Abstract

Expressions are derived for diffraction by the triangular Ising antiferromagnet, a disordered lattice system consisting of two kinds of scatterer and exhibiting geometric frustration. Analysis of the expressions shows characteristics of the diffraction patterns, including the presence of Bragg and diffuse diffraction, superlattice reflections, and their behavior with temperature. These characteristics are illustrated by numerical simulations. The results have application to diffraction imaging of disordered systems.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. M. Woolfson, An Introduction to X-ray Crystallography, 2nd ed. (Cambridge University, 1997).
  2. T. R. Welberry, Diffuse X-Ray Scattering and Models of Disorder (Oxford University, 2004).
  3. S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
    [CrossRef]
  4. D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
    [CrossRef]
  5. R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
    [CrossRef]
  6. Y. Qi, T. Brintlinger, and J. Cumings, “Direct observations of the ice rule in artificial kagome spin ice,” Phys. Rev. B 77, 094418 (2008).
    [CrossRef]
  7. E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
    [CrossRef]
  8. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010).
    [CrossRef]
  9. Y. Han, “Geometric frustration in buckled colloidal monolayers,” Nature 456, 898–903 (2008).
    [CrossRef]
  10. P. K. Luther and J. M. Squire, “Three-dimensional structure of the vertebrate muscle A-band. II. The myosin filament superlattice,” J. Mol. Biol. 141, 409–439 (1980).
    [CrossRef]
  11. C. H. Yoon, “Image analysis and diffraction by the myosin lattice of vertebrate muscle,” Ph.D. thesis (University of Canterbury, 2008).
  12. T. R. Welberry, A. P. Heerdegen, D. C. Goldstone, and I. A. Taylor, “Diffuse scattering resulting from macromolecular frustration,” Acta Crystallogr. B67, 516–524 (2011).
    [CrossRef]
  13. M. Bretz, “Ordered helium films on highly uniform graphite—finite size effects, critical parameters, and the three-state Potts model,” Phys. Rev. Lett. 38, 501–505 (1977).
    [CrossRef]
  14. A. N. Berker, S. Ostlund, and F. A. Putnam, “Renormalization-group treatment of a Potts lattice gas for krypton adsorbed onto graphite,” Phys. Rev. B 17, 3650–3665 (1978).
    [CrossRef]
  15. J. L. Jacobsen and H. C. Fogedby, “Monte Carlo study of correlations near the ground state of the triangular antiferromagnetic Ising model,” Phys. A 246, 563–575 (1997).
    [CrossRef]
  16. R. Liebmann, Statistical Mechanics of Periodic Frustrated Ising Systems (Springer-Verlag, 1986).
  17. A. P. Ramirez, “Geometric frustration: magic moments,” Nature 421, 483 (2003).
    [CrossRef]
  18. A. P. Ramirez, “Geometrically frustrated matter—magnets to molecules,” MRS Bull. 30, 447–451 (2005).
    [CrossRef]
  19. M. Harris, “The eternal triangle,” Nature 456, 886–887 (2008).
    [CrossRef]
  20. G. H. Wannier, “Antiferromagnetism: the triangular Ising net,” Phys. Rev. 79, 357–364 (1950).
    [CrossRef]
  21. J. Stephenson, “Ising-model spin correlations on the triangular lattice,” J. Math. Phys. 5, 1009–1024 (1964).
    [CrossRef]
  22. J. Stephenson, “Ising-model spin correlations on the triangular lattice. III. Isotropic antiferromagnetic lattice,” J. Math. Phys. 11, 413–419 (1970).
    [CrossRef]
  23. D. J. Wojtas and R. P. Millane, “The two-point correlation function for the triangular Ising antiferromagnet,” Phys. Rev. E 79, 041123 (2009).
    [CrossRef]
  24. W. J. Stroud and R. P. Millane, “Cylindrically averaged diffraction by distorted lattices,” Proc. R. Soc. London Ser. A 452, 151–173 (1996).
    [CrossRef]
  25. E. Rastelli, S. Regina, and A. Tassi, “Monte Carlo simulations on a triangular Ising antiferromagnet with nearest and next-nearest interactions,” Phys. Rev. B 71, 174406 (2005).
    [CrossRef]

2011 (1)

T. R. Welberry, A. P. Heerdegen, D. C. Goldstone, and I. A. Taylor, “Diffuse scattering resulting from macromolecular frustration,” Acta Crystallogr. B67, 516–524 (2011).
[CrossRef]

