Proc. Natl. Acad. Sci. USA Vol. 96, pp. 3502–3506, March 1999 Physics

Screw rotations and glide mirrors: Crystallography in Fourier space

ANJA KO¨NIG AND N. DAVID MERMIN

Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501

Contributed by N. David Mermin, January 22, 1999

ABSTRACT The traditional crystallographic symmetry Fourier-space structure that makes them nonsymmorphic in elements of screw axes and glide planes are subdivided into spite of the absence of screw rotations. those that are removable and those that are essential. A simple real-space criterion, depending only on Bravais class, deter- Screws and Glides in Three-Dimensional Real Space mines which types can be present in any space group. This terminological refinement is useful in expressing the comple- The traditional description of crystal symmetry as summarized mentary relation between the real-space and Fourier-space in the International Tables of Crystallography (1) specifies two formulations of crystal symmetry, particularly in the case of kinds of rotation axes and mirror planes. (a) Axes or planes for the two nonsymmorphic space groups that have no systematic which the rotation or mirror is a symmetry of the crystal extinctions (I212121 and I213). A simple analysis in Fourier without an accompanying translation. We shall call such space demonstrates the nonsymmorphicity of these two space rotation axes or mirror planes simple.(b) Axes or planes for groups, which finds its physical expression not in a charac- which the rotation or mirror is only a symmetry of the crystal teristic absence of Bragg peaks, but in a characteristic pres- when accompanied by a translation parallel to the axis or ence of electronic level degeneracies. plane.a Such rotation axes or mirror planes are called screw axes or glide planes. This paper serves two related purposes. (i) We show how the The space group of a crystal is said to be symmorphic if there geometric language of conventional crystallography can ben- is a single point through which the axis of every rotation and efit from a subdivision of screw axes and glide planes into two the plane of every mirror is simple; if there is no such point the varieties, which we call essential and removable, and we note the space group is nonsymmorphic. elementary geometric criterion that determines which varieties In terms of these two kinds of axes or planes, a given rotation can be found in a given crystal structure. (ii) We discuss, in the (specified by the angle and direction, but not the location of its comparatively new framework of Fourier space crystallogra- axis) or a given mirror (specified by the orientation but not the I phy, the peculiar space group 212121 (no. 24) and its cubic location of its plane) can come in three varieties: (i) a given counterpart, I2 3 (no. 199). Alone among all the 157 nonsym- 1 rotation (mirror) has only simple axes (simple planes); (ii)a morphic three-dimensional space groups, these two have no given rotation (mirror) has both simple and screw axes (simple systematic extinctions in their diffraction patterns: every wave and glide planes); and (iii) a given rotation [mirror] has only vector in the face centered reciprocal lattice is associated with a Bragg peak of nonzero intensity. screw axes (glide planes). The relation between these matters is this: while crystals Cases ii and iii provide grounds for an elementary but nontraditional distinction between two kinds of screw axes or characterized by space-groups I212121 and I213 contain 2-fold mirror planes. We shall call the screw axes or mirror planes screw axes (21 axes), those 2-fold rotations fail to satisfy the Fourier-space criterion for a screw. This clash of Fourier-space occurring in case ii removable because their rotations or and conventional nomenclature occurs for none of the other mirrors can be made simple by a mere translation of the axis space groups with nj in their International space-group sym- or plane. We call the screw axes and mirror planes occurring bols. The