Abstract

A technique is described for determining the reciprocal lattice basis vectors of randomly oriented colloidal crystals from optical Fourier transform images using a parallel incident beam. This approach is demonstrated by comparing information gathered using this technique with results from real-space images for a single colloidal crystal grain with a random hexagonal close-packed structure. The reciprocal space and available real-space results agreed to within experimental error. The complete set of reciprocal lattice basis vectors was determined using the proposed technique in contrast to the partial lattice information available from real-space images. This technique appears to be generally capable of measuring lattice parameters to within 1% and orientation to better than 1°.

© 2008 Optical Society of America

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References

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
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  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
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  5. 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. F. Hull, and J. -P. Ryckaert, eds. (Kluwer Academic Publishers, 1995), pp. 3-51.
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    [CrossRef] [PubMed]
  7. P. Bartlett, R. H. Ottewill, and P. N. Pusey, "Superlattice formation in binary-mixtures of hard-sphere colloids," Phys. Rev. Lett. 68, 3801-3804 (1992).
    [CrossRef] [PubMed]
  8. Z. D. Cheng, J. X. Zhu, W. B. Russel, W. V. Meyer, and P. M. Chaikin, "Colloidal hard-sphere crystallization kinetics in microgravity and normal gravity," Appl. Opt. 40, 4146-4151 (2001).
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    [CrossRef]
  13. N. A. M. Verhaegh, J. S. Vanduijneveldt, A. van Blaaderen, and H. N. W. Lekkerkerker, "Direct observation of stacking disorder in a colloidal crystal," J. Chem. Phys. 102, 1416-1421 (1995).
    [CrossRef]
  14. M. S. Elliot, B. T. F. Bristol, and W. C. K. Poon, "Direct measurement of stacking disorder in hard-sphere colloidal crystals," Physica A 235, 216-223 (1997).
    [CrossRef]
  15. A. van Blaaderen and P. Wiltzius, "Real-space structure of colloidal hard-sphere glasses," Science 270, 1177-1179 (1995).
    [CrossRef]
  16. E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, "Three-dimensional direct imaging of structural relaxation near the colloidal glass transition," Science 287, 627-631 (2000).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  18. M. S. Elliot and W. C. K. Poon, "Conventional optical microscopy of colloidal suspensions," Adv. Colloid Interface Sci. 92, 133-194 (2001).
    [CrossRef] [PubMed]
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  20. L. Azaroff, Elements of X-Ray Crystallography (McGraw-Hill, 1968).
  21. M. Pluta, Advanced Light Microscopy. Vol. 1. Principles and Basic Properties (Elsevier, 1988).
  22. S. Inoue and K. R. Spring, Video Microscopy: The Fundamentals (Plenum, 1997).
    [CrossRef]
  23. S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge U. Press, 1995).
  24. 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]
  25. P. N. Pusey and W. van Megen, "Phase-behavior of concentrated suspensions of nearly hard colloidal spheres," Nature 320, 340-342 (1986).
    [CrossRef]
  26. S. E. Paulin and B. J. Ackerson, "Observation of a phase-transition in the sedimentation-velocity of hard-spheres," Phys. Rev. Lett. 64, 2663-2666 (1990).
    [CrossRef] [PubMed]
  27. E. Hecht, Optics (Addison-Wesley, 1987).
  28. J. C. Russ, The Image Processing Handbook (CRC Press, 1999).
  29. J. V. Sanders, "Colour of precious opal," Nature 204, 1151-1153 (1964).
    [CrossRef]
  30. Y. Monovoukas and A. P. Gast, "A study of colloidal crystal morphology and orientation via polarizing microscopy," Langmuir 7, 460-468 (1991).
    [CrossRef]
  31. 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]
  32. V. C. Martelozzo, A. B. Schofield, W. C. K. Poon, and P. N. Pusey, "Structural aging of crystals of hard-sphere colloids," Phys. Rev. E 66, 021408 (2002).
    [CrossRef]
  33. A. Guinier, X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies (W. H. Freeman, 1963).

2002 (1)

V. C. Martelozzo, A. B. Schofield, W. C. K. Poon, and P. N. Pusey, "Structural aging of crystals of hard-sphere colloids," Phys. Rev. E 66, 021408 (2002).
[CrossRef]

2001 (3)

Z. D. Cheng, J. X. Zhu, W. B. Russel, W. V. Meyer, and P. M. Chaikin, "Colloidal hard-sphere crystallization kinetics in microgravity and normal gravity," Appl. Opt. 40, 4146-4151 (2001).
[CrossRef]

U. Gasser, E. R. Weeks, A. Schofield, P. N. Pusey, and D. A. Weitz, "Real-space imaging of nucleation and growth in colloidal crystallization," Science 292, 258-262 (2001).
[CrossRef] [PubMed]

