Abstract

We present a new method to study light scattering on nonabsorbing spherical particles. This method is based on the Ramsauer approach, a model known in atomic and nuclear physics. Its main advantage is its intuitive understanding of the underlying physics phenomena. We show that although the approximations are numerous, the Ramsauer analytical solutions describe fairly well the scattering phase function and the total cross section. Then this model is applied to the Henyey–Greenstein parameterization of the scattering phase function to give a relation between its asymmetry parameter and the mean particle size.

© 2012 Optical Society of America

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  1. H. C. Van De Hulst, Light Scattering by Small Particles(Dover, 1981).
  2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
  3. G. Mie, “Beiträge zur Optik Trüber-Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
    [CrossRef]
  4. C. W. Ramsauer, “Über den Wirkungsquerschnitt der Gasmoleküle gegenüber langsamen Elektronen,” Ann. Phys. 369, 513–540 (1921).
    [CrossRef]
  5. W. F. Egelhoff, “Semiclassical explanation of the generalized Ramsauer–Townsend minima in electron-atom scattering,” Phys. Rev. Lett. 71, 2883–2886 (1993).
    [CrossRef]
  6. D. E. Golden and H. W. Bandel, “Low-energy e−-Ar total scattering cross sections: the Ramsauer–Townsend effect,” Phys. Rev. 149, 58–59 (1966).
    [CrossRef]
  7. G. P. Karwasz, “Positrons—an alternative probe to electron scattering,” Eur. Phys. J. D 35, 267–278 (2005).
    [CrossRef]
  8. R. S. Grace, W. M. Pope, D. L. Johnson, and J. G. Skofronick, “Ramsauer–Townsend effect in the total cross section of He4+He4 and He3+He3,” Phys. Rev. A 14, 1006–1008 (1976).
    [CrossRef]
  9. R. W. Bauer, J. D. Anderson, S. M. Grimes, and V. A. Madsen, Application of Simple Ramsauer Model to Neutron Total Cross Sections (Lawrence Livermore National Laboratory, 1997), preprint.
  10. S. Fernbach, R. Serber, and T. B. Taylor, “The scattering of high energy neutrons by nuclei,” Phys. Rev. 75, 1352–1355 (1949).
    [CrossRef]
  11. R. S. Gowda, S. V. Surya Narayan, and S. Ganesan, “The Ramsauer model for the total cross sections of neutron nucleus scattering,” http://arxiv.org/abs/nucl-th/0506004 .
  12. J. M. Peterson, “Neutron giant resonances—nuclear Ramsauer effect,” Phys. Rev. 125, 955–963 (1962).
    [CrossRef]
  13. W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
    [CrossRef]
  14. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef]
  15. I. Weiner, M. Rust, and T. D. Donnelly, “Particle size determination: an undergraduate lab in Mie scattering,” Am. J. Phys. 69, 129–136 (2001).
    [CrossRef]
  16. http://www.philiplaven.com/mieplot.htm .
  17. B. Barkey, M. Bailey, K.-N. Liou, and J. Hallett, “Light-scattering properties of plate and column ice crystals generated in a laboratory cold chamber,” Appl. Opt. 41, 5792–5796 (2002).
    [CrossRef]
  18. J.-L. Castagner and I. J. Bigio, “Particle sizing with a fast polar nephelometer,” Appl. Opt. 46, 527–532 (2007).
    [CrossRef]
  19. http://www.sigmaaldrich.com .
  20. S. V. Surya Narayan, R. S. Gowda, and S. Ganesan, “Empirical estimates of the neutron-nucleus scattering cross sections,” http://arxiv.org/abs/nucl-th/0409005 .
  21. D. Toublanc, “Henyey–Greenstein and Mie phase functions in Monte Carlo radiative transfer computations,” Appl. Opt. 35, 3270–3274 (1996).
    [CrossRef]
  22. O. Boucher, “On aerosol shortwave forcing and the Henyey–Greenstein phase function,” J. Atmos. Sci. 55, 128–134(1998).
    [CrossRef]
  23. T. Binzoni, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
    [CrossRef]
  24. L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  25. The Pierre Auger Collaboration, “The fluorescence detector of the Pierre Auger Observatory,” Nucl. Instrum. Methods Phys. Res. A 620, 227–251 (2010).
  26. CTA Consortium, “Design concepts for the Cherenkov Telescope Array CTA: an advanced facility for ground-based high-energy gamma-ray astronomy,” Exp. Astron. 32, 193–316(2011).
  27. The Pierre Auger Collaboration, “A study of the effect of molecular and aerosol conditions in the atmosphere on air fluorescence measurements at the Pierre Auger Observatory,” Astropart. Phys. 33, 108–129 (2010).
    [CrossRef]
  28. B. Keilhauer and M. Will, for the Pierre Auger Collaboration, “Description of atmospheric conditions at the Pierre Auger Observatory using meteorological measurements and models,” Eur. Phys. Plus J. 127, 96 (2012).
    [CrossRef]
  29. K. Louedec for the Pierre Auger Collaboration and R. Losno, “Atmospheric aerosols at the Pierre Auger Observatory and environmental implications,” Eur. Phys. J. Plus 127, 97 (2012).
    [CrossRef]
  30. S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
    [CrossRef]
  31. K. Louedec, for the Pierre Auger Collaboration, “Atmospheric monitoring at the Pierre Auger Observatory—Status and update,” in Proceedings of 32nd ICRC (2011), Vol. 2, pp. 63–66.