reason is that the nj axes appearing in all space-group in case iii essential because there is no parallel axis or plane symbols, with the sole exceptions of these two, are essential about which the rotation or mirror is simple. screw axes. The space groups I212121 and I213 are unique In terms of this distinction, there are evidently two different among these nonsymmorphic space groups in having only ways in which the space group of a crystal can be nonsym- removable screw axes, a feature the International nomencla- morphic: (i) the crystal has at least one essential screw axis or ture obscures. In the Fourier-space scheme, rotations with at least one essential glide plane and (ii) all screw axes and removable screw axes never are regarded as screws. Nomen- glide planes are removable, but there is no single origin about clatural confusion can be avoided by using the terms screw which all rotations and mirrors have simple axes and planes. rotation and glide mirror to characterize rotations with essen- Somewhat surprisingly, among all the 230 space groups in tial screw axes and mirrors with essential glide planes. three dimensions, there are only two instances of nonsymmor- Below we define essential and removable screw axes and phic space groups of type ii: the nonsymmorphic space groups glide planes and give the simple connection between the I212121 (no. 24) and I213 (no. 199) have only removable screw Bravais class of a crystal and whether or not it can have them. axes, but there is no single origin through which every axis is We show how these elementary geometric criteria for the simple. Every one of the remaining 155 nonsymmorphic space removability of screw axes and glide planes lead directly in Fourier space to the absence of the corresponding screw aTranslations parallel to the rotation axis or mirror plane are special rotations and glide mirrors. We construct the space groups because such translations are restricted to a small number of discrete I212121 and I213 directly in Fourier space, identifying the values. In contrast, given an axis A for a rotation r and any translation d perpendicular to that axis, there is another axis AЈ parallel to A such Ј The publication costs of this article were defrayed in part by page charge that application of r about A is the same as application of r about A followed by translation through d; and given a mirror plane P and any payment. This article must therefore be hereby marked ‘‘advertisement’’ in translation d perpendicular to that plane, there is another mirror accordance with 18 U.S.C. §1734 solely to indicate this fact. plane PЈ parallel to P such that mirroring in PЈ is the same as mirroring PNAS is available online at www.pnas.org. in P followed by a translation through d.

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groups has at least one essential screw axis or essential glide vector. As a result, if there are nonlattice projections onto a plane. 4-fold axis (which happens in the centered cubic and tetragonal The 21 occurring in the space-group symbols I213 and Bravais classes) these prohibit essential 42 axes and produce I212121 are unique among all such subscripted rotation axes in removable 42 axes. They are compatible with the existence of being associated with removable 2-fold screw axes. This devi- essential 4-fold screw axes, but these must always occur as b ant notation is needed to distinguish I213 and I212121 from the coexisting parallel 41 and 43 axes. corresponding symmorphic space groups I23 and I222. The In the case of a mirror, all nonlattice projections of lattice notation is potentially misleading, because every screw axis in translations into the plane of the mirror are the same, modulo the nonsymmorphic cases is removable, and the symmorphic the sublattice of translations in the plane of the mirror.e Simple cases also contain (removable) 2-fold screw axes. mirrors must coexist with removable glide planes having such There are elementary geometric criteria that determine special nonlattice