 474–479 (1963).
[CrossRef]

Z. Phys. (1)

F. Zernike and J. A. Prins, "Die Beuging von Rontgenstrahlen in Flussigkeiten als Effekt der Molekulanordnung," Z. Phys. 41, 184–194 (1927).
[CrossRef]

Other (7)

V. Twersky, "Scattering by quasi-periodic and quasi-random distributions," IRE Trans. AP-7, S307–S319 (1959); "Multiple scattering of waves by planar random distributions of cylinders and bosses," Rep. No. EM58 (New York U. Press, New York, 1953).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

G. J. Throop and R. J. Bearman, "Numerical solution of the Percus—Yevick equation for the hard-sphere potential," J. Chem. Phys. 42, 2408–2411 (1965); F. Mandell, R. J. Bearman, and M. Y. Bearman, "Numerical solution of the Percus—Yevick equation for the Lennard—Jones (6–12) and hard sphere potentials," J. Chem. Phys. 52, 3315–3323 (1970); D. Levesque, J. J. Weis, and J. P. Hansen, "Simulation of classial fluids," in Monte Carlo Methods in Statistical Physics, K. Binder, ed. (Springer-Verlag, New York, 1979), pp. 121–144.
[CrossRef]

V. Twersky, "Transparency of pair-correlated, random distributions of small scatterers with applications to the cornea," J. Opt. Soc. Am. 65, 524–530 (1975); "Propagation in pair-correlated distributions of small-spaced lossy scatterers," J. Opt. Soc. Am. 69, 1567–1572 (1979). The text cites the equations of the second paper.
[CrossRef] [PubMed]

H. L. Frisch and J. L. Lebowitz, The Equilibrium Theory of Fluids (Benjamin, New York, 1964); R. L. Baxter, "Distribution functions," in Physical Chemistry, H. Eyring, D. Henderson, and W. Jost, eds. (Academic, New York, 1971), Vol. VIII A, Chap. 4, pp. 267–334.

M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287–310 (1951); "The effective field in dense systems," Phys. Rev. 88, 621–629 (1952).
[CrossRef]

J. C. Maxwell, A Treatise on Electricity and Magnetism (Cambridge, 1873; Dover, New York, 1954); spheres are considered in Sec. 314 and slabs in Sec. 321.

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Equations (68)