M. S. Elliot and W. C. K. Poon, "Conventional optical microscopy of colloidal suspensions," Adv. Colloid Interface Sci. 92, 133-194 (2001).
[CrossRef] [PubMed]

2000 (2)

E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, "Three-dimensional direct imaging of structural relaxation near the colloidal glass transition," Science 287, 627-631 (2000).
[CrossRef] [PubMed]

R. M. Amos, J. G. Rarity, P. R. Tapster, T. J. Shepherd, and S. C. Kitson, "Fabrication of large-area face-centered-cubic hard-sphere colloidal crystals by shear alignment," Phys. Rev. E 61, 2929-2935 (2000).
[CrossRef]

1997 (1)

M. S. Elliot, B. T. F. Bristol, and W. C. K. Poon, "Direct measurement of stacking disorder in hard-sphere colloidal crystals," Physica A 235, 216-223 (1997).
[CrossRef]

1996 (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]

1995 (2)

A. van Blaaderen and P. Wiltzius, "Real-space structure of colloidal hard-sphere glasses," Science 270, 1177-1179 (1995).
[CrossRef]

N. A. M. Verhaegh, J. S. Vanduijneveldt, A. van Blaaderen, and H. N. W. Lekkerkerker, "Direct observation of stacking disorder in a colloidal crystal," J. Chem. Phys. 102, 1416-1421 (1995).
[CrossRef]

1992 (1)

P. Bartlett, R. H. Ottewill, and P. N. Pusey, "Superlattice formation in binary-mixtures of hard-sphere colloids," Phys. Rev. Lett. 68, 3801-3804 (1992).
[CrossRef] [PubMed]

1991 (1)

Y. Monovoukas and A. P. Gast, "A study of colloidal crystal morphology and orientation via polarizing microscopy," Langmuir 7, 460-468 (1991).
[CrossRef]

1990 (1)

S. E. Paulin and B. J. Ackerson, "Observation of a phase-transition in the sedimentation-velocity of hard-spheres," Phys. Rev. Lett. 64, 2663-2666 (1990).
[CrossRef] [PubMed]

1989 (1)

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 (2)

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]

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]

1981 (1)

B. J. Ackerson and N. A. Clark, "Shear-induced melting," Phys. Rev. Lett. 46, 123-126 (1981).
[CrossRef]

1968 (1)

J. V. Sanders, "Diffraction of light by opals," Acta Crystallogr. 24, 427-434 (1968).
[CrossRef]

1964 (1)

J. V. Sanders, "Colour of precious opal," Nature 204, 1151-1153 (1964).
[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]

Acta Crystallogr. (1)

J. V. Sanders, "Diffraction of light by opals," Acta Crystallogr. 24, 427-434 (1968).
[CrossRef]

Adv. Colloid Interface Sci. (1)

M. S. Elliot and W. C. K. Poon, "Conventional optical microscopy of colloidal suspensions," Adv. Colloid Interface Sci. 92, 133-194 (2001).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Chem. Phys. (1)

N. A. M. Verhaegh, J. S. Vanduijneveldt, A. van Blaaderen, and H. N. W. Lekkerkerker, "Direct observation of stacking disorder in a colloidal crystal," J. Chem. Phys. 102, 1416-1421 (1995).
[CrossRef]

Langmuir (1)

Y. Monovoukas and A. P. Gast, "A study of colloidal crystal morphology and orientation via polarizing microscopy," Langmuir 7, 460-468 (1991).
[CrossRef]

Nature (2)

J. V. Sanders, "Colour of precious opal," Nature 204, 1151-1153 (1964).
[CrossRef]

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

Phys. Rev. E (3)

R. M. Amos, J. G. Rarity, P. R. Tapster, T. J. Shepherd, and S. C. Kitson, "Fabrication of large-area face-centered-cubic hard-sphere colloidal crystals by shear alignment," Phys. Rev. E 61, 2929-2935 (2000).
[CrossRef]

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]

V. C. Martelozzo, A. B. Schofield, W. C. K. Poon, and P. N. Pusey, "Structural aging of crystals of hard-sphere colloids," Phys. Rev. E 66, 021408 (2002).
[CrossRef]

Phys. Rev. Lett. (6)

B. J. Ackerson and N. A. Clark, "Shear-induced melting," Phys. Rev. Lett. 46, 123-126 (1981).
[CrossRef]

S. E. Paulin and B. J. Ackerson, "Observation of a phase-transition in the sedimentation-velocity of hard-spheres," Phys. Rev. Lett. 64, 2663-2666 (1990).
[CrossRef] [PubMed]

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. Bartlett, R. H. Ottewill, and P. N. Pusey, "Superlattice formation in binary-mixtures of hard-sphere colloids," Phys. Rev. Lett. 68, 3801-3804 (1992).
[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]