translations. Essential glide planes are those when a simple rotation axis or simple mirror plane can or with nonlattice translations that are not of this special kind. cannot be accompanied by parallel screw axes or glide planes, and when a screw axis or glide plane is removable. The criteria The View from Fourier Space depend only on the Bravais lattice of translations that leave the crystal invariantc:(a) a simple rotation axis (simple mirror Fourier-space crystallographyf starts with a point group whose plane) has screw axes (glide planes) parallel to it if and only if operations act on the reciprocal lattice L of wave vectors. All there are vectors in the lattice of translations whose compo- point group operations act about the same origin, k ϭ 0. Under nents parallel to the axis (plane) are not in the lattice; and (b) point group operations, density Fourier coefficients acquire a a screw axis (glide plane) is essential if and only if its associated phase: nonlattice translation is not the component of a lattice vector ␲ ⌽ ͑ ͒ ␳͑ ͒ ϭ 2 i g k ␳͑ ͒ parallel to the axis (plane). These criteria follow straightfor- gk e k . [1] wardly from two facts: (i) the nonlattice translation associated ⌽ g with a screw axis or glide plane is only specified to within an The phase functions g are linear on L and satisfy, as a direct 1 additive vector from the lattice of translations; and (ii)a consequence of , the group compatibility condition change of origin can shift that nonlattice translation by an ⌽ ͑ ͒ ϵ ⌽ ͑ ͒ ϩ ⌽ ͑ ͒ gh k g hk h k . [2] arbitrary translation perpendicular to the rotation axis or mirror plane. Two sets of phase functions related by Several elementary geometric facts about screw axes and ⌽Ј͑ ͒ ϵ ⌽ ͑ ͒ ϩ ␹͑ Ϫ ͒ glide planes follow directly from these criteria. g k g k gk k , [3] The simplest consequences are for 2- or 3-fold rotations. The ␹ only possible nonlattice translations (modulo the sublattice where is linear on L, are said to be gauge equivalent. The ␹ ⌽Ј along the axis) for a 2- or 3-fold screw axis is a half or a third function is called a gauge function and is said to differ ⌽ of a lattice vector that primitively generates that sublattice. But from by a gauge transformation. Such gauge equivalent phase nonlattice projections of lattice vectors on the axis also must functions can characterize densities that differ only by differ from a lattice vector on the axis by a half or a third of ␲ ␹͑ ͒ ␳Ј͑k͒ ϭ e2 i k ␳͑k͒. [4] a primitive generating vector for the sublattice on the axis. Therefore, if there are any lattice vectors with nonlattice The significance of 4 with ␹ linear on L is that ␳Ј and ␳ have projections onto a 2- or 3-fold screw axis, then that screw axis identical positionally averaged real space autocorrelation func- is removable. Thus, for example, because all rhombohedral tions. Symmetry types of densities therefore are characterized and cubic Bravais classes have lattice vectors whose projections not by individual phase functions, but by gauge equivalence on the 3-fold axes are not lattice vectors, no space groups in classes of phase functions. Note that it follows from 3 that the these Bravais classes can have essential 3-fold screw axes, but ⌽ value of a phase function g at wave vectors k in the invariant all have removable 3-fold screw axes. And because the body- subspace of the point group operation g (i.e., with gk ϭ k)is centered orthorhombic lattice has nonlattice projections on its the same throughout the entire gauge equivalence class. These 2-fold axes, both of its space groups have removable 2-fold gauge invariant values of phase functions play a central role in screw axes. Fourier-space crystallography. Importantly, it follows from 1 Because a 6-fold axis (present only in the hexagonal system) ⌽ that if g(k) is not zero (modulo 1) at a vector in the invariant is also an axis of both 2- and 3-fold symmetry, all lattice vectors ␳ d subspace of g, then (k) must vanish—i.e., the Bragg peak must project to lattice vectors on the axis. Therefore 