2012 (2)

B. Keilhauer and M. Will, for the Pierre Auger Collaboration, “Description of atmospheric conditions at the Pierre Auger Observatory using meteorological measurements and models,” Eur. Phys. Plus J. 127, 96 (2012).
[CrossRef]

K. Louedec for the Pierre Auger Collaboration and R. Losno, “Atmospheric aerosols at the Pierre Auger Observatory and environmental implications,” Eur. Phys. J. Plus 127, 97 (2012).
[CrossRef]

K. Louedec for the Pierre Auger Collaboration and R. Losno, “Atmospheric aerosols at the Pierre Auger Observatory and environmental implications,” Eur. Phys. J. Plus 127, 97 (2012).
[CrossRef]

2011 (1)

CTA Consortium, “Design concepts for the Cherenkov Telescope Array CTA: an advanced facility for ground-based high-energy gamma-ray astronomy,” Exp. Astron. 32, 193–316(2011).

2010 (2)

The Pierre Auger Collaboration, “A study of the effect of molecular and aerosol conditions in the atmosphere on air fluorescence measurements at the Pierre Auger Observatory,” Astropart. Phys. 33, 108–129 (2010).
[CrossRef]

The Pierre Auger Collaboration, “The fluorescence detector of the Pierre Auger Observatory,” Nucl. Instrum. Methods Phys. Res. A 620, 227–251 (2010).

2007 (2)

S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
[CrossRef]

J.-L. Castagner and I. J. Bigio, “Particle sizing with a fast polar nephelometer,” Appl. Opt. 46, 527–532 (2007).
[CrossRef]

2006 (1)

T. Binzoni, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

2005 (1)

G. P. Karwasz, “Positrons—an alternative probe to electron scattering,” Eur. Phys. J. D 35, 267–278 (2005).
[CrossRef]

2002 (1)

2001 (2)

I. Weiner, M. Rust, and T. D. Donnelly, “Particle size determination: an undergraduate lab in Mie scattering,” Am. J. Phys. 69, 129–136 (2001).
[CrossRef]

W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
[CrossRef]

1998 (1)

O. Boucher, “On aerosol shortwave forcing and the Henyey–Greenstein phase function,” J. Atmos. Sci. 55, 128–134(1998).
[CrossRef]

1996 (1)

1993 (1)

W. F. Egelhoff, “Semiclassical explanation of the generalized Ramsauer–Townsend minima in electron-atom scattering,” Phys. Rev. Lett. 71, 2883–2886 (1993).
[CrossRef]

1980 (1)

1976 (1)

R. S. Grace, W. M. Pope, D. L. Johnson, and J. G. Skofronick, “Ramsauer–Townsend effect in the total cross section of He4+He4 and He3+He3,” Phys. Rev. A 14, 1006–1008 (1976).
[CrossRef]

1966 (1)