2010 (1)

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010).
[CrossRef]

2009 (1)

D. J. Wojtas and R. P. Millane, “The two-point correlation function for the triangular Ising antiferromagnet,” Phys. Rev. E 79, 041123 (2009).
[CrossRef]

2008 (4)

Y. Han, “Geometric frustration in buckled colloidal monolayers,” Nature 456, 898–903 (2008).
[CrossRef]

Y. Qi, T. Brintlinger, and J. Cumings, “Direct observations of the ice rule in artificial kagome spin ice,” Phys. Rev. B 77, 094418 (2008).
[CrossRef]

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

M. Harris, “The eternal triangle,” Nature 456, 886–887 (2008).
[CrossRef]

2006 (1)

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

2005 (2)

A. P. Ramirez, “Geometrically frustrated matter—magnets to molecules,” MRS Bull. 30, 447–451 (2005).
[CrossRef]

E. Rastelli, S. Regina, and A. Tassi, “Monte Carlo simulations on a triangular Ising antiferromagnet with nearest and next-nearest interactions,” Phys. Rev. B 71, 174406 (2005).
[CrossRef]

2003 (1)

A. P. Ramirez, “Geometric frustration: magic moments,” Nature 421, 483 (2003).
[CrossRef]

1997 (2)

J. L. Jacobsen and H. C. Fogedby, “Monte Carlo study of correlations near the ground state of the triangular antiferromagnetic Ising model,” Phys. A 246, 563–575 (1997).
[CrossRef]

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

1996 (2)

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

W. J. Stroud and R. P. Millane, “Cylindrically averaged diffraction by distorted lattices,” Proc. R. Soc. London Ser. A 452, 151–173 (1996).
[CrossRef]

1980 (1)

P. K. Luther and J. M. Squire, “Three-dimensional structure of the vertebrate muscle A-band. II. The myosin filament superlattice,” J. Mol. Biol. 141, 409–439 (1980).
[CrossRef]

1978 (1)

A. N. Berker, S. Ostlund, and F. A. Putnam, “Renormalization-group treatment of a Potts lattice gas for krypton adsorbed onto graphite,” Phys. Rev. B 17, 3650–3665 (1978).
[CrossRef]

1977 (1)

M. Bretz, “Ordered helium films on highly uniform graphite—finite size effects, critical parameters, and the three-state Potts model,” Phys. Rev. Lett. 38, 501–505 (1977).
[CrossRef]

1970 (1)

J. Stephenson, “Ising-model spin correlations on the triangular lattice. III. Isotropic antiferromagnetic lattice,” J. Math. Phys. 11, 413–419 (1970).
[CrossRef]

1964 (1)

J. Stephenson, “Ising-model spin correlations on the triangular lattice,” J. Math. Phys. 5, 1009–1024 (1964).
[CrossRef]

1950 (1)

G. H. Wannier, “Antiferromagnetism: the triangular Ising net,” Phys. Rev. 79, 357–364 (1950).
[CrossRef]

Berker, A. N.

A. N. Berker, S. Ostlund, and F. A. Putnam, “Renormalization-group treatment of a Potts lattice gas for krypton adsorbed onto graphite,” Phys. Rev. B 17, 3650–3665 (1978).
[CrossRef]

Bisig, A.

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

Bramwell, S. T.

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

Branford, W. R.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010).
[CrossRef]

Braun, H. B.

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

Bretz, M.

M. Bretz, “Ordered helium films on highly uniform graphite—finite size effects, critical parameters, and the three-state Potts model,” Phys. Rev. Lett. 38, 501–505 (1977).
[CrossRef]

Brintlinger, T.

Y. Qi, T. Brintlinger, and J. Cumings, “Direct observations of the ice rule in artificial kagome spin ice,” Phys. Rev. B 77, 094418 (2008).
[CrossRef]

Carling, S. G.

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

Cohen, L. F.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010).
[CrossRef]

Cooley, B. J.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Crespi, V. H.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Cumings, J.

Y. Qi, T. Brintlinger, and J. Cumings, “Direct observations of the ice rule in artificial kagome spin ice,” Phys. Rev. B 77, 094418 (2008).
[CrossRef]

Davidovic, D.

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Field, S. B.

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Fogedby, H. C.

J. L. Jacobsen and H. C. Fogedby, “Monte Carlo study of correlations near the ground state of the triangular antiferromagnetic Ising model,” Phys. A 246, 563–575 (1997).
[CrossRef]

Freitas, R. S.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Goldstone, D. C.