6-fold associated with the reciprocal lattice vector k is extinguished.h screw axes are always essential. The above paragraph may be viewed as just a re-expression Because 4-fold axes are also axes of 2-fold symmetry, twice of well-known elementary facts about the symmetries of the the projection of any lattice vector onto the axis is a lattice real space density ␳(r) in Fourier-space language. Thus for a point group operation g acting on a crystal in real space to bIt is cited in section 4.1 of volume A of the International Tables of leave the crystal invariant it must in general be accompanied Crystallography (1) as a violation of the ‘‘priority rule,’’ which decrees by a translation a that depends on the origin about which g acts. that when a rotation has parallel 2 (simple) and 21 (screw) axes (i.e., when a screw axis is removable) it should be identified in the space-group symbol bya2without a subscript. The two space groups eView the lattice as a family of lattice planes parallel to the mirror also are cited in section 3.5 of the International Tables as a second very plane. The special translation is (any) one taking neighboring planes special instance (the many examples of the first being provided by into one another. Friedel’s law) of when a space group cannot be determined from the fFormulated by Bienenstock and Ewald (4), refined and extended by diffraction diagram. This feature of I212121 and I213 seems first to Rokhsar, Wright, and Mermin (5), expounded in some detail by have been noted independently by M. J. Buerger (2) and G. Zhdanov Rabson, Mermin, Rokhsar, and Wright (6) and Mermin (7), and given and V. Popelov (3). a rigorous and concise formulation by Dra¨gerand Mermin (8). This cWe know of no explicit statement of these simple rules in the paragraph states those definitions of Fourier-space crystallography crystallographic literature, perhaps because the distinction between that are essential for what follows. essential and removable screw axes or glide planes has received so gAs a consequence of the requirement that point group operations little attention. preserve all positionally averaged real space density autocorrelation dFor a lattice vector that projected to a nonlattice vector on the axis functions of all orders. would have to be both a third and a half of a primitive sublattice hMermin (9) gives a concise discussion of how Fourier-space crystal- vector, modulo the sublattice on the axis, which is impossible. lography treats extinctions. Downloaded by guest on September 29, 2021 3504 Physics: Ko¨nig and Mermin Proc. Natl. Acad. Sci. USA 96 (1999)

⌽ ⅐ ͞ ␲ The phase function g(k) is nothing more than a k 2 . The compatibility condition (2), which, applied repeatedly to the different phase functions in a gauge equivalence class simply identity gn ϭ e requires for arbitrary k that correspond to the different values of a associated with differ- ϵ ⌽ ͑ ͒ ϵ ⌽ ͑ ͒ ϵ ⌽ ͑ ϩ ϩ ϩ nϪ1 ͒ ϭ ⌽ ͑ ͒ ent choices of real space origin. The power of the Fourier- 0 e k gn k g k gk ··· g k g nPk , space formulation is that nothing changes when the lattice of [6] wave vectors is generalized from a three-dimensional recipro- cal lattice to a rank-DZ-module of wave vectors consisting of where P projects into the invariant subspace of g. Necessarily, the integral linear combinations of D three-dimensional wave- i nPk is a reciprocal lattice vector. Therefore when n is2or3, vectors that are linearly independent over the integers. This if Pk is not a reciprocal lattice vector, then Pk must differ from more general case applies to an enormous variety of aperiodic an integral multiple of b by Ϯb͞n, where b is a primitive vector crystals whose densities have neither translational nor rota- for the (one-dimensional) reciprocal sublattice along the axis. tional symmetry in real space, but whose density autocorre- Consequently nPk differs from an integral multiple of nb by lation functions continue to possess rotational symmetries. Ϯb. But because ⌽ (nb) ϵ 0 as a special case of 