Physica A (1)

M. S. Elliot, B. T. F. Bristol, and W. C. K. Poon, "Direct measurement of stacking disorder in hard-sphere colloidal crystals," Physica A 235, 216-223 (1997).
[CrossRef]

Proc. R. Soc. London (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]

Science (3)

A. van Blaaderen and P. Wiltzius, "Real-space structure of colloidal hard-sphere glasses," Science 270, 1177-1179 (1995).
[CrossRef]

E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, "Three-dimensional direct imaging of structural relaxation near the colloidal glass transition," Science 287, 627-631 (2000).
[CrossRef] [PubMed]

U. Gasser, E. R. Weeks, A. Schofield, P. N. Pusey, and D. A. Weitz, "Real-space imaging of nucleation and growth in colloidal crystallization," Science 292, 258-262 (2001).
[CrossRef] [PubMed]

Other (12)

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

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

M. Pluta, Advanced Light Microscopy. Vol. 1. Principles and Basic Properties (Elsevier, 1988).

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

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

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

P. N. Pusey, "Colloidal suspensions," in Les Houches Session L1: Liquids, Freezing, and the Glass Transition, J. P. Hansen, D. Levesque, and J. Zinn-Justin, eds. (North-Holland, 1991), pp. 765-942.

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. F. Hull, and J. -P. Ryckaert, eds. (Kluwer Academic Publishers, 1995), pp. 3-51.

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).
[PubMed]

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

E. Hecht, Optics (Addison-Wesley, 1987).

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

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Figures (7)

Fig. 1
Fig. 1

Geometric relationships pertinent to extracting reciprocal lattice data from experimental OFT images. (a) Representational sketch of an OFT image having hexagonal symmetry about the direct beam; (b) geometry of the Ewald sphere construction; and (c) sketch showing the relationship between the elements of part (b) and the reciprocal lattice coordinate system.

Fig. 2
Fig. 2

Micrographs of crystallite studied in this paper: (a) dark-field image and (b) phase contrast image of the same region. Portions of adjacent grains can also be seen.

Fig. 3
Fig. 3

OFT images of crystallite shown in Fig. 2. (a) White light OFT image. The dark ring intersecting the central beam is the phase plate in the exit pupil of the phase contrast objective. (b) Corresponding OFT image with a 546 nm filter.

Fig. 4
Fig. 4

Geometry of tilted close-packed planes. FP, focal plane; d cp, distance between close-packed planes; and γ, angle between focal plane and close-packed planes.

Fig. 5
Fig. 5

Indexed OFT data. Data are represented by filled circles. The origin of the reciprocal lattices is shown as an open circle. The vertical lines are the linear fits to Bragg rods as discussed in the text. For the ( 2 ¯ 20 l ) and ( 3 ¯ 21 l ) rods, which contain only one data point each, the rss average direction of the other four rods was used. (a) View in the direction of c * ; (b) view in the direction of ( c * × a 1 * ) ; (c) view in the direction of a 1 * .

Fig. 6
Fig. 6

Stereographic projection of crystal orientation. The figure shows the standard (0001) hcp projection for ideal close-packed c∕a. The three-symbol indices show the corresponding fcc planes. The small circle marks the orientation of the OA, and the large circle marks the orientation of the +x-axis in the microscope frame of reference. The filled circle is in the (0001) hemisphere, and the open circle is in the ( 000 1 ¯ ) hemisphere.

Fig. 7
Fig. 7

Kossel lines in parallel-beam image. (a) Light and dark Kossel line pair in a 436   nm OFT image; (b) same image with calculated position of (0002) Kossel line pair superimposed using the crystal orientation determined using the method described in the text. The calculated edges of the phase plate in the exit pupil of the microscope are also shown.

Tables (1)

Tables Icon

Table 1 Comparison of Data from Real Space and Reciprocal Space Measurements

Equations (18)

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λ 0 = 2 n d h k l sin θ B ,
σ h k l = S S 0 .
r = L sin ψ ,
x = x x OA , y = y OA y ,
ψ = sin 1 ( r L ) ,
ε = { sin 1 ( y r ) f o r y 0 sin 1 ( y r ) f o r y < 0 ,
ρ = n λ 0 ,
x R = ρ sin ψ cos ε ,
y R = ρ sin ψ sin ε ,
z R = ρ cos ψ .
m λ 0 = a n sin 2 θ ,
L = r a n m λ 0 ,
ϕ = 2 3 π ( d a ) 3 ,
α = 1 κ N 2 ,
v a * = 0.
y = cos β ( x x 1 ) cos α + y 1 ,
z = cos γ ( x x 1 ) cos α + z 1 ,
x = x 1 ( cos 2 β + cos 2 γ ) cos α ( y 1 cos β + z 1 cos γ ) ,

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