The relation between these matters is this: while crystals characterized by space-groups I2₁2₁2₁ and I2₁3 contain 2-fold screw axes (2₁ axes), those 2-fold rotations fail to satisfy the Fourier-space criterion for a screw. This clash of Fourier-space and conventional nomenclature occurs for none of the other space groups with n_j in their International space-group symbols. The reason is that the n_j axes appearing in all space-group symbols, with the sole exceptions of these two, are essential screw axes. The space groups I2₁2₁2₁ and I2₁3 are unique among these nonsymmorphic space groups in having only removable screw axes, a feature the International nomenclature obscures. In the Fourier-space scheme, rotations with removable screw axes never are regarded as screws. Nomenclatural confusion can be avoided by using the terms screw rotation and glide mirror to characterize rotations with essential screw axes and mirrors with essential glide planes.

Below we define essential and removable screw axes and glide planes and give the simple connection between the Bravais class of a crystal and whether or not it can have them. We show how these elementary geometric criteria for the removability of screw axes and glide planes lead directly in Fourier space to the absence of the corresponding screw rotations and glide mirrors. We construct the space groups I2₁2₁2₁ and I2₁3 directly in Fourier space, identifying the Fourier-space structure that makes them nonsymmorphic in spite of the absence of screw rotations.

Screws and Glides in Three-Dimensional Real Space The traditional description of crystal symmetry as summarized in the International Tables of Crystallography (1) specifies two kinds of rotation axes and mirror planes. (a) Axes or planes for which the rotation or mirror is a symmetry of the crystal without an accompanying translation. We shall call such rotation axes or mirror planes simple. (b) Axes or planes for which the rotation or mirror is only a symmetry of the crystal when accompanied by a translation parallel to the axis or plane.a Such rotation axes or mirror planes are called screw axes or glide planes.

The space group of a crystal is said to be symmorphic if there is a single point through which the axis of every rotation and the plane of every mirror is simple; if there is no such point the space group is nonsymmorphic.

In terms of these two kinds of axes or planes, a given rotation (specified by the angle and direction, but not the location of its axis) or a given mirror (specified by the orientation but not the location of its plane) can come in three varieties: (i) a given rotation (mirror) has only simple axes (simple planes); (ii) a given rotation (mirror) has both simple and screw axes (simple and glide planes); and (iii) a given rotation [mirror] has only screw axes (glide planes).

Cases ii and iii provide grounds for an elementary but nontraditional distinction between two kinds of screw axes or mirror planes. We shall call the screw axes or mirror planes occurring in case ii removable because their rotations or mirrors can be made simple by a mere translation of the axis or plane. We call the screw axes and mirror planes occurring in case iii essential because there is no parallel axis or plane about which the rotation or mirror is simple.

In terms of this distinction, there are evidently two different ways in which the space group of a crystal can be nonsymmorphic: (i) the crystal has at least one essential screw axis or at least one essential glide plane and (ii) all screw axes and glide planes are removable, but there is no single origin about which all rotations and mirrors have simple axes and planes.

Somewhat surprisingly, among all the 230 space groups in three dimensions, there are only two instances of nonsymmorphic space groups of type ii: the nonsymmorphic space groups I2₁2₁2₁ (no. 24) and I2₁3 (no. 199) have only removable screw axes, but there is no single origin through which every axis is simple. Every one of the remaining 155 nonsymmorphic space groups has at least one essential screw axis or essential glide plane.

The 2₁ occurring in the space-group symbols I2₁3 and I2₁2₁2₁ are unique among all such subscripted rotation axes in being associated with removable 2-fold screw axes. This deviant notationb is needed to distinguish I2₁3 and I2₁2₁2₁ from the corresponding symmorphic space groups I23 and I222. The notation is potentially misleading, because every screw axis in the nonsymmorphic cases is removable, and the symmorphic cases also contain (removable) 2-fold screw axes.

There are elementary geometric criteria that determine when a simple rotation axis or simple mirror plane can or cannot be accompanied by parallel screw axes or glide planes, and when a screw axis or glide plane is removable. The criteria depend only on the Bravais lattice of translations that leave the crystal invariantc: (a) a simple rotation axis (simple mirror plane) has screw axes (glide planes) parallel to it if and only if there are vectors in the lattice of translations whose components parallel to the axis (plane) are not in the lattice; and (b) a screw axis (glide plane) is essential if and only if its associated nonlattice translation is not the component of a lattice vector parallel to the axis (plane). These criteria follow straightforwardly from two facts: (i) the nonlattice translation associated with a screw axis or glide plane is only specified to within an additive vector from the lattice of translations; and (ii) a change of origin can shift that nonlattice translation by an arbitrary translation perpendicular to the rotation axis or mirror plane.

Several elementary geometric facts about screw axes and glide planes follow directly from these criteria.