6, it follows The Fourier-space language is essential if one wishes to g from 6 and the linearity of ⌽ that 0 ϵ⌽(nPk) ϵϮ⌽(b). explore in a three-dimensional space the three-dimensional g g g symmetries of aperiodic crystals. It also can be a useful and Fourier-Space Treatment of Space Groups Nos. 24 and 199 powerful tool for characterizing the more familiar symmetries of ordinary periodic crystals, as we illustrate below. In conventional crystallographic language a space group is Because all point group operations act through the origin of symmorphic if there is a single origin about which every point Fourier space, the terms screw and glide cannot apply to a point group operation in combination with a real-space origin group operation is a symmetry of the crystal without an through which it acts; the terms give global characterizations accompanying translation. In Fourier-space language this of the point group operation itself. Whether a point group translates into the requirement that there should be a gauge in operation g is a screw or glide depends on the associated phase which every phase function vanishes (modulo unity). A nec- ⌽ essary condition for a space group to be symmorphic is thus function g. A point group operation g is a screw rotation or a ⌽ that the phase function for each point group operation van- glide mirror if and only if the associated phase function g has a nonzero (modulo unity) value for some reciprocal lattice ishes on the invariant subspace of that operation. Is this also vector k in the invariant subspace of g. As noted above, such sufficient? values are invariant under gauge transformations (3). We show in the Appendix that if a phase function vanishes It is a basic theorem of Fourier-space crystallography that a on its invariant subspace then there is a gauge in which it ⌽ phase function g can vanish on the invariant subspace of g,if vanishes everywhere. But for a space group to be symmorphic, and only if there is a gauge in which it vanishes everywhere.j there must be at least one gauge in which every phase function In such a gauge 1 reduces to vanishes everywhere. Space groups nos. 24 and 199 are the unique examples of space groups in which all phase functions ␳͑ ͒ ϭ ␳͑ ͒ gk k , [5] vanish on their invariant subspaces but there is no gauge in ϭ which they all vanish everywhere. Nevertheless, their nonsym- and g is a simple rotation or simple mirror about the point r morphicity is established by the existence of certain gauge 0. Thus a Fourier-space screw rotation or glide mirror is one invariant, nonvanishing linear combinations of phase func- that in real space has only essential screw axes or essential glide tions. planes. Removable real-space screw axes or glide planes are We show below how this follows directly from the rules of not associated with Fourier-space screw rotations or glide Fourier space crystallography summarized above, but first we mirrors. note that it is a result of general interest, rather than a mere We noted above that essential 2- and 3-fold screw axes are isolated curiosity. Within any given arithmetic crystal class, the incompatible with the existence of vectors in the real-space lattice of translations with nonlattice projections on the axis. absence of a nonsymmorphic space group without extinctions It is instructive to examine how the existence of such vectors is the necessary and sufficient condition for the space-group leads directly in Fourier space to the vanishing of the associ- types of that class to be entirely determined by the values of ated phase function on the axis. Note first that such vectors their phase functions on the invariant subspaces of their exist in the real-space lattice of translations if and only if they associated point group elements. Clearly the condition is exist in the reciprocal lattice of wave vectors.k For let P project necessary: if an arithmetic crystal class contains a nonsym- into the invariant subspace, and let a be a direct lattice vector morphic space group without