D. E. Golden and H. W. Bandel, “Low-energy e−-Ar total scattering cross sections: the Ramsauer–Townsend effect,” Phys. Rev. 149, 58–59 (1966).
[CrossRef]

1962 (1)

J. M. Peterson, “Neutron giant resonances—nuclear Ramsauer effect,” Phys. Rev. 125, 955–963 (1962).
[CrossRef]

1949 (1)

S. Fernbach, R. Serber, and T. B. Taylor, “The scattering of high energy neutrons by nuclei,” Phys. Rev. 75, 1352–1355 (1949).
[CrossRef]

1941 (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

1921 (1)

C. W. Ramsauer, “Über den Wirkungsquerschnitt der Gasmoleküle gegenüber langsamen Elektronen,” Ann. Phys. 369, 513–540 (1921).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik Trüber-Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Abfalterer, W. P.

W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
[CrossRef]

Anderson, J. D.

R. W. Bauer, J. D. Anderson, S. M. Grimes, and V. A. Madsen, Application of Simple Ramsauer Model to Neutron Total Cross Sections (Lawrence Livermore National Laboratory, 1997), preprint.

Bailey, M.

Bandel, H. W.

D. E. Golden and H. W. Bandel, “Low-energy e−-Ar total scattering cross sections: the Ramsauer–Townsend effect,” Phys. Rev. 149, 58–59 (1966).
[CrossRef]

Barkey, B.

Bateman, F. B.

W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
[CrossRef]

Bauer, R. W.

R. W. Bauer, J. D. Anderson, S. M. Grimes, and V. A. Madsen, Application of Simple Ramsauer Model to Neutron Total Cross Sections (Lawrence Livermore National Laboratory, 1997), preprint.

BenZvi, S.

S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
[CrossRef]

Bigio, I. J.

Binzoni, T.

T. Binzoni, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Boucher, O.

O. Boucher, “On aerosol shortwave forcing and the Henyey–Greenstein phase function,” J. Atmos. Sci. 55, 128–134(1998).
[CrossRef]

Castagner, J.-L.

Connolly, B. M.

S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
[CrossRef]

Delpy, D. T.

T. Binzoni, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Dietrich, F. S.

W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
[CrossRef]

Donnelly, T. D.

I. Weiner, M. Rust, and T. D. Donnelly, “Particle size determination: an undergraduate lab in Mie scattering,” Am. J. Phys. 69, 129–136 (2001).
[CrossRef]

Egelhoff, W. F.

W. F. Egelhoff, “Semiclassical explanation of the generalized Ramsauer–Townsend minima in electron-atom scattering,” Phys. Rev. Lett. 71, 2883–2886 (1993).
[CrossRef]

Fernbach, S.

S. Fernbach, R. Serber, and T. B. Taylor, “The scattering of high energy neutrons by nuclei,” Phys. Rev. 75, 1352–1355 (1949).
[CrossRef]

Finlay, R. W.

W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
[CrossRef]

Gandjbakhche, A. H.

T. Binzoni, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Golden, D. E.

D. E. Golden and H. W. Bandel, “Low-energy e−-Ar total scattering cross sections: the Ramsauer–Townsend effect,” Phys. Rev. 149, 58–59 (1966).
[CrossRef]

Grace, R. S.

R. S. Grace, W. M. Pope, D. L. Johnson, and J. G. Skofronick, “Ramsauer–Townsend effect in the total cross section of He4+He4 and He3+He3,” Phys. Rev. A 14, 1006–1008 (1976).
[CrossRef]

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Grimes, S. M.

R. W. Bauer, J. D. Anderson, S. M. Grimes, and V. A. Madsen, Application of Simple Ramsauer Model to Neutron Total Cross Sections (Lawrence Livermore National Laboratory, 1997), preprint.

Haight, R. C.

W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
[CrossRef]

Hallett, J.

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Johnson, D. L.

R. S. Grace, W. M. Pope, D. L. Johnson, and J. G. Skofronick, “Ramsauer–Townsend effect in the total cross section of He4+He4 and He3+He3,” Phys. Rev. A 14, 1006–1008 (1976).
[CrossRef]

Karwasz, G. P.

G. P. Karwasz, “Positrons—an alternative probe to electron scattering,” Eur. Phys. J. D 35, 267–278 (2005).
[CrossRef]

Keilhauer, B.