T. R. Welberry, A. P. Heerdegen, D. C. Goldstone, and I. A. Taylor, “Diffuse scattering resulting from macromolecular frustration,” Acta Crystallogr. B67, 516–524 (2011).
[CrossRef]

Han, Y.

Y. Han, “Geometric frustration in buckled colloidal monolayers,” Nature 456, 898–903 (2008).
[CrossRef]

Harding, C. J.

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

Harris, K. D. M.

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

Harris, M.

M. Harris, “The eternal triangle,” Nature 456, 886–887 (2008).
[CrossRef]

Heerdegen, A. P.

T. R. Welberry, A. P. Heerdegen, D. C. Goldstone, and I. A. Taylor, “Diffuse scattering resulting from macromolecular frustration,” Acta Crystallogr. B67, 516–524 (2011).
[CrossRef]

Hey, R.

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Heyderman, L. J.

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

Jacobsen, J. L.

J. L. Jacobsen and H. C. Fogedby, “Monte Carlo study of correlations near the ground state of the triangular antiferromagnetic Ising model,” Phys. A 246, 563–575 (1997).
[CrossRef]

Kariuki, B. M.

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

Kumar, S.

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Ladak, S.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010).
[CrossRef]

Le Guyader, L.

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

Leighton, C.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Li, J.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Liebmann, R.

R. Liebmann, Statistical Mechanics of Periodic Frustrated Ising Systems (Springer-Verlag, 1986).

Lund, M. S.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Luther, P. K.

P. K. Luther and J. M. Squire, “Three-dimensional structure of the vertebrate muscle A-band. II. The myosin filament superlattice,” J. Mol. Biol. 141, 409–439 (1980).
[CrossRef]

McConville, W.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Mengotti, E.

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

Millane, R. P.

D. J. Wojtas and R. P. Millane, “The two-point correlation function for the triangular Ising antiferromagnet,” Phys. Rev. E 79, 041123 (2009).
[CrossRef]

W. J. Stroud and R. P. Millane, “Cylindrically averaged diffraction by distorted lattices,” Proc. R. Soc. London Ser. A 452, 151–173 (1996).
[CrossRef]

Nisoli, C.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Nixon, L.

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

Nolting, F.

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

Ostlund, S.

A. N. Berker, S. Ostlund, and F. A. Putnam, “Renormalization-group treatment of a Potts lattice gas for krypton adsorbed onto graphite,” Phys. Rev. B 17, 3650–3665 (1978).
[CrossRef]

Parkin, I. P.

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

Perkins, G. K.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010).
[CrossRef]

Ploog, K.

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Putnam, F. A.

A. N. Berker, S. Ostlund, and F. A. Putnam, “Renormalization-group treatment of a Potts lattice gas for krypton adsorbed onto graphite,” Phys. Rev. B 17, 3650–3665 (1978).
[CrossRef]

Qi, Y.

Y. Qi, T. Brintlinger, and J. Cumings, “Direct observations of the ice rule in artificial kagome spin ice,” Phys. Rev. B 77, 094418 (2008).
[CrossRef]

Ramirez, A. P.

A. P. Ramirez, “Geometrically frustrated matter—magnets to molecules,” MRS Bull. 30, 447–451 (2005).
[CrossRef]

A. P. Ramirez, “Geometric frustration: magic moments,” Nature 421, 483 (2003).
[CrossRef]

Rastelli, E.

E. Rastelli, S. Regina, and A. Tassi, “Monte Carlo simulations on a triangular Ising antiferromagnet with nearest and next-nearest interactions,” Phys. Rev. B 71, 174406 (2005).
[CrossRef]

Read, D. E.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010).
[CrossRef]

Regina, S.

E. Rastelli, S. Regina, and A. Tassi, “Monte Carlo simulations on a triangular Ising antiferromagnet with nearest and next-nearest interactions,” Phys. Rev. B 71, 174406 (2005).
[CrossRef]

Reich, D. H.

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Rodríguez, A. F.

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

Samarth, N.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Schiffer, P.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Siegel, J.

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Squire, J. M.

P. K. Luther and J. M. Squire, “Three-dimensional structure of the vertebrate muscle A-band. II. The myosin filament superlattice,” J. Mol. Biol. 141, 409–439 (1980).
[CrossRef]

Stephenson, J.

