Symmetry, extinctions, and band sticking

American Journal of Physics

Volumn 68, Issue 6, 2000, Pages 525-530

  König, Anja ^a   Mermin, N David ^a  

^a Laboratory of Atomic and Solid State Physics and Cornell Center for Materials Research   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034397448     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19479     Document Type: Article

References (22)

1. A. Bienenstock and P. P. Ewald, "Symmetry of Fourier space," Acta Crystallogr. 15, 1253-1261 (1962).
2. D. S. Rokhsar, D. C. Wright, and N. D. Mermin, "The Two-Dimensional Quasicrystallographic Space Groups with Rotational Symmetries Less than 23-Fold," Acta Cryst. A 44, 197-211 (1998); "Scale Equivalence of Quasicrystallographic Space Groups," Phys. Rev. B 37, 8145-8149 (1988).
3. D. S. Rokhsar, D. C. Wright, and N. D. Mermin, "The Two-Dimensional Quasicrystallographic Space Groups with Rotational Symmetries Less than 23-Fold," Acta Cryst. A 44, 197-211 (1998); "Scale Equivalence of Quasicrystallographic Space Groups," Phys. Rev. B 37, 8145-8149 (1988).
4. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
5. A systematic Fourier-space treatment of all the crystallographic and many quasicrystallographic space-group types can be found in D. A. Rabson, N. D. Mermin, D. S. Rokhsar, and D. C. Wright, "The space groups of axial crystals and quasicrystals," Rev. Mod. Phys. 63, 699-733 (1991), and N. D. Mermin, "The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals." Rev. Mod. Phys. 64, 3-49 (1992).
6. A systematic Fourier-space treatment of all the crystallographic and many quasicrystallographic space-group types can be found in D. A. Rabson, N. D. Mermin, D. S. Rokhsar, and D. C. Wright, "The space groups of axial crystals and quasicrystals," Rev. Mod. Phys. 63, 699-733 (1991), and N. D. Mermin, "The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals." Rev. Mod. Phys. 64, 3-49 (1992).
7. L are linear homogeneous transformations on L.
8. It is this second requirement that distinguishes, for example, among the face-centered, body-centered, and simple cubic Bravais classes.
9. Although we make no use of it here, we remark that a crucial difference between periodic and aperiodic crystals is that in the aperiodic case the function X linear on L cannot be extended to a function (like d·K) linear on all of k-space.
10. We take ≡ to indicate equality modulo unity - i.e., to within an additive integer.
11. The other important features, less frequently encountered, are gauge invariant values of linear combinations of phase functions whose values are not individually gauge invariant. These play a central role in the discussion of band sticking in Sec. IV.
12. We always use the term "invariant subspace" to mean the subspace of individually invariant vectors.
13. e≡0.
14. When we specify the value of a phase function it should always be understood as specified only to within an additive integer.
15. For more on the geometric and terminological subtleties of screw rotations and glide mirrors, see A. König and N. D. Mermin, "Screw rotations and glide mirrors: Crystallography in Fourier space," Proc. Natl. Acad. Sci. USA 96, 3502-3506 (1999).
16. What follows is based on A. König and N. D. Mermin, "Electronic level degeneracy in non-symmorphic periodic or aperiodic crystals," Phys. Rev. B 56, 13607-13610 (1997).
17. All we require for what follows is that t and u be invariant under all operations in the point group G of the crystal.
18. Treatments of space group representations can be found in: G. F. Koster, Space Groups and their Representations, Solid State Physics (Academic, New York, 1957), Vol. 5, p. 173; A. C. Hurley, "Ray Representations of Point Groups and the Irreducible Representations of Space Groups and Double Space Groups," Philos. Trans. R. Soc. London, Ser. A 260, 1108 (1966); J. Zak, ed., The Irreducible Representations of Space Groups (Benjamin, New York, 1969); M. Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, New York. 1974).
19. Treatments of space group representations can be found in: G. F. Koster, Space Groups and their Representations, Solid State Physics (Academic, New York, 1957), Vol. 5, p. 173; A. C. Hurley, "Ray Representations of Point Groups and the Irreducible Representations of Space Groups and Double Space Groups," Philos. Trans. R. Soc. London, Ser. A 260, 1108 (1966); J. Zak, ed., The Irreducible Representations of Space Groups (Benjamin, New York, 1969); M. Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, New York. 1974).
20. Treatments of space group representations can be found in: G. F. Koster, Space Groups and their Representations, Solid State Physics (Academic, New York, 1957), Vol. 5, p. 173; A. C. Hurley, "Ray Representations of Point Groups and the Irreducible Representations of Space Groups and Double Space Groups," Philos. Trans. R. Soc. London, Ser. A 260, 1108 (1966); J. Zak, ed., The Irreducible Representations of Space Groups (Benjamin, New York, 1969); M. Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, New York. 1974).
21. Treatments of space group representations can be found in: G. F. Koster, Space Groups and their Representations, Solid State Physics (Academic, New York, 1957), Vol. 5, p. 173; A. C. Hurley, "Ray Representations of Point Groups and the Irreducible Representations of Space Groups and Double Space Groups," Philos. Trans. R. Soc. London, Ser. A 260, 1108 (1966); J. Zak, ed., The Irreducible Representations of Space Groups (Benjamin, New York, 1969); M. Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, New York. 1974).
22. Although it is convenient to call the mirrors horizontal and vertical, the argument that follows applies to any two perpendicular mirrors.