B. Keilhauer and M. Will, for the Pierre Auger Collaboration, “Description of atmospheric conditions at the Pierre Auger Observatory using meteorological measurements and models,” Eur. Phys. Plus J. 127, 96 (2012).
[CrossRef]

Leung, T. S.

T. Binzoni, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Liou, K.-N.

Losno, R.

K. Louedec for the Pierre Auger Collaboration and R. Losno, “Atmospheric aerosols at the Pierre Auger Observatory and environmental implications,” Eur. Phys. J. Plus 127, 97 (2012).
[CrossRef]

Louedec, K.

K. Louedec for the Pierre Auger Collaboration and R. Losno, “Atmospheric aerosols at the Pierre Auger Observatory and environmental implications,” Eur. Phys. J. Plus 127, 97 (2012).
[CrossRef]

K. Louedec, for the Pierre Auger Collaboration, “Atmospheric monitoring at the Pierre Auger Observatory—Status and update,” in Proceedings of 32nd ICRC (2011), Vol. 2, pp. 63–66.

Madsen, V. A.

R. W. Bauer, J. D. Anderson, S. M. Grimes, and V. A. Madsen, Application of Simple Ramsauer Model to Neutron Total Cross Sections (Lawrence Livermore National Laboratory, 1997), preprint.

Matthews, J. A. J.

S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik Trüber-Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Morgan, G. L.

W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
[CrossRef]

Peterson, J. M.

J. M. Peterson, “Neutron giant resonances—nuclear Ramsauer effect,” Phys. Rev. 125, 955–963 (1962).
[CrossRef]

Pope, W. M.

R. S. Grace, W. M. Pope, D. L. Johnson, and J. G. Skofronick, “Ramsauer–Townsend effect in the total cross section of He4+He4 and He3+He3,” Phys. Rev. A 14, 1006–1008 (1976).
[CrossRef]

Prouza, M.

S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
[CrossRef]

Ramsauer, C. W.

C. W. Ramsauer, “Über den Wirkungsquerschnitt der Gasmoleküle gegenüber langsamen Elektronen,” Ann. Phys. 369, 513–540 (1921).
[CrossRef]

Rüfenacht, D.

T. Binzoni, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Rust, M.

I. Weiner, M. Rust, and T. D. Donnelly, “Particle size determination: an undergraduate lab in Mie scattering,” Am. J. Phys. 69, 129–136 (2001).
[CrossRef]

Serber, R.

S. Fernbach, R. Serber, and T. B. Taylor, “The scattering of high energy neutrons by nuclei,” Phys. Rev. 75, 1352–1355 (1949).
[CrossRef]

Skofronick, J. G.

R. S. Grace, W. M. Pope, D. L. Johnson, and J. G. Skofronick, “Ramsauer–Townsend effect in the total cross section of He4+He4 and He3+He3,” Phys. Rev. A 14, 1006–1008 (1976).
[CrossRef]

Taylor, T. B.

S. Fernbach, R. Serber, and T. B. Taylor, “The scattering of high energy neutrons by nuclei,” Phys. Rev. 75, 1352–1355 (1949).
[CrossRef]

Toublanc, D.

Van De Hulst, H. C.

H. C. Van De Hulst, Light Scattering by Small Particles(Dover, 1981).

Visbal, E. F.

S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
[CrossRef]

Weiner, I.

I. Weiner, M. Rust, and T. D. Donnelly, “Particle size determination: an undergraduate lab in Mie scattering,” Am. J. Phys. 69, 129–136 (2001).
[CrossRef]

Westerhoff, S.

S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
[CrossRef]

Will, M.

B. Keilhauer and M. Will, for the Pierre Auger Collaboration, “Description of atmospheric conditions at the Pierre Auger Observatory using meteorological measurements and models,” Eur. Phys. Plus J. 127, 96 (2012).
[CrossRef]

Wiscombe, W. J.

