Abstract

A technique has been developed to extract quantitative crystallographic data from randomly oriented colloidal crystals using a divergent-beam approach. This technique was tested on a series of diverse experimental images of colloidal crystals formed from monodisperse suspensions of sterically stabilized poly-(methyl methacrylate) spheres suspended in organic index-matching solvents. Complete sets of reciprocal lattice basis vectors were extracted in all but one case. When data extraction was successful, results appeared to be accurate to about 1% for lattice parameters and to within ~2° for orientation. This approach is easier to implement than a previously developed parallel-beam approach with the drawback that the divergent-beam approach is not as robust in certain situations with random hexagonal close-packed crystals. The two techniques are therefore complimentary to each other, and between them it should be possible to extract quantitative crystallographic data with a conventional optical microscope from any closely index-matched colloidal crystal whose lattice parameters are compatible with visible wavelengths.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25-73(1983).
    [CrossRef]
  8. T. Yoshiyama, I. Sogami, and N. Ise, “Kossel line analysis on colloidal crystals in semidilute aqueous-solutions,” Phys. Rev. Lett. 53, 2153-2156 (1984).
    [CrossRef]
  9. T. Yoshiyama and I. S. Sogami, “Photographic records of ordered and disordered structures in polymer latex dispersions,” Langmuir 3, 851-853 (1987).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. T. Shinohara, H. Yamada, I. S. Sogami, N. Ise, and T. Yoshiyama, “Gravitational, vertical compression of colloidal crystals as studied by the Kossel diffraction method,” Langmuir 20, 5141-5144 (2004).
    [CrossRef]
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    [CrossRef]
  22. P. N. Pusey and W. van Megen, “Phase-behavior of concentrated suspensions of nearly hard colloidal spheres,” Nature 320, 340-342 (1986).
    [CrossRef]
  23. J. C. Russ, The Image Processing Handbook (CRC Press, 1999).
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    [CrossRef]
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    [CrossRef]
  28. P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
    [CrossRef] [PubMed]
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    [CrossRef]
  30. A. Guinier, X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies (W.H. Freeman, 1963).

2008 (1)

2004 (1)

T. Shinohara, H. Yamada, I. S. Sogami, N. Ise, and T. Yoshiyama, “Gravitational, vertical compression of colloidal crystals as studied by the Kossel diffraction method,” Langmuir 20, 5141-5144 (2004).
[CrossRef]

1999 (1)

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476-480 (1999).
[CrossRef]

1997 (1)

R. D. Pradhan, J. A. Bloodgood, and G. H. Watson, “Photonic band structure of bcc colloidal crystals,” Phys. Rev. B 55, 9503-9507 (1997).
[CrossRef]

1996 (2)

I. I. Tarhan and G. H. Watson, “Photonic band structure of fcc colloidal crystals,” Phys. Rev. Lett. 76, 315-318 (1996).
[CrossRef] [PubMed]

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

1990 (3)

Y. Monovoukas, G. G. Fuller, and A. P. Gast, “Optical anisotropy in colloidal crystals,” J. Chem. Phys. 93, 8294-8299(1990).
[CrossRef]

Y. Monovoukas and A. P. Gast, “Microstructure identification during crystallization of charged colloidal suspensions,” Phase Transitions 21, 183-195 (1990).
[CrossRef]

I. S. Sogami and T. Yoshiyama, “Kossel line analysis on crystallization in colloidal suspensions,” Phase Transitions 21, 171-182 (1990).
[CrossRef]

1989 (2)

Y. Monovoukas and A. P. Gast, “The experimental phase-diagram of charged colloidal suspensions,” J. Colloid Interface Sci. 128, 533-548 (1989).
[CrossRef]

P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
[CrossRef] [PubMed]

1987 (3)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062(1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489(1987).
[CrossRef] [PubMed]

T. Yoshiyama and I. S. Sogami, “Photographic records of ordered and disordered structures in polymer latex dispersions,” Langmuir 3, 851-853 (1987).
[CrossRef]

1986 (1)

P. N. Pusey and W. van Megen, “Phase-behavior of concentrated suspensions of nearly hard colloidal spheres,” Nature 320, 340-342 (1986).
[CrossRef]

1984 (1)

T. Yoshiyama, I. Sogami, and N. Ise, “Kossel line analysis on colloidal crystals in semidilute aqueous-solutions,” Phys. Rev. Lett. 53, 2153-2156 (1984).
[CrossRef]

1983 (1)

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25-73(1983).
[CrossRef]

1942 (1)

A. J. C. Wilson, “Imperfections in the structure of cobalt II. Mathematical treatment of proposed structure,” Proc. R. Soc. London Ser. A 180, 277-285 (1942).
[CrossRef]

Ackerson, B. J.