J. Stephenson, “Ising-model spin correlations on the triangular lattice. III. Isotropic antiferromagnetic lattice,” J. Math. Phys. 11, 413–419 (1970).
[CrossRef]

J. Stephenson, “Ising-model spin correlations on the triangular lattice,” J. Math. Phys. 5, 1009–1024 (1964).
[CrossRef]

Stroud, W. J.

W. J. Stroud and R. P. Millane, “Cylindrically averaged diffraction by distorted lattices,” Proc. R. Soc. London Ser. A 452, 151–173 (1996).
[CrossRef]

Tassi, A.

E. Rastelli, S. Regina, and A. Tassi, “Monte Carlo simulations on a triangular Ising antiferromagnet with nearest and next-nearest interactions,” Phys. Rev. B 71, 174406 (2005).
[CrossRef]

Taylor, I. A.

T. R. Welberry, A. P. Heerdegen, D. C. Goldstone, and I. A. Taylor, “Diffuse scattering resulting from macromolecular frustration,” Acta Crystallogr. B67, 516–524 (2011).
[CrossRef]

Tiberio, R. C.

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Wang, R. F.

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

Wannier, G. H.

G. H. Wannier, “Antiferromagnetism: the triangular Ising net,” Phys. Rev. 79, 357–364 (1950).
[CrossRef]

Welberry, T. R.

T. R. Welberry, A. P. Heerdegen, D. C. Goldstone, and I. A. Taylor, “Diffuse scattering resulting from macromolecular frustration,” Acta Crystallogr. B67, 516–524 (2011).
[CrossRef]

T. R. Welberry, Diffuse X-Ray Scattering and Models of Disorder (Oxford University, 2004).

Wojtas, D. J.

D. J. Wojtas and R. P. Millane, “The two-point correlation function for the triangular Ising antiferromagnet,” Phys. Rev. E 79, 041123 (2009).
[CrossRef]

Woolfson, M. M.

M. M. Woolfson, An Introduction to X-ray Crystallography, 2nd ed. (Cambridge University, 1997).

Yoon, C. H.

C. H. Yoon, “Image analysis and diffraction by the myosin lattice of vertebrate muscle,” Ph.D. thesis (University of Canterbury, 2008).

Acta Crystallogr. (1)

T. R. Welberry, A. P. Heerdegen, D. C. Goldstone, and I. A. Taylor, “Diffuse scattering resulting from macromolecular frustration,” Acta Crystallogr. B67, 516–524 (2011).
[CrossRef]

J. Math. Phys. (2)

J. Stephenson, “Ising-model spin correlations on the triangular lattice,” J. Math. Phys. 5, 1009–1024 (1964).
[CrossRef]

J. Stephenson, “Ising-model spin correlations on the triangular lattice. III. Isotropic antiferromagnetic lattice,” J. Math. Phys. 11, 413–419 (1970).
[CrossRef]

J. Mol. Biol. (1)

P. K. Luther and J. M. Squire, “Three-dimensional structure of the vertebrate muscle A-band. II. The myosin filament superlattice,” J. Mol. Biol. 141, 409–439 (1980).
[CrossRef]

J. Phys. Condens. Matter (1)

S. T. Bramwell, S. G. Carling, C. J. Harding, K. D. M. Harris, B. M. Kariuki, L. Nixon, and I. P. Parkin, “The anhydrous alums as model triangular-lattice magnets,” J. Phys. Condens. Matter 8, L123–L129 (1996).
[CrossRef]

MRS Bull. (1)

A. P. Ramirez, “Geometrically frustrated matter—magnets to molecules,” MRS Bull. 30, 447–451 (2005).
[CrossRef]

Nat. Phys. (1)

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010).
[CrossRef]

Nature (4)

Y. Han, “Geometric frustration in buckled colloidal monolayers,” Nature 456, 898–903 (2008).
[CrossRef]

R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, “Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands,” Nature 439, 303–306 (2006).
[CrossRef]

A. P. Ramirez, “Geometric frustration: magic moments,” Nature 421, 483 (2003).
[CrossRef]

M. Harris, “The eternal triangle,” Nature 456, 886–887 (2008).
[CrossRef]

Phys. A (1)

J. L. Jacobsen and H. C. Fogedby, “Monte Carlo study of correlations near the ground state of the triangular antiferromagnetic Ising model,” Phys. A 246, 563–575 (1997).
[CrossRef]

Phys. Rev. (1)

G. H. Wannier, “Antiferromagnetism: the triangular Ising net,” Phys. Rev. 79, 357–364 (1950).
[CrossRef]

Phys. Rev. B (5)