Am. J. Phys. (1)

I. Weiner, M. Rust, and T. D. Donnelly, “Particle size determination: an undergraduate lab in Mie scattering,” Am. J. Phys. 69, 129–136 (2001).
[CrossRef]

Ann. Phys. (2)

G. Mie, “Beiträge zur Optik Trüber-Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

C. W. Ramsauer, “Über den Wirkungsquerschnitt der Gasmoleküle gegenüber langsamen Elektronen,” Ann. Phys. 369, 513–540 (1921).
[CrossRef]

Appl. Opt. (4)

Astropart. Phys. (2)

The Pierre Auger Collaboration, “A study of the effect of molecular and aerosol conditions in the atmosphere on air fluorescence measurements at the Pierre Auger Observatory,” Astropart. Phys. 33, 108–129 (2010).
[CrossRef]

S. BenZvi, B. M. Connolly, J. A. J. Matthews, M. Prouza, E. F. Visbal, and S. Westerhoff, “Measurement of the aerosol phase function at the Pierre Auger Observatory,” Astropart. Phys. 28, 312–320 (2007).
[CrossRef]

Astrophys. J. (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Eur. Phys. J. D (1)

G. P. Karwasz, “Positrons—an alternative probe to electron scattering,” Eur. Phys. J. D 35, 267–278 (2005).
[CrossRef]

Eur. Phys. J. Plus (1)

K. Louedec for the Pierre Auger Collaboration and R. Losno, “Atmospheric aerosols at the Pierre Auger Observatory and environmental implications,” Eur. Phys. J. Plus 127, 97 (2012).
[CrossRef]

Eur. Phys. Plus J. (1)

B. Keilhauer and M. Will, for the Pierre Auger Collaboration, “Description of atmospheric conditions at the Pierre Auger Observatory using meteorological measurements and models,” Eur. Phys. Plus J. 127, 96 (2012).
[CrossRef]

Exp. Astron. (1)

CTA Consortium, “Design concepts for the Cherenkov Telescope Array CTA: an advanced facility for ground-based high-energy gamma-ray astronomy,” Exp. Astron. 32, 193–316(2011).

J. Atmos. Sci. (1)

O. Boucher, “On aerosol shortwave forcing and the Henyey–Greenstein phase function,” J. Atmos. Sci. 55, 128–134(1998).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

The Pierre Auger Collaboration, “The fluorescence detector of the Pierre Auger Observatory,” Nucl. Instrum. Methods Phys. Res. A 620, 227–251 (2010).

Phys. Med. Biol. (1)

T. Binzoni, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. 51, N313–N322 (2006).
[CrossRef]

Phys. Rev. (3)

S. Fernbach, R. Serber, and T. B. Taylor, “The scattering of high energy neutrons by nuclei,” Phys. Rev. 75, 1352–1355 (1949).
[CrossRef]

D. E. Golden and H. W. Bandel, “Low-energy e−-Ar total scattering cross sections: the Ramsauer–Townsend effect,” Phys. Rev. 149, 58–59 (1966).
[CrossRef]

J. M. Peterson, “Neutron giant resonances—nuclear Ramsauer effect,” Phys. Rev. 125, 955–963 (1962).
[CrossRef]

Phys. Rev. A (1)

R. S. Grace, W. M. Pope, D. L. Johnson, and J. G. Skofronick, “Ramsauer–Townsend effect in the total cross section of He4+He4 and He3+He3,” Phys. Rev. A 14, 1006–1008 (1976).
[CrossRef]

Phys. Rev. C (1)

W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W. Finlay, R. C. Haight, and G. L. Morgan, “Measurement of neutron total cross sections up to 560 MeV,” Phys. Rev. C 63, 044608 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

W. F. Egelhoff, “Semiclassical explanation of the generalized Ramsauer–Townsend minima in electron-atom scattering,” Phys. Rev. Lett. 71, 2883–2886 (1993).
[CrossRef]

Other (8)

H. C. Van De Hulst, Light Scattering by Small Particles(Dover, 1981).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

R. W. Bauer, J. D. Anderson, S. M. Grimes, and V. A. Madsen, Application of Simple Ramsauer Model to Neutron Total Cross Sections (Lawrence Livermore National Laboratory, 1997), preprint.

R. S. Gowda, S. V. Surya Narayan, and S. Ganesan, “The Ramsauer model for the total cross sections of neutron nucleus scattering,” http://arxiv.org/abs/nucl-th/0506004 .

http://www.sigmaaldrich.com .