P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
[CrossRef] [PubMed]

Azaroff, L.

L. Azaroff, ,i>Elements of X-Ray Crystallography (McGraw-Hill, 1968).

Bartlett, P.

P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
[CrossRef] [PubMed]

Bloodgood, J. A.

R. D. Pradhan, J. A. Bloodgood, and G. H. Watson, “Photonic band structure of bcc colloidal crystals,” Phys. Rev. B 55, 9503-9507 (1997).
[CrossRef]

Carter, C. B.

D. B. Williams and C. B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science (Plenum, 1996).

Chaikin, P. M.

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

Cheng, Z. D.

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

Dunsmuir, J. H.

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

Fisher, R. B.

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476-480 (1999).
[CrossRef]

Fitzgibbon, A.

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476-480 (1999).
[CrossRef]

Fuller, G. G.

Y. Monovoukas, G. G. Fuller, and A. P. Gast, “Optical anisotropy in colloidal crystals,” J. Chem. Phys. 93, 8294-8299(1990).
[CrossRef]

Gast, A. P.

Y. Monovoukas, G. G. Fuller, and A. P. Gast, “Optical anisotropy in colloidal crystals,” J. Chem. Phys. 93, 8294-8299(1990).
[CrossRef]

Y. Monovoukas and A. P. Gast, “Microstructure identification during crystallization of charged colloidal suspensions,” Phase Transitions 21, 183-195 (1990).
[CrossRef]

Y. Monovoukas and A. P. Gast, “The experimental phase-diagram of charged colloidal suspensions,” J. Colloid Interface Sci. 128, 533-548 (1989).
[CrossRef]

Guinier, A.

A. Guinier, X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies (W.H. Freeman, 1963).

Inoue, S.

S. Inoue and K. R. Spring, Video Microscopy:The Fundamentals (Plenum, 1997).
[CrossRef]

Ise, N.

T. Shinohara, H. Yamada, I. S. Sogami, N. Ise, and T. Yoshiyama, “Gravitational, vertical compression of colloidal crystals as studied by the Kossel diffraction method,” Langmuir 20, 5141-5144 (2004).
[CrossRef]

T. Yoshiyama, I. Sogami, and N. Ise, “Kossel line analysis on colloidal crystals in semidilute aqueous-solutions,” Phys. Rev. Lett. 53, 2153-2156 (1984).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489(1987).
[CrossRef] [PubMed]

Lagerlöf, K. P. D.

Lichtblau, D.

D. Lichtblau, “MathGroup Archive 2001, Re: MATLAB to Mathematica 4.1,” http://forums.wolfram.com/mathgroup/archive/2001/Sep/msg00179.html.

Lipson, H.

S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge U. Press, 1995).

Lipson, S. G.

S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge U. Press, 1995).

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Monovoukas, Y.

Y. Monovoukas and A. P. Gast, “Microstructure identification during crystallization of charged colloidal suspensions,” Phase Transitions 21, 183-195 (1990).
[CrossRef]

Y. Monovoukas, G. G. Fuller, and A. P. Gast, “Optical anisotropy in colloidal crystals,” J. Chem. Phys. 93, 8294-8299(1990).
[CrossRef]

Y. Monovoukas and A. P. Gast, “The experimental phase-diagram of charged colloidal suspensions,” J. Colloid Interface Sci. 128, 533-548 (1989).
[CrossRef]

Ottewill, R. H.

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

Phan, S. E.

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

Pieranski, P.

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25-73(1983).
[CrossRef]

Pilu, M.

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476-480 (1999).
[CrossRef]

Pluta, M.

M. Pluta, ,i>Advanced Light Microscopy: Volume 2--Specialized Methods (Elsevier, 1989).

Poon, W. C.K.

W. C.K. Poon and P. N. Pusey, “Phase transition of spherical colloids,” in Observation, Prediction and Simulation of Phase Transitions in Complex Fluids, M. Baus, L. FRull, and J. -P. Ryckaert, eds. (Kluwer Academic, 1995), pp. 3-51.