E. Rastelli, S. Regina, and A. Tassi, “Monte Carlo simulations on a triangular Ising antiferromagnet with nearest and next-nearest interactions,” Phys. Rev. B 71, 174406 (2005).
[CrossRef]

A. N. Berker, S. Ostlund, and F. A. Putnam, “Renormalization-group treatment of a Potts lattice gas for krypton adsorbed onto graphite,” Phys. Rev. B 17, 3650–3665 (1978).
[CrossRef]

Y. Qi, T. Brintlinger, and J. Cumings, “Direct observations of the ice rule in artificial kagome spin ice,” Phys. Rev. B 77, 094418 (2008).
[CrossRef]

E. Mengotti, L. J. Heyderman, A. F. Rodríguez, A. Bisig, L. Le Guyader, F. Nolting, and H. B. Braun, “Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands,” Phys. Rev. B 78, 144402 (2008).
[CrossRef]

D. Davidovic, S. Kumar, D. H. Reich, J. Siegel, S. B. Field, R. C. Tiberio, R. Hey, and K. Ploog, “Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings,” Phys. Rev. B 55, 6518–6540 (1997).
[CrossRef]

Phys. Rev. E (1)

D. J. Wojtas and R. P. Millane, “The two-point correlation function for the triangular Ising antiferromagnet,” Phys. Rev. E 79, 041123 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

M. Bretz, “Ordered helium films on highly uniform graphite—finite size effects, critical parameters, and the three-state Potts model,” Phys. Rev. Lett. 38, 501–505 (1977).
[CrossRef]

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

W. J. Stroud and R. P. Millane, “Cylindrically averaged diffraction by distorted lattices,” Proc. R. Soc. London Ser. A 452, 151–173 (1996).
[CrossRef]

Other (4)

C. H. Yoon, “Image analysis and diffraction by the myosin lattice of vertebrate muscle,” Ph.D. thesis (University of Canterbury, 2008).

R. Liebmann, Statistical Mechanics of Periodic Frustrated Ising Systems (Springer-Verlag, 1986).

M. M. Woolfson, An Introduction to X-ray Crystallography, 2nd ed. (Cambridge University, 1997).

T. R. Welberry, Diffuse X-Ray Scattering and Models of Disorder (Oxford University, 2004).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1.
Fig. 1.

Triangular lattice with coordinate systems (x,y) and (m,n).

Fig. 2.
Fig. 2.

Frustration on a triangular plaquette with the two states represented by up and down arrows.

Fig. 3.
Fig. 3.

Sublattice structure of the TIA with the sites of the three sublattices labeled 0, 1, and 2. The origin is at the lower left corner.

Fig. 4.
Fig. 4.

Correlation functions at (a) T=0 and (b) T=1. The three curves are for sublattice 0 (+), sublattice 1 (×), and sublattice 2 (o).

Fig. 5.
Fig. 5.

Example lattice configurations for (a) T=0 and (b) T=1. Superlattice cells are shown by the rhombi.

Fig. 6.
Fig. 6.

Lattices and the corresponding reciprocal lattices. (a) Triangular lattice and (c) reciprocal lattice. (b) Lattice sites shown by the filled and open circles, and the superlattice sites shown by open circles. (d) Reciprocal superlattice (gray filled circles) and the reciprocal lattice (black circles).

Fig. 7.
Fig. 7.

Ordered triangular lattice in a circular crystallite (left) and its diffraction pattern (right).

Fig. 8.
Fig. 8.

TIA realizations of point scatterers and no scatterers at T=0 (left) and T= (right).

Fig. 9.
Fig. 9.

Full diffraction patterns from a lattice with point and no scatterers (left column) and the amplitude profile along the horizontal axis (denoted u) through the origin (right column) for T=0,1,2,3 and (top to bottom).

Fig. 10.
Fig. 10.

Diffuse diffraction patterns from a lattice with point and no scatterers (left column) and the amplitude profile along the horizontal axis through the origin (right column) for T=0,1,2,3, and (top to bottom). The diffraction patterns in the left column are normalized to their minimum and maximum values in order to show the weak diffraction more clearly.

Fig. 11.
Fig. 11.

Full diffraction patterns (left) and amplitude profiles (right) from a lattice with point and no scatterers at T=0 for crystallite sizes rc=16 (top) and rc=4 (bottom).

Fig. 12.
Fig. 12.

