The space-group symmetry of a crystal has a simple form in Fourier space, which provides a natural language for the basic facts about x-ray extinctions and band sticking in nonsymmorphic crystals.
REFERENCES
1.
A.
Bienenstock
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N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
4.
A systematic Fourier-space treatment of all the crystallographic and many quasicrystallographic space-group types can be found in
D. A.
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, D. S.
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We take all the rotations in to leave invariant, so all elements in the point group are linear homogeneous transformations on L.
6.
It is this second requirement that distinguishes, for example, among the face-centered, body-centered, and simple cubic Bravais classes.
7.
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 χ linear on L cannot be extended to a function (like ) linear on all of k-space.
8.
We take ≡ to indicate equality modulo unity—i.e., to within an additive integer.
9.
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.
10.
We always use the term “invariant subspace” to mean the subspace of individually invariant vectors.
11.
The identity e leaves all K invariant and has the gauge invariant phase function
12.
When we specify the value of a phase function it should always be understood as specified only to within an additive integer.
13.
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
).14.
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
).15.
All we require for what follows is that t and be invariant under all operations in the point group G of the crystal.
16.
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).
17.
Although it is convenient to call the mirrors horizontal and vertical, the argument that follows applies to any two perpendicular mirrors.
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