Pradhan, R. D.

R. D. Pradhan, J. A. Bloodgood, and G. H. Watson, “Photonic band structure of bcc colloidal crystals,” Phys. Rev. B 55, 9503-9507 (1997).
[CrossRef]

Pusey, P. N.

P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
[CrossRef] [PubMed]

P. N. Pusey and W. van Megen, “Phase-behavior of concentrated suspensions of nearly hard colloidal spheres,” Nature 320, 340-342 (1986).
[CrossRef]

W. C.K. Poon and P. N. Pusey, “Phase transition of spherical colloids,” in Observation, Prediction and Simulation of Phase Transitions in Complex Fluids, M. Baus, L. FRull, and J. -P. Ryckaert, eds. (Kluwer Academic, 1995), pp. 3-51.

P. N. Pusey, “Colloidal suspensions,” in Les Houches Session LI: Liquids, Freezing, and the Glass Transition, J. P. Hansen, D. Levesque, and J. Zinn-Justin, eds. (North-Holland, 1991), pp. 765-942.

Rarity, J. G.

P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
[CrossRef] [PubMed]

Rogers, R. B.

R. B. Rogers and K. P. D. Lagerlöf, “Crystallography of ordered colloids using optical microscopy. 1. Parallel-beam technique,” Appl. Opt. 47, 284-295 (2008).
[CrossRef] [PubMed]

R. B. Rogers, “The measurement of solid-liquid interfacial energy in colloidal suspensions using grain boundary grooves,” Ph.D. dissertation (Case Western Reserve University, 2006).

Russ, J. C.

J. C. Russ, The Image Processing Handbook (CRC Press, 1999).

Russel, W. B.

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

Shinohara, T.

T. Shinohara, H. Yamada, I. S. Sogami, N. Ise, and T. Yoshiyama, “Gravitational, vertical compression of colloidal crystals as studied by the Kossel diffraction method,” Langmuir 20, 5141-5144 (2004).
[CrossRef]

Sogami, I.

T. Yoshiyama, I. Sogami, and N. Ise, “Kossel line analysis on colloidal crystals in semidilute aqueous-solutions,” Phys. Rev. Lett. 53, 2153-2156 (1984).
[CrossRef]

Sogami, I. S.

T. Shinohara, H. Yamada, I. S. Sogami, N. Ise, and T. Yoshiyama, “Gravitational, vertical compression of colloidal crystals as studied by the Kossel diffraction method,” Langmuir 20, 5141-5144 (2004).
[CrossRef]

I. S. Sogami and T. Yoshiyama, “Kossel line analysis on crystallization in colloidal suspensions,” Phase Transitions 21, 171-182 (1990).
[CrossRef]

T. Yoshiyama and I. S. Sogami, “Photographic records of ordered and disordered structures in polymer latex dispersions,” Langmuir 3, 851-853 (1987).
[CrossRef]

Spring, K. R.

S. Inoue and K. R. Spring, Video Microscopy:The Fundamentals (Plenum, 1997).
[CrossRef]

Tannhauser, D. S.

S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge U. Press, 1995).

Tarhan, I. I.

I. I. Tarhan and G. H. Watson, “Photonic band structure of fcc colloidal crystals,” Phys. Rev. Lett. 76, 315-318 (1996).
[CrossRef] [PubMed]

Underwood, S. M.

P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
[CrossRef] [PubMed]

van Megen, W.

P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
[CrossRef] [PubMed]

P. N. Pusey and W. van Megen, “Phase-behavior of concentrated suspensions of nearly hard colloidal spheres,” Nature 320, 340-342 (1986).
[CrossRef]

Watson, G. H.

R. D. Pradhan, J. A. Bloodgood, and G. H. Watson, “Photonic band structure of bcc colloidal crystals,” Phys. Rev. B 55, 9503-9507 (1997).
[CrossRef]

I. I. Tarhan and G. H. Watson, “Photonic band structure of fcc colloidal crystals,” Phys. Rev. Lett. 76, 315-318 (1996).
[CrossRef] [PubMed]

Williams, D. B.

D. B. Williams and C. B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science (Plenum, 1996).

Wilson, A. J. C.