The simplest consequences are for 2- or 3-fold rotations. The only possible nonlattice translations (modulo the sublattice along the axis) for a 2- or 3-fold screw axis is a half or a third of a lattice vector that primitively generates that sublattice. But nonlattice projections of lattice vectors on the axis also must differ from a lattice vector on the axis by a half or a third of a primitive generating vector for the sublattice on the axis. Therefore, if there are any lattice vectors with nonlattice projections onto a 2- or 3-fold screw axis, then that screw axis is removable. Thus, for example, because all rhombohedral and cubic Bravais classes have lattice vectors whose projections on the 3-fold axes are not lattice vectors, no space groups in these Bravais classes can have essential 3-fold screw axes, but all have removable 3-fold screw axes. And because the body-centered orthorhombic lattice has nonlattice projections on its 2-fold axes, both of its space groups have removable 2-fold screw axes.

Because a 6-fold axis (present only in the hexagonal system) is also an axis of both 2- and 3-fold symmetry, all lattice vectors must project to lattice vectors on the axis.d Therefore 6-fold screw axes are always essential.

Because 4-fold axes are also axes of 2-fold symmetry, twice the projection of any lattice vector onto the axis is a lattice vector. As a result, if there are nonlattice projections onto a 4-fold axis (which happens in the centered cubic and tetragonal Bravais classes) these prohibit essential 4₂ axes and produce removable 4₂ axes. They are compatible with the existence of essential 4-fold screw axes, but these must always occur as coexisting parallel 4₁ and 4₃ axes.

In the case of a mirror, all nonlattice projections of lattice translations into the plane of the mirror are the same, modulo the sublattice of translations in the plane of the mirror.e Simple mirrors must coexist with removable glide planes having such special nonlattice translations. Essential glide planes are those with nonlattice translations that are not of this special kind.

The View from Fourier Space Fourier-space crystallographyf starts with a point group whose operations act on the reciprocal lattice L of wave vectors. All point group operations act about the same origin, k = 0. Under point group operations, density Fourier coefficients acquire a phase:

1

2

3

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The above paragraph may be viewed as just a re-expression of well-known elementary facts about the symmetries of the real space density ρ(r) in Fourier-space language. Thus for a point group operation g acting on a crystal in real space to leave the crystal invariant it must in general be accompanied by a translation a that depends on the origin about which g acts. The phase function Φ_g(k) is nothing more than ak/2π. The different phase functions in a gauge equivalence class simply correspond to the different values of a associated with different choices of real space origin. The power of the Fourier-space formulation is that nothing changes when the lattice of wave vectors is generalized from a three-dimensional reciprocal lattice to a rank-D Z-module of wave vectors consisting of the integral linear combinations of D three-dimensional wave-vectors that are linearly independent over the integers.i This more general case applies to an enormous variety of aperiodic crystals whose densities have neither translational nor rotational symmetry in real space, but whose density autocorrelation functions continue to possess rotational symmetries. The Fourier-space language is essential if one wishes to explore in a three-dimensional space the three-dimensional symmetries of aperiodic crystals. It also can be a useful and powerful tool for characterizing the more familiar symmetries of ordinary periodic crystals, as we illustrate below.

Because all point group operations act through the origin of Fourier space, the terms screw and glide cannot apply to a point group operation in combination with a real-space origin through which it acts; the terms give global characterizations of the point group operation itself. Whether a point group operation g is a screw or glide depends on the associated phase function Φ_g. A point group operation g is a screw rotation or a glide mirror if and only if the associated phase function Φ_g has a nonzero (modulo unity) value for some reciprocal lattice vector k in the invariant subspace of g. As noted above, such values are invariant under gauge transformations (3).

It is a basic theorem of Fourier-space crystallography that a phase function Φ_g can vanish on the invariant subspace of g, if and only if there is a gauge in which it vanishes everywhere.j In such a gauge 1 reduces to

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We noted above that essential 2- and 3-fold screw axes are incompatible with the existence of vectors in the real-space lattice of translations with nonlattice projections on the axis. It is instructive to examine how the existence of such vectors leads directly in Fourier space to the vanishing of the associated phase function on the axis. Note first that such vectors exist in the real-space lattice of translations if and only if they exist in the reciprocal lattice of wave vectors.k For let P project into the invariant subspace, and let a be a direct lattice vector with Pa not in the direct lattice. Because Pa is not a direct lattice vector there must be some vector of the reciprocal lattice k for which (Pa, k) is not an integral multiplel of 2π. But because (Pa, k) = (a, Pk) it follows that (a, Pk) is not an integral multiple of 2π, which means that Pk is not in the reciprocal lattice.

So it is enough to understand why the existence of a reciprocal lattice vector whose projection on a 2- or 3-fold axis is not in the reciprocal lattice, should guarantee the vanishing of the phase function associated with that 2- or 3-fold rotation at wave vectors on the axis. This follows from the group compatibility condition (2), which, applied repeatedly to the identity gⁿ = e requires for arbitrary k that

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Fourier-Space Treatment of Space Groups Nos. 24 and 199 In conventional crystallographic language a space group is symmorphic if there is a single origin about which every point group operation is a symmetry of the crystal without an accompanying translation. In Fourier-space language this translates into the requirement that there should be a gauge in which every phase function vanishes (modulo unity). A necessary condition for a space group to be symmorphic is thus that the phase function for each point group operation vanishes on the invariant subspace of that operation. Is this also sufficient?