S. V. Surya Narayan, R. S. Gowda, and S. Ganesan, “Empirical estimates of the neutron-nucleus scattering cross sections,” http://arxiv.org/abs/nucl-th/0409005 .

http://www.philiplaven.com/mieplot.htm .

K. Louedec, for the Pierre Auger Collaboration, “Atmospheric monitoring at the Pierre Auger Observatory—Status and update,” in Proceedings of 32nd ICRC (2011), Vol. 2, pp. 63–66.

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Figures (12)

Fig. 1.
Fig. 1.

Extinction efficiency factor Qext versus the size parameter x, for nonabsorbing spherical particles with relative refractive indices n=1.05, 1.33, and 2.00. x is given by the relation x=2πR/λ=koutR. The vertical scale applies only to the lowest curve, the others being successively shifted upward by 2.

Fig. 2.
Fig. 2.

Angular scattering for water droplets illuminated by unpolarized light, for three different radii. The incident wavelength is fixed at 0.6 μm. The vertical scale applies only to the lowest curve, the others being successively multiplied by 100.

Fig. 3.
Fig. 3.

Angular scattering for a distribution of water droplets illuminated by unpolarized light, for three different standard deviations. The mean radius R¯ is fixed at 15 μm and the incident wavelength at 0.6 μm. The vertical scale applies only to the lowest curve, the others being successively multiplied by 100.

Fig. 4.
Fig. 4.

(a) Electron–krypton normalized total cross section versus energy [7]. (b) Normalized neutron-lead total scattering cross section versus energy [13].

Fig. 5.
Fig. 5.

Qualitative picture of the Ramsauer phenomenon. The wavelength of the light is supposed to be reduced in the dielectric. This picture shows the case where the contraction of the wavelength between (a) outside and (b) inside the medium (dashed line) is such that they come out in phase. Thus the sphere looks invisible, resulting in an almost zero cross section.

Fig. 6.
Fig. 6.

Definition of the variables used in the text to calculate the phase function for light scattering by a spherical particle of radius R. kin and kout are the wavenumbers for the light inside and outside, respectively, and b is the impact parameter.

Fig. 7.
Fig. 7.

Characteristic behaviors of physics quantities from the phase function. The curves are obtained by Mie calculations (using [16]) for an incident wavelength of 0.6 μm. Three different refractive indices are shown: n=1.05 (blue circles), n=1.33 (red squares), and n=1.60 (magenta triangles). The vertical scale applies only to the lowest curve, the others being successively multiplied by 100.

Fig. 8.
Fig. 8.

Simplified Ramsauer solution, independent of the refractive index, compared to the Ramsauer solution and Mie calculations. (a) SR versus Ramsauer. (b) SR versus Mie theory. Three different refractive indices are shown: n=1.05, 1.33, and 1.60. The simplified solution, independent of the refractive index, is in black thick line. The incident wavelength is fixed at 0.6 μm and the sphere radius is equal to 10 μm. The vertical scale applies only to the lowest curve, the others being successively multiplied by 100.

Fig. 9.
Fig. 9.

Extinction efficiency factor versus the phase shift ΔR, for nonabsorbing spherical particles with relative refractive indices n=1.05, 1.33, and 2.00. The Ramsauer solution from Eq. (30) for n=1.05 is also given. The vertical scale applies only to the lowest curve, the others being successively shifted upward by 2.

Fig. 10.
Fig. 10.

Total cross section as a function of the phase shift ΔR=2x(n1) for a refractive index fixed at (a) n=1.05 and (b) n=1.33. The simplified Ramsauer solution from Eq. (31) is in blue continuous line, the Mie prediction in red dashed line (using [16]), and the formula from Eq. (28) in black thick line. The incident wavelength is fixed at 0.6 μm.

Fig. 11.
Fig. 11.

Henyey–Greenstein functions representing the scattering phase function for different asymmetry parameters gHG and backward factors f. The Rayleigh phase function, proportional to (1+cos2θ), is also plotted.

Fig. 12.
Fig. 12.

Relation between the asymmetry parameter of the HG function and the mean radius of the particle size distribution. The equivalence is plotted for two different incident wavelengths: 0.4 μm (blue circles) and 0.8 μm (red squares).