A. J. C. Wilson, “Imperfections in the structure of cobalt II. Mathematical treatment of proposed structure,” Proc. R. Soc. London Ser. A 180, 277-285 (1942).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062(1987).
[CrossRef] [PubMed]

Yamada, H.

T. Shinohara, H. Yamada, I. S. Sogami, N. Ise, and T. Yoshiyama, “Gravitational, vertical compression of colloidal crystals as studied by the Kossel diffraction method,” Langmuir 20, 5141-5144 (2004).
[CrossRef]

Yoshiyama, T.

T. Shinohara, H. Yamada, I. S. Sogami, N. Ise, and T. Yoshiyama, “Gravitational, vertical compression of colloidal crystals as studied by the Kossel diffraction method,” Langmuir 20, 5141-5144 (2004).
[CrossRef]

I. S. Sogami and T. Yoshiyama, “Kossel line analysis on crystallization in colloidal suspensions,” Phase Transitions 21, 171-182 (1990).
[CrossRef]

T. Yoshiyama and I. S. Sogami, “Photographic records of ordered and disordered structures in polymer latex dispersions,” Langmuir 3, 851-853 (1987).
[CrossRef]

T. Yoshiyama, I. Sogami, and N. Ise, “Kossel line analysis on colloidal crystals in semidilute aqueous-solutions,” Phys. Rev. Lett. 53, 2153-2156 (1984).
[CrossRef]

Zhu, J. X.

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

Appl. Opt. (1)

Contemp. Phys. (1)

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25-73(1983).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476-480 (1999).
[CrossRef]

J. Chem. Phys. (1)

Y. Monovoukas, G. G. Fuller, and A. P. Gast, “Optical anisotropy in colloidal crystals,” J. Chem. Phys. 93, 8294-8299(1990).
[CrossRef]

J. Colloid Interface Sci. (1)

Y. Monovoukas and A. P. Gast, “The experimental phase-diagram of charged colloidal suspensions,” J. Colloid Interface Sci. 128, 533-548 (1989).
[CrossRef]

Langmuir (2)

T. Yoshiyama and I. S. Sogami, “Photographic records of ordered and disordered structures in polymer latex dispersions,” Langmuir 3, 851-853 (1987).
[CrossRef]

T. Shinohara, H. Yamada, I. S. Sogami, N. Ise, and T. Yoshiyama, “Gravitational, vertical compression of colloidal crystals as studied by the Kossel diffraction method,” Langmuir 20, 5141-5144 (2004).
[CrossRef]

Nature (1)

P. N. Pusey and W. van Megen, “Phase-behavior of concentrated suspensions of nearly hard colloidal spheres,” Nature 320, 340-342 (1986).
[CrossRef]

Phase Transitions (2)

Y. Monovoukas and A. P. Gast, “Microstructure identification during crystallization of charged colloidal suspensions,” Phase Transitions 21, 183-195 (1990).
[CrossRef]

I. S. Sogami and T. Yoshiyama, “Kossel line analysis on crystallization in colloidal suspensions,” Phase Transitions 21, 171-182 (1990).
[CrossRef]

Phys. Rev. B (1)

R. D. Pradhan, J. A. Bloodgood, and G. H. Watson, “Photonic band structure of bcc colloidal crystals,” Phys. Rev. B 55, 9503-9507 (1997).
[CrossRef]

Phys. Rev. E (1)

S. E. Phan, W. B. Russel, Z. D. Cheng, J. X. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, “Phase transition, equation of state, and limiting shear viscosities of hard sphere dispersions,” Phys. Rev. E 54, 6633-6645 (1996).
[CrossRef]

Phys. Rev. Lett. (5)

P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres,” Phys. Rev. Lett. 63, 2753-2756 (1989).
[CrossRef] [PubMed]

T. Yoshiyama, I. Sogami, and N. Ise, “Kossel line analysis on colloidal crystals in semidilute aqueous-solutions,” Phys. Rev. Lett. 53, 2153-2156 (1984).
[CrossRef]

I. I. Tarhan and G. H. Watson, “Photonic band structure of fcc colloidal crystals,” Phys. Rev. Lett. 76, 315-318 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062(1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489(1987).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

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Figures (13)

Fig. 1
Fig. 1

Geometry of the Ewald sphere construction.

Fig. 2
Fig. 2

Laue cone geometry constructed by rotating vectors S and S 0 about σ h k l .