We show in the Appendix that if a phase function vanishes on its invariant subspace then there is a gauge in which it vanishes everywhere. But for a space group to be symmorphic, there must be at least one gauge in which every phase function vanishes everywhere. Space groups nos. 24 and 199 are the unique examples of space groups in which all phase functions vanish on their invariant subspaces but there is no gauge in which they all vanish everywhere. Nevertheless, their nonsymmorphicity is established by the existence of certain gauge invariant, nonvanishing linear combinations of phase functions.

We show below how this follows directly from the rules of Fourier space crystallography summarized above, but first we note that it is a result of general interest, rather than a mere isolated curiosity. Within any given arithmetic crystal class, the absence of a nonsymmorphic space group without extinctions is the necessary and sufficient condition for the space-group types of that class to be entirely determined by the values of their phase functions on the invariant subspaces of their associated point group elements. Clearly the condition is necessary: if an arithmetic crystal class contains a nonsymmorphic space group without extinctions then there are two space groups within that class (the second being the symmorphic one) whose phase functions vanish (and therefore agree) on the invariant subspaces of their point group operations. The space-group type in such an arithmetic crystal class is thus not determined by the phase functions on the invariant subspaces.

But the condition is also sufficient, for if Φ_g⁽¹⁾ and Φ_g⁽²⁾ are two sets of phase functions associated with two different space-group types, then the differences Φ_g⁽¹⁾ − Φ_g⁽²⁾ also will satisfy the group compatibility condition (2), and are therefore themselves a set of phase functions for some space-group type. But if Φ_g⁽¹⁾ and Φ_g⁽²⁾ agree on the invariant subspaces of their point-group operations, then their differences will vanish on all those invariant subspaces. Because Φ_g⁽¹⁾ and Φ_g⁽²⁾ describe distinct space-group types they cannot be gauge equivalent, and therefore their differences cannot be gauge equivalent to a set of phase functions that are zero everywhere. Their differences are thus a set of phase functions describing a nonsymmorphic space group, in the same arithmetic crystal class, without extinctions.

We now show that the phase functions for space groups nos. 24 and 199 have this peculiar property:

(a) I2₁2₁2₁ (No. 24). The basis of the face-centered orthorhombic reciprocal lattice consists of the three vectors b₁, b₂, b₃ that can be expressed in terms of three mutually orthogonal vectors a, b, c of different lengths as:

7

8

9

Because phase functions are linear on the reciprocal lattice, for the conditions (9) to hold for all k it suffices for them to hold for each of the reciprocal lattice primitive vector b_i given in 7. Because rk + k gives twice the projection of k onto the axis of a 2-fold rotation r, each of the three relations (9) gives just a single condition, when applied to each of the three primitive vectors in 7.

10

The distinct gauge equivalence classes of phase functions are determined by the possible values for the two independent point-group generators r_a and r_b at the primitive reciprocal-lattice generating vectors b_i. Expanding the third of the conditions (10) using r_c = r_ar_b and the group compatibility condition (2) gives

11

12

13

The peculiar combination of phases appearing in 13 has no obvious geometric significance. Interestingly, however, it is precisely the combination of phases that determines that the electronic levels in the nonsymmorphic space group I2₁2₁2₁ have 2-fold degeneracies at the points ½(±a ± b ± c). This is shown in König and Mermin (10), which explains how the theory of electronic level degeneracies in crystals (the theory of space-group representations in the periodic case) is directly related to the phase functions of Fourier-space crystallography.

(b) I2₁3 (No. 199). We can continue to take 7 to give the primitive generating vectors for the face-centered-cubic reciprocal lattice, with the understanding that the mutually orthogonal 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 rotation r_a about a and the 3-fold rotation r₃ that takes a → b, b → c, and c → a. The generating relations can be take to be

14

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This work was supported by the National Science Foundation Grant DMR9531430.

The value of a phase function Φ_g is gauge invariant on the invariant subspace of g, so if there is a gauge in which Φ_g vanishes, it must vanish on that invariant subspace. We prove here that this vanishing is also sufficient: if Φ_g(k) 0 whenever gk = k, then there is a gauge in which Φ_g vanishes for all k.

We must show that if

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Let b₁,…, b_n be a basis for a rank n module L. Because L_g is a submodule, it also has a basis c₁,…, c_m and a rank m ≤ n, (see for example theorem 7.8 of Hartley and Hawkes, ref. 11). Because b₁,…, b_n is a basis for the whole module we have

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Because both the c and the b are linearly independent over the rationals, it follows that the m n-dimensional row vectors of α are linearly independent over the rationals. Because the column rank of any matrix over a field is equal to its row rank, it follows that the n m-dimensional column vectors of α span a space of dimension m.

To extend the function χ linear on L_g to one linear on all of L, it suffices to specify values χ(b₁),…, χ(b_n) satisfying

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