extinctions then there are two with Pa not in the direct lattice. Because Pa is not a direct space groups within that class (the second being the symmor- lattice vector there must be some vector of the reciprocal phic one) whose phase functions vanish (and therefore agree) lattice k for which (Pa, k) is not an integral multiplel of 2␲. But on the invariant subspaces of their point group operations. The because (Pa, k) ϭ (a, Pk) it follows that (a, Pk) is not an space-group type in such an arithmetic crystal class is thus not integral multiple of 2␲, which means that Pk is not in the determined by the phase functions on the invariant subspaces. ⌽(1) ⌽(2) reciprocal lattice. But the condition is also sufficient, for if g and g are So it is enough to understand why the existence of a two sets of phase functions associated with two different ⌽(1) Ϫ⌽(2) reciprocal lattice vector whose projection on a 2- or 3-fold axis space-group types, then the differences g g also will is not in the reciprocal lattice, should guarantee the vanishing satisfy the group compatibility condition (2), and are therefore of the phase function associated with that 2- or 3-fold rotation themselves a set of phase functions for some space-group type. ⌽(1) ⌽(2) at wave vectors on the axis. This follows from the group But if g and g agree on the invariant subspaces of their point-group operations, then their differences will vanish on ⌽(1) ⌽(2) iNow, however, functions linear on L can no longer be extended to all those invariant subspaces. Because g and g describe functions linear on all of three-dimensional wave-vector space. distinct space-group types they cannot be gauge equivalent, jThe ‘‘if’’ part is obvious; the ‘‘only if’’ part has not been explicitly and therefore their differences cannot be gauge equivalent to stated in the literature; we give a proof (valid for periodic or aperiodic a set of phase functions that are zero everywhere. Their crystals) in the Appendix. kThis fact also holds for 4- and 6-fold rotations and for mirrors. differences are thus a set of phase functions describing a lBecause otherwise Pa would be in the reciprocal of the reciprocal nonsymmorphic space group, in the same arithmetic crystal lattice, which is the direct lattice. class, without extinctions. Downloaded by guest on September 29, 2021 Physics: Ko¨nig and Mermin Proc. Natl. Acad. Sci. USA 96 (1999) 3505

We now show that the phase functions for space groups nos. theory of electronic level degeneracies in crystals (the theory 24 and 199 have this peculiar property: of space-group representations in the periodic case) is directly (a) I212121 (No. 24). The basis of the face-centered orthor- related to the phase functions of Fourier-space crystallogra- hombic reciprocal lattice consists of the three vectors b1, b2, b3 phy. that can be expressed in terms of three mutually orthogonal (b) I213 (No. 199). We can continue to take 7 to give the vectors a, b, c of different lengths as: primitive generating vectors for the face-centered-cubic recip- rocal lattice, with the understanding that the mutually orthog- ϭ ϩ ϭ ϩ ϭ ϩ b1 b c, b2 c a, b3 a b. [7] onal vectors a, b, and c now have equal lengths. The generators of the tetrahedral point group 23 can be taken to be the 2-fold The possible phase functions are determined by requiring 3 rotation ra about a and the 3-fold rotation r3 that takes a b, the group compatibility condition (2) to hold throughout the b 3 c, and c 3 a. The generating relations can be take to be point group. To ensure this it suffices to impose 2 on the point ϭ 2 ϭ 3 ϭ ͑ ͒3 ϭ group generating relations. The generators of G 222 can be ra e, r3 e, r3ra e. [14] taken to be the two 2-fold rotations ra and rb about the axes a and b, with generating relations: As in the orthorhombic case, applying the group compatibility condition (2) to each of these gives the conditions 2 ϭ 2 ϭ ͑ ͒2 ϭ ra e, rb e, rarb e. [8] ⌽ ͑r k ϩ k͒ ϵ 0, ⌽ ͑r2k ϩ r k ϩ k͒ ϵ 0, ⌽ ϵ ra a r3 3 3 Because e(k) 0, application of the group compatibility 