Equations (35)

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I(θ)=I0r2×σsca(R,λ)×P(θ),
Qsca=σsca(R,λ)πR2,
x=2πRλ=koutR,
Qext=2x2=1(2+1)Re(a+b),
n(R|R¯,σ)=dN(R)dR=N2πlog10σ1Rexp(log102(R/R¯)2log102σ),
iλ(1+cosθ)2AincidentdS,
Adisk(ρ)=iλ[ei2koutR+ein2koutR2ρ2eikout(2R2R2ρ2)]1+cosθ2=ikout2πei2koutR[1ei2koutR2ρ2(n1)]1+cosθ2.
fsphere(θ)=ei2koutR02πdϕ0RAdisk(ρ)ρeikoutρcosϕsinθdρ=ikout2π1+cosθ202πdϕ0Reikoutρcosϕsinθ[1ei2koutR2ρ2(n1)]ρdρ,
J0(u)=12π02πeiucosϕdϕ.
fsphere(θ)=ikout1+cosθ20RρJ0(koutρsinθ)[1ei2koutR2ρ2(n1)]dρ.
[dσdθ]sphere=|fsphere(θ)|2,
PR(θ)=1σ[dσdθ]sphere=|fsphere(θ)|202πdΦ0πdθsinθ|fsphere(θ)|2.
fcyl(θ)=ikout1+cosθ20RρJ0(koutρsinθ)[1ei2koutR(n1)]dρ=ikout1+cosθ2[1ei2(n1)koutR]0RρJ0(koutρsinθ)dρ=i1+cosθ2ei2(n1)koutR2(2i)sin[2(n1)koutR2]RsinθJ1(koutRsinθ),
0RuJ0(u)du=RJ1(R).
[dσdθ]cyl=|fcyl(θ)|2=R2[koutR1+cosθ2sin[(n1)koutR]2J1(koutRsinθ)koutRsinθ]2.
σtotcyl=02πdΦ0πdθsinθ[dσdθ]cyl=4πR2sin2[(n1)2koutL]=4πR2sin2[(n1)koutR],
[dσdθ]cyl=14πσtotcyl[koutR(1+cosθ)22J1(koutRsinθ)koutRsinθ]2.
[dσdθ]sphere=14πσtotsphere[koutR(1+cosθ)22J1(koutRsinθ)koutRsinθ]2.
PSR(θ)=14π[koutR(1+cosθ)22J1(koutRsinθ)koutRsinθ]2.
1σdσdθ(θ=0°)R2.
ΔθFWHM=2koutR.
Ψ(r⃗)=eik⃗outr⃗+f(θ)eikoutrr.
σtot=4πkoutImf(θ=0deg),
f(θ)=12ikout=0(2+1)P(cosθ)[ηe2iδ1],
dkoutdb,
f(θ=0deg)=kouti0R(ei2(kinkout)Rcosψ1)bdb,
f(θ=0deg)=koutR22i01(1ei2(kinkout)Rcosψ)dcos2ψ.
σtot=Im[i4πR201w(1ei(kinkout)2Rw)dw]=Im[i2πR2i4πR201wei(kinkout)2Rwdw].
σtot=Im[i2πR2i4πR2([wei(kinkout)2Rwi(kinkout)2R]0101ei(kinkout)2Rwi(kinkout)2Rdw)]=Im[i2πR2i4πR2(ei(kinkout)2Ri(kinkout)2R+ei(kinkout)2R1(2R)2(kinkout)2)]=2πR24πR2[sin(2R(kinkout))2R(kinkout)12[sin(R(kinkout))R(kinkout)]2].
σtot=2πR2[12sinΔRΔR+(sinΔR/2ΔR/2)2].
σtot=2π(R+λ4π)2[12sinΔRΔR+(sinΔR/2ΔR/2)2].
Qext,R=2(1+n1ΔR)2[12sinΔRΔR+(sinΔR/2ΔR/2)2],
σtot=02πdΦ0πdθsinθdσdθ.
PHG(θ|gHG,f)=1gHG24π[1(1+gHG22gHGcosθ)3/2+f3cos2θ12(1+gHG2)3/2],
gHG=cosθ=0πcosθPHG(θ|gHG,f)2πsinθdθ.

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