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ϕ = e ^ e i k z ,             e ^ · z ^ = 0 ,             k = 2 π / λ = 2 π η e / λ ,
K = k η b / η e = k η ,             η 2 =
- Re e ^ · g ( z ^ , z ^ ) = - Re g = M g ( r ^ , z ^ ) 2 ,
g = a n ,             a n = a n ( , x ) ,             = p / e ,             x k a .
= 1 + c + i s = E 1 + x 2 E c + i x m E s ,
η R - 1 = - c g / 2 ,             R = η R 2 ;             c = i 4 π ρ k 3 ,             i 4 ρ k 2 ,             i 2 ρ k .
η 2 - 1 = - c G ,             η 2 = ,
a 1 i x 3 ( - 1 ) + 2 [ 1 - x 2 3 ( 2 - ) 5 ( + 2 ) ] - x 6 2 ( - 1 ) 2 3 ( + 2 ) 2 ,
a 1 M i x 5 ( - 1 ) 30 ,             a 2 i x 5 ( - 1 ) 6 ( 2 + 3 ) ,
a 1 i π x 2 ( - 1 ) 2 ( + 1 ) { 1 - x 2 [ 3 + - 4 ( - 1 ) L ] 8 ( + 1 ) } - x 4 π 2 ( - 1 ) 2 8 ( + 1 ) 2 ,
a 0 i π x 4 ( - 1 ) 32 ,             a 2 i π x 4 ( - 1 ) 16 ( + 1 ) ,
a 0 i π 2 ( - 1 ) 4 { 1 - x 2 8 [ 3 - - ( - 1 ) 4 L ] } - x 4 π 2 ( - 1 ) 2 16 ,
a 1 i π x 4 ( - 1 ) / 16
a 0 i x ( - 1 ) [ 1 - x 2 ( 2 - 1 ) 3 ] - x 2 ( - 1 ) 2 ,
a 1 i x 3 ( - 1 ) / 3 ,
G = A 0 + A 1 ,             c = i 2 ρ / k = i w / x , w = ρ 2 a = ρ v ( a ) ,
A 0 = a 0 ( 1 + A 0 H 0 + A 1 H 1 / η ) , A 1 = η 2 a 1 ( 1 + A 0 H 1 / η + A 1 H 11 / η 2 ) ,
H 0 2 ρ 0 F d R + i k 2 ρ 0 F R d R W - 1 + i x N , H 1 - i x η N ,
= [ 1 - c a 0 ( 1 + a 0 H 0 ) ] ( 1 - c a 1 )
= 1 + w δ - x 2 w δ 2 [ 2 - w + 3 N ] / 3 + i x δ 2 w W ,             δ = - 1 ,
F 0 = 0 F d R = b ( - 2 + W ) / 2 - b F ¯ 0 ,             W = ρ b = ρ v ( b / 2 ) = w b / 2 a , F 1 = 0 F R d R = - b 2 ( 1 - 4 W / 3 + W / 2 ) / 2 - b 2 F ¯ 1 ,
W = ( 1 - W ) 2 ,             N = - ( b / a ) W ( 1 - 4 W / 3 + W 2 / 2 ) ,
[ ] = 2 - 7 w + 8 w 2 - 3 w 3 ,             b = 2 a .
c = i 4 ρ / k 2 = i 4 w / π x 2 ,             w = ρ π a 2 = ρ v ( a ) ,
H 0 2 π ρ F R d R + i 4 ρ F ln ( c k R / 2 ) R d R W - 1 + i N , H n - i ( η n 2 π ρ / n π ) F r d R = - i η n ( W - 1 ) / n π ,
= 1 + w δ + x 2 w δ 2 ( 1 + 2 w + 4 M ) / 8 + i π x 2 δ 2 w W / 4 ,             δ = - 1.
W = 1 + 2 π ρ F R d R = 1 + 2 π ρ F 1 = 1 - 8 W F ¯ 1 , W = π ρ ( b / 2 ) 2 + ρ v ( b / 2 ) = w ( b / 2 a ) 2 ,
M = L - π N / 2 = ln ( b / a ) + W ln ( 2 / c kb ) - 2 π ρ F l n ( R / b ) R d R ln ( b / a ) + W L b + 8 W F ¯ l ,             F ¯ l = - 0 F ( ln u ) u d u ,
F = 8 W π { cos - 1 u 2 - u 2 [ 1 - ( u 2 ) 2 ] 1 / 2 } ,             1 u 2 ,
W = ( 1 - W ) 3 / ( 1 + W ) ,
W = 1 - 4 W + 3 12 W 2 / π 1 - 4 W + 6.6159 W 2 .
- ( - 1 ) / c = a 0 + A 1 + 2 a 2 ( 1 + A 1 c / 2 ) 2 ; A 1 = a 1 / ( 1 - a 1 H 11 ) , 2 H 11 = c + ( W - 1 ) + i N - i ( W - 1 ) / 2 π ,
1 = 1 + 2 w ( - 1 ) 1 + - w ( - 1 ) = 1 + w δ D ,             D = 1 + ( 1 - w ) δ 2 ,             δ = - 1.
= 1 - x 2 w δ 8 { 3 + - δ [ 4 M + 1 ( W - 1 ) ] 2 D 2 - 1 - 1 ( 1 + 1 ) 2 2 ( + 1 ) } + i π x 2 δ 2 w W 8 D 2 .
1 + w δ - w ( 1 - w ) δ 2 / 2 + x 2 w δ 2 ( 1 + 2 w + 4 M + W ) / 16 + i π x 2 δ 2 w W / 8.
l - t w ( 1 - w ) δ 2 / 2 + x 2 w δ 2 ( 1 + 2 w + 4 M - W ) / 16 + i π x 2 δ 2 w W / 8 ,