2 condition (2) to each of 8 gives ⌽ ͑ Ј ϩ Ј ϩ ͒ ϵ r3Ј r3 k r3 k k 0, [15] ⌽ ͑ ϩ ͒ ϵ ⌽ ͑ ϩ ͒ ϵ ⌽ ͑ ϩ ͒ ϵ r rak k 0, r rbk k 0, r rck k 0, [9] ϭ 3 3 Ϫ Ϫ 3 a b c where r3Ј r3ra, which takes a b, b c, and c a. Each ϭ part of 15 gives a single relation when applied to the bi: where we have used rc rarb to write 9 in a symmetric form. Because phase functions are linear on the reciprocal lattice, ⌽ ͑ ͒ ϵ ⌽ ͑ ϩ ϩ ͒ ϵ ⌽ Ј͑ ϩ Ϫ ͒ ϵ ra 2a 0, r3 2a 2b 2c 0, r3 2a 2b 2c 0. for the conditions (9) to hold for all k it suffices for them to [16] hold for each of the reciprocal lattice primitive vector bi given ⌽ in 7. Because rk ϩ k gives twice the projection of k onto the Each of these relations sets a g equal to zero at a vector that axis of a 2-fold rotation r, each of the three relations (9) gives primitively generates the invariant subspace of g, so there are just a single condition, when applied to each of the three no extinctions along the axes of ra, r3,orr3Ј. There are primitive vectors in 7. therefore no extinctions at all, because the group compatibility condition (2) insures that if ⌽ vanishes on its invariant ⌽ ͑ ͒ ϵ ⌽ ͑ ͒ ϵ ⌽ ͑ ͒ ϵ g ra 2a 0, rb 2b 0, rc 2c 0. [10] subspace, the same is true for all point group operations m Ј ϭ conjugate to g. But because r3 r3ra, under the group Because 2a,2b, and 2c are primitive generating vectors for the compatibility condition (2) the third relation in 16 expands to three sublattices on the three 2-fold axes, every phase function ⌽ ͑ Ϫ ϩ ͒ ϩ ⌽ ͑ ϩ Ϫ ͒ ϵ vanishes on the invariant subspace of its point-group opera- r3 2a 2b 2c ra 2a 2b 2c 0. [17] tion—i.e., there are no extinctions. The distinct gauge equivalence classes of phase functions are Subtracting the first relation of 16 from 17 and adding the determined by the possible values for the two independent second, converts 16 to point-group generators ra and rb at the primitive reciprocal- ⌽ ͑ ͒ ϩ ⌽ ͑ Ϫ ͒ ϵ r 4b2 r 2b3 2b2 0. [18] lattice generating vectors bi. Expanding the third of the 3 a ϭ conditions (10) using rc rarb and the group compatibility condition (2) gives This allows for two distinct gauge equivalence classes of phase functions, differing by the gauge invariant relationsn Ϫ⌽ ͑ ͒ ϩ ⌽ ͑ ͒ ϵ ra 2c rb 2c 0. [11] ⌽ ͑2b ͒ ϩ ⌽ ͑b Ϫ b ͒ ϵ 0or1 . [19] r3 2 ra 3 2 2 Subtracting the first of the two relations of 10 from 11 and adding the second, converts it into a relation between primitive The way in which these two nonsymmorphic space groups generating vectors: without extinctions emerge from this analysis is so simple and natural, that what is surprising is not their existence, but the ⌽ ͑2b ͒ Ϫ ⌽ ͑2b ͒ ϵ 0. [12] rb 1 ra 2 fact that in every one of the remaining cases, the vanishing of every phase function on its invariant subspace is incompatible This allows for two classes of phase functions, satisfying with the existence of any nonzero gauge invariant linear combination of phase functions. It would be interesting to ⌽ ͑b ͒ Ϫ ⌽ ͑b ͒ ϵ 0or1 . [13] rb 1 ra 2 2 know whether such possibilities are more common in higher dimensional spaces.o Because 12 is a linear combination of relations derived from the (gauge invariant) group compatibility conditions (2), the Appendix combination of phases on the left side of 13 is gauge invariant ⌽ (as can also be verified directly). The nonzero choice for this The value of a phase function g is gauge invariant on the ⌽ combination therefore gives a nonsymmorphic space group. invariant subspace of g, so if there is a gauge in which g ⌽ ⌽ Even though ra and rb both vanish on the invariant subspaces m of their point-group operations, there is no gauge in which they For it follows from Eq. 2 that ⌽rgrϪ1 (rk) ϵ⌽r(gk) ϩ⌽g(k) ϩ both vanish everywhere. ⌽rϪ1(rk) and that 0 ϵ⌽rϪ1(rk) ϩ⌽r(k). Therefore, when gk ϭ k, ⌽ ϵ⌽ The peculiar combination of phases appearing in 13 has no rgrϪ1(rk) g(k). obvious geometric significance. Interestingly, however, it is nAs in the orthorhombic case, the nonvanishing of this combination of precisely the combination of