Fig. 3
Fig. 3

Comparison of Kossel patterns from sin ψ and tan ψ scaling relationships. Center to left, sin ψ ; center to right, tan ψ . Solid curves, circles or ellipses; short dashes, hyperbolae; long dashes, straight line. Outer circle, scattering at 90 ° for the sin ψ relationship. Scaling of the tan ψ curves is arbitrary with respect to the sin ψ curves.

Fig. 4
Fig. 4

Geometry of divergent-beam forward scattering patterns for a Kossel pair. (a) divergent-beam pattern with the OA outside of the Kossel cones. The x y reference system used for analysis is shown along with the points of the Kossel patterns used for analysis, ( x 1 , y 1 ) and ( x 2 , y 2 ) . (b) scattering geometry in the plane containing the OA, σ, S 0 , and S for the divergent-beam pattern shown in (a). (c) and (d) correspond to (a) and (b) for the case where the OA falls within one of the Kossel cones.

Fig. 5
Fig. 5

Geometry of divergent-beam backscatter patterns for a single Kossel cone. (a) divergent-beam pattern with OA outside of the Kossel cone. (b) scattering geometry in the plane containing the OA, σ, S 0 , and S for the divergent-beam pattern shown in (a). (c) and (d) correspond to (a) and (b) for the case where the OA falls within the Kossel cone.

Fig. 6
Fig. 6

Divergent-beam image with a swab stem in the front focal plane of the condenser, blocking a portion of the scattering. Kossel pairs are marked with pairs of white arrows having the same orientation.

Fig. 7
Fig. 7

Comparison of divergent-beam imaging modes: (a) bright-field optics, (b) optimized DIC optics.

Fig. 8
Fig. 8

Examples of divergent-beam patterns seen in colloidal crystals formed from 0.504 μm diameter spheres. The white arrows in (a) show examples of Kossel pairs used in analysis. Arrows having the same orientation mark corresponding pairs, and each arrow is perpendicular to the feature being identified.

Fig. 9
Fig. 9

Divergent-beam pattern from the same crystal examined with the parallel-beam technique of part 1 [6]. Two different camera exposures are shown because the intensity range of the scattered light exceeded the dynamic range of the camera. The dark annular ring is the phase plate in the back focal plane of the phase contrast microscope objective. (a) Shorter exposure emphasizing the bright Kossel pair running diagonally across the image. (b) Longer exposure showing further detail in the weaker Kossel patterns.

Fig. 10
Fig. 10

Reciprocal lattice data extracted from Fig. 8a. Filled circles are extracted Bragg rod data points. Open circles are select reciprocal lattice points derived from the extracted data. Views in direction of (a)  c * , (b)  ( c * × a 1 * ) , (c)  a 1 * .

Fig. 11
Fig. 11

Indexed Kossel patterns corresponding to Fig. 8b.

Fig. 12
Fig. 12

Indexed Kossel patterns corresponding to Fig. 9. The center circles mark the edges of the phase plate seen in Fig. 9.

Fig. 13
Fig. 13

Stereographic projection of the orientation of the crystallite in Fig. 9, showing the standard ( 0001 ) hcp projection for ideal close-packed c / a . The three-symbol indices show the corresponding fcc planes. The small symbols mark the orientation of the OA and the large symbols mark the orientation of the + x axis in the microscope frame of reference. Filled symbols are in the ( 0001 ) hemisphere and open symbols are in the ( 000 1 ¯ ) hemisphere. Circles, parallel-beam results from [6]; squares, divergent-beam results from this work.

Equations (9)

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λ 0 = 2 n d h k l sin θ B ,
φ = 90 ° θ B .
| σ h k l | = K 1 d h k l ,
| σ h k l | = 2 n λ 0 sin θ B .
σ h k l = S S 0 .
r = L sin ψ ,
δ = ϕ + ψ 1 if the OA lies outside the Kossel cone , δ = ϕ ψ 1 if the OA lies inside the Kossel cone ,
δ = φ ± ψ 1 , ε = sin 1 ( y r ) for   y 0 , ε = sin 1 ( y r ) for   y < 0 , ρ = | σ h k l | = 2 n λ 0 sin θ B ,
x R = ρ sin δ cos ε , y R = ρ sin δ sin ε , z R = ρ cos δ .

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