G = A 1 + A 1 M + A 2 ,             c = i 4 π ρ / k 3 = i 3 w / x 3 ,             w = ρ 4 π a 3 / 3 = ρ v ( a ) ,
- ( η 2 - 1 ) / c = η 2 A 1 + η 2 a 1 M [ 1 + A 1 c η / ( η + 1 ) ] 2 + η 4 a 2 ( 1 + A 1 c 3 / 5 ) 2 ; A 1 = a 1 / ( 1 - a 1 H 11 ) , 3 H 11 = 2 c + 2 H 0 + H 2 2 c + 2 ( W - 1 ) + i N ( 2 + η 2 / 5 ) / x ,
W = 1 + 4 π ρ 0 F R 2 d R = 1 + 4 π ρ F 2 = 1 - 24 W F ¯ 2 , W = ρ 4 π ( b / 2 ) 3 / 3 = ρ v ( b / 2 ) = w ( b / 2 a ) 3 ,
N = - 4 π ρ a 0 F R d R = - 4 π ρ a F 1 = 24 ( a / b ) W F ¯ 1 .
1 = 1 + 3 w ( - 1 ) 2 + - w ( - 1 ) = 1 + w δ D ,             D = 1 - ( 1 - w ) δ 3 ,             δ = - 1 ,
= 1 - x 2 w δ { 1 D 2 [ 2 - 5 + N 9 ( 2 + 1 5 ) ] - 1 10 [ 1 + ( 2 1 + 3 ) 2 5 ( 2 + 3 ) ] } + i x 3 2 δ 2 w W 9 D 2 .
W = ( 1 - W ) 4 / ( 1 + 2 W ) 2 ,
N = 2 a b 6 W 1 + 2 W ( 1 - W 5 + W 2 10 ) .
1 + w δ - w ( 1 - w ) δ 2 / 3 + x 2 w δ 2 [ 6 + 2 w - 5 N ] 11 / ( 15 ) 2 + i x 3 2 δ 2 w W / 9.
[ ] = 3 ( 2 - 5 w + 4 w 2 - w 3 ) / ( 1 + 2 w ) .
= 1 + w δ 2 P ( x 2 ) + i w δ 2 S ( x m ) , δ = - 1 ,
1 = 1 + w δ / ( 1 + δ D ) ,             D = ( 1 - w ) Q ,             Q 1 = Q 2 l = 0 ,             Q 2 t = ½ ,             Q 3 =
η = [ 1 + w δ 2 ( P + i S ) ] 1 / 2 η 1 + w δ 2 ( P + i S ) / 2 η 1 ,
{ η r η i } = [ ± r 2 ] 1 / 2 ,             = [ r 2 + i 2 ] 1 / 2 .
( - 1 ) / w = Δ / w = δ ( 1 - δ D + δ 2 D 2 ) + δ 2 ( P + i S )
Δ r / w = δ r ( 1 - δ r D + δ r 2 D 2 ) + δ i 2 D ( 1 - 3 δ r D ) - 2 δ r δ i S + ( δ r 2 - δ i 2 ) P
Δ i / w = δ i [ 1 - 2 δ r D + ( 3 δ r 2 - δ i 2 ) D 2 ] + ( δ r 2 - δ i 2 ) S + 2 δ r δ i P .
( η - 1 ) / w = ν + ν 2 ( A + 4 P + i 4 S ) / 2 + ν 3 B / 2 , A = 1 - ( w + 4 D ) ,             B = - ( 1 - w ) ( w + 4 D ) + 8 D 2 .
( η r - 1 ) / w = ν r ( 1 + ν r A / 2 + ν r 2 B / 2 ) - ν i 2 ( A + 3 ν r B ) / 2 - 4 ν i ν r S + 2 ( ν r 2 - ν i 2 ) P ,
η i / w = ν i [ 1 + ν r A + ( 3 ν r 2 - ν i 2 ) B / 2 ] + 2 ( ν r 2 - ν i 2 ) S + 4 ν i ν r P .
D 1 = 0 ,             A 1 = 1 - w ,             B 1 = - w ( 1 - w ) ,
P 1 = - x 2 ( 2 - W + 3 N ) / 3 ,             S 1 = x W ,
P 2 l = x 2 ( 1 + 2 w + 4 M ) / 8 ,             S 2 l = x 2 π W / 4.
D 2 t = ( 1 - w ) / 2 ,             A 2 t = - ( 1 - w ) = - A 2 l ,             B 2 t = - w ( 1 - w ) = B 2 l ,
P 2 t = x 2 ( 1 + 2 w + 4 M + W ) / 16 = P 2 l / 2 + x 2 W / 16 ,             S 2 t = π x 2 W / 8 = S 2 l / 2 ,
D 3 = ( 1 - w ) / 3 ,             A 3 = - ( 1 - w ) / 3 ,             B 3 = - ( 1 - w ) ( 4 + 5 w ) / 9 ,
P 3 = x 2 ( 6 + 3 w - 5 N ) 11 / ( 15 ) 2 ,             S 3 = 2 x 3 W / 9 ,
η l - η t = w ν 2 [ A l + P l - x 2 W / 16 + i S l ] = w ν 2 [ 1 - w + x 2 ( 1 + 2 w + M - W ) / 16 + i π x 2 W / 4 ] ν 2 ( R + i I ) .
Re ( η l - η t ) = ( ν r 2 - ν i 2 ) R - 2 ν r ν i I
Im ( η l - η t ) = 2 ν r ν i R + ( ν r 2 - ν i 2 ) I .
Re ( η b l - η b t ) / η p r [ ( ξ 2 - μ 2 ) R + 2 ξ μ I ] ( 1 - ξ ) ,
Im ( η b l - η b t ) / η p r [ - 2 ξ μ R + ( ξ 2 - μ 2 ) I ] ( 1 - ξ ) ,