phases that determines that the phases is directly related to the existence of electronic level degen- eracies, as shown in Ko¨nig and Mermin (10). electronic levels in the nonsymmorphic space group I212121 oNo examples that are not trivially related to these two have yet turned 1 Ϯ Ϯ Ϯ have 2-fold degeneracies at the points ⁄2( a b c). This is up in any of the many investigations of space groups for aperiodic shown in Ko¨nig and Mermin (10), which explains how the three-dimensional crystals. Downloaded by guest on September 29, 2021 3506 Physics: Ko¨nig and Mermin Proc. Natl. Acad. Sci. USA 96 (1999)

vanishes, it must vanish on that invariant subspace. We prove n, (see for example theorem 7.8 of Hartley and Hawkes, ref. ⌽ ϵ here that this vanishing is also sufficient: if g(k) 0 whenever 11). Because b1,...,bn is a basis for the whole module we have gk ϭ k, then there is a gauge in which ⌽ vanishes for all k. g ϭ ␣ ϩ ϩ ␣ We must show that if c1 11b1 ··· 1nbn · · [27] ⌽ ͑ ͒ ϵ ϭ · , g k 0 whenever gk k, [20] ϭ ␣ ϩ ϩ ␣ cm m1b1 ··· mnbn ␹ then there is a function linear on the lattice of wave-vectors where ␣ is an m ϫ n matrix of integers. L such that Because both the c and the b are linearly independent over ⌽ ͑ ͒ ϩ ␹͑ Ϫ ͒ ϵ the rationals, it follows that the mn-dimensional row vectors g k gk k 0. [21] of ␣ are linearly independent over the rationals. Because the Let Lg be the set of all vectors q of L that are of the form column rank of any matrix over a field is equal to its row rank, it follows that the nm-dimensional column vectors of ␣ span ϭ Ϫ q gk k [22] a space of dimension m. ␹ To extend the function linear on Lg to one linear on all of for some vector k in L. Clearly Lg is a sublattice of L. Define ␹ ␹ ␹ L, it suffices to specify values (b1),..., (bn) satisfying on Lg by ␹͑ ͒ ϭ ␣ ␹͑ ͒ ϩ ϩ ␣ ␹͑ ͒ ␹͑ ͒ ϵ Ϫ⌽ ͑ ͒ c1 11 b1 ··· 1n bn q g k . [23] · · . [28] ␹͑ ͒ ϭ ␣ ␹͑ ͒·ϩ ϩ ␣ ␹͑ ͒ The definition is unique even if the correspondence (22) cm m1 b1 ··· mn bn between q and k is not, because if ␹ The existence of the (bi) is ensured by the fact that the n q ϭ gk؅ Ϫ k؅ [24] m-dimensional column vectors of ␣ span an m-dimensional space over the rationals. with kЈÞk, then g͑k Ϫ k؅͒ ϭ k Ϫ k؅. [25] This work was supported by the National Science Foundation Grant DMR9531430. Ϫ Ј ⌽ Therefore, k k is in the invariant subspace of g where g vanishes. Because ⌽ is linear on L, 1. Hahn, T., ed. (1995) International Tables for Crystallography g (Kluwer, Dordrecht The Netherlands), Vol. A, 4th Ed. .(⌽ ͑ ͒ ϵ ⌽ ͑ ؅͒ 2. Buerger, M. J. (1942) X-Ray Crystallography (Wiley, New York g k g k . [26] 3. Zhdanov, G. & Popelov, V. (1945), Zh. Eksp. Teor. Fiz. 12, This leads to 21 holding for all k in L. Because ␹ is linear on 709–720. 3 Ϫ 4. Bienenstock, A. & Ewald, P. P. (1962) Acta Crystallogr. 15, Lg as a consequence of the linearity of k gk k and the ⌽ ␹ 1253–1261. linearity of g, it only remains to show that can be extended from L to a function linear on the whole lattice L. This follows 5. Rokhsar, D. S., Wright, D. C. & Mermin, N. D. (1988) Phys. Rev. g B 37, 8145–8149. from the fact that any function linear on a submodule of a 6. Rabson, D. A., Mermin, N. D., Rokhsar, D. S. & Wright, D. C. Z-module can be extended to a function linear on the entire (1991) Rev. Mod. Phys. 63, 699–733. Z-module. (We state this in the language of Z-modules—linear 7. Mermin, N. D. (1992) Rev. Mod. Phys. 64, 3–49. vector spaces over the integers—to establish that the proof 8. Dra¨ger,J. & Mermin, N. D. (1996) Phys. Rev. Lett. 76, 1489–1492. applies even in the aperiodic case, where the rank of L exceeds 9. Mermin, N. D. (1995) Phys. Stat. Sol. 151, 275–279. the dimension of physical space.) 10. Ko¨nig, A. & Mermin, N. D. (1997) Phys. Rev. B 56, 13607–13610. Let b1,...,bn be a basis for a rank n module L. Because Lg 11. Hartley, B. & Hawkes, T. O. (1970) Rings, Modules, and Linear Յ is a submodule, it also has a basis c1,...,cm and a rank m Algebra (Chapman and Hall, London). Downloaded by guest on September 29, 2021