(a) Up and down triangular scatterers, (b) their diffraction amplitudes, and (c) functions |F(R)| (left) and |F(R)|2|F(R)|2 (right).

Fig. 13.
Fig. 13.

Full diffraction patterns (left) and the diffuse component (right) for a TIA consisting of up and down triangles for T=0,1,2,3 and (top to bottom). Both components are shown at their correct values, i.e. the figure shows the correct relative amplitude between the full and diffuse diffraction.

Tables (1)

Tables Icon

Table 1. Values of the Parameters Used in Eq. (3)

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

rmn=ma+nb,
a=(1,0),b=(1/2,3/2).
ρmn(T)=ρ(r,T)=αi(T)r1/2[β(T)]r,
ρ(0)(r,T)2ρ(1)(r,T)2ρ(2)(r,T),
as=(3/2,3/2),bs=(3/2,3/2).
as=2ab,bs=a+b
a=13(as+bs),b=13(2bsas).
Fjk(R)=fjk(r)exp(i2πR·r)dr,
A(R)=jks(rjk)Fjk(R)exp(i2πR·rjk),
I(R)=|A(R)|2=jkjks(rjk)s(rjk)Fjk*(R)Fjk(R)exp(i2πR·[rjkrjk]),
I(R)=jkmns(rjk)s(rj+m,k+n)Fjk*(R)Fj+m,k+n(R)exp(i2πR·rmn).
Fjk*(R)Fj+m,k+n(R)=F00*(R)Fmn(R),
I(R)=mnF00*(R)Fmn(R)exp(i2πR·rmn)×jks(rjk)s(rj+m,k+n).
jks(rjk)s(rj+m,k+n)1|a×b|t(rmn),
I(R)=mnt(rmn)F00*(R)Fmn(R)exp(i2πR·rmn),
F00*(R)Fmn(R)=F00*(R)Fmn(R)+ρmn[(|F00*(R)|2|F00*(R)|2)(|Fmn(R)|2|Fmn(R)|2)]1/2.
F00*(R)Fmn(R)=|F(R)|2+ρmn(|F(R)|2|F(R)|2),
I(R)=mnt(rmn)[|F(R)|2+ρmn(|F(R)|2|F(R)|2)]exp(i2πR·rmn).
I(R)=IB(R)+ID(R),
IB(R)=|F(R)|2mnt(rmn)exp(i2πR·rmn)
ID(R)=(|F(R)|2|F(R)|2)mnρmnt(rmn)exp(i2πR·rmn).
Rpq=pa˘+qb˘.
a·a˘=b·b˘=1,a·b˘=b·a˘=0.
a˘=(1,1/3),b˘=(0,2/3).
I(R)=|F(R)|2mnt(rmn)exp(i2πR·rmn),
I(R)=IB(R)+N(|F(R)|2|F(R)|2),
|F(R)|2=14(F1(R)+F2(R))2,|F(R)|2=12(|F1(R)|2+|F2(R)|2).
a˘s=(1/3,1/3),b˘s=(1/3,1/3).
Rs,pq=pa˘s+qb˘s,
Rs,pq{Rpq}p=3mq,m.
ZD(R)=(m,n)S0ρmnt(rmn)exp(i2πR·rmn)+(m,n)S1S2ρmnt(rmn)exp(i2πR·rmn).
ZD(R)(m,n)S0{ρmnt(rmn)exp(i2πR·rmn)+ρm+1,nt(rm+1,n)exp(i2πR·(rmn+a))+ρm,n+1t(rm,n+1)exp(i2πR·(rmn+b))}.
ρmn2ρm+1,n2ρm,n+1,(m,n)S0,
ZD(R)B(R)(m,n)S0ρmnt(rmn)exp(i2πR·rmn),
B(R)=112exp(i2πR·a)12exp(i2πR·b).
B(Rs,pq)=112exp(i2πRs,pq·a)12exp(i2πRs,pq·b).
Rs,pq·a=(pa˘s+qb˘s)·a=13(p+q).
Rs,pq·b=13(2qp).
B(Rs,pq)=112exp(i2π13(p+q))12exp(i2π13(2qp)).
F1(u,v)=τ23τ30τ3y3exp(i2π(ux+vy))dxdy+τ23τ3y3τ30exp(i2π(ux+vy))dxdy=exp(iπτ(uv/3))exp(i2πτv/3)4π2u(vu/3)+exp(i2πτv/3)exp(iπτ(u+v/3))4π2u(v+u/3).

Metrics