4. Critical Nets on Orbifolds

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In Sect 3. we saw that critical net drawings can become rather complex even for very simple examples such as the body-centered cubic (bcc) structure. In the present section, we discuss critical nets on orbifolds, which reduce both the graphical and interpretation complexity associated with critical nets while including valuable space group topology information as well. Lattice complex information also is integrated into the model to clarify the global topology. Finally, we compare the series of bcc lattice complexes on orbifolds that develop as symmetry is systematically reduced.

**Body-Centered Cubic Orbifold**

The orbifold for Imm, the parent space group for bcc structures, is derived from the fundamental domain shown in the lower left of Fig. 4.1. The space group coordinates for the vertices of the fundamental domain are given in parentheses as fractions of the unit cell lengths. The arrows denote the down density critical net paths leading from the peak at (a) to the pit at (b). Wyckoff identification letters (a-k) are shown on the asymmetric unit drawing, and the ITCr information on most of those Wyckoff sites is listed in the columns labeled "Wyckoff set" in the middle of the figure. The tetrahedral fundamental domain has three sides bounded with the top (k) and bottom (j) mirrors with (k) bridged over the 3-fold axis as described in Sect. 1, but the fourth side is open (unbounded) with a 2-fold axis (i) extending from one corner of the open end (c) to the center (d) of the opposite face, which contains another 2-fold axis (g).

Visualize the tetrahedral asymmetric unit as a single-pole pup tent, covered by a silvered rubber reflective sheet, with a support pole (i) in the entrance. A horizontal "threshold" pole (g) with a hinge in the middle (d) lies across the front of the tent floor with the hinge attached to the bottom of the support pole. To close the tent, we grab the two corners of the rubber sheets (j and k) at the two ends (b) and (b') of the hinged threshold pole (g) and bring them together stretching the extensible and flexible tent floor poles (e) and (h) in the process. We then zipper the edges of the sheet (k) together to form the bounded orbifold shown in the lower right drawings of Fig. 4.1.

The underlying topological space of this 3-orbifold is a 3-disk with silvered boundary (i.e., a silvered 3-ball). The orbifold has two trivalent dihedral corners, 4'3'2' at (a) and 4'2'2' at (b), and two cone points, 23' at (c) and 22' at (d).

**Linearized Critical Net on Orbifold**

Critical nets are actually Morse functions that are defined in terms of a mathematical mapping from Euclidean 3-space to Euclidean 1-space (i.e., a single valued 3-dimensional function). Taking this requirement literally, we deform the orbifold so that the Euclidean 1-space of density is vertical in the page (i.e., peak height > pass height > pale height > pit height). This adds a welcome constraint to the drawing of orbifolds that in general have no inherent topological constraints to guide the illustrator. The topologist would probably tend to draw it as a solid sphere as shown later, but we are not violating any topological principles in forming the linearized critical net on orbifold shown at the top of Fig. 4.1.

The multiplicity for each Wyckoff site is given as a column in the table and the preceding column shows the integer ratios of the multiplicities in adjacent rows, which are by design the adjacent elements in the critical net graph. These ratios tell us the coordination numbers of critical net components around other critical net components, thus summarizing much of the structural topology information you would obtain by examining ORTEP critical net stereo drawings or calculating and evaluating long tables of intercomponent distances and angles.

**Resolution of the Critical Net Versus Tiling Discrepancy**

The coordination numbers also provide a method for applying topological constraints in that there must be exactly two peaks around a pass and two pits around a pale. This particular combinatorial constraint holds for the tiling approach of Dress, Huson, and Molnar (1993) as well as for our critical net Morse function approach. Fig. 4.2 shows two solutions satisfying that constraint based on the orbifold topology for space group Fdm with atoms (i.e., tiling vertices in the Dress approach, peaks in the critical net approach) on the two 3m sites of Fdm. Fig. 4.2 compares the two configurations assuming both are linearized critical nets on the Fdm orbifold. The columns of numbers are sums of Wyckoff set multiplicities for each level of the critical net and integer ratios of neighboring rows. Only the connections between adjacent levels are summed. An ORTEP drawing of the configuration labeled bcc derivative is shown in Fig. 3.5. A similar drawing cannot be made for the special rhombohedral tiling given by the second configuration since the two pales are far from collinear with the pit.

What's going on here? First, we note that the left configuration has seven nodes while the right has only six, but the six in common are on the same Wyckoff sites and point positions. We then note that on the orbifold drawing, in the lower right of the figure, the h2 axis lies directly between the (e) and (f) sites. Since a separatrix line can never traverse more than one isometry zone (i.e. Wyckoff site zone), there has to be another critical point at point (h). According to the special rhombohedral indexing, this point would have to be a degenerate critical point with a cubic (triple point) algebraic dependence rather than quadratic along the (e) to (f) vector since the density is heading downhill along that vector. We can always decompose a degenerate critical point into several nondegenerate critical points, but then we would be in trouble satisfying the Euler-Poincare relationship described in Sect. 3. The obviously related (c) and (d) Wyckoff sites must be assigned to the same Morse function levels, which then produces the correct configuration shown in the lefthand drawing.

In other situations, missed critical points may make one of the critical points found appear to be degenerate. In our experience to date, a critical net that is not a Morse function has always been traceable to misindexing caused by the omission of valid critical points. Once the peak positions have been assigned by positioning atoms and assigning their Gaussian thermal motion parameters, the rest of the critical net is fixed; it is just a case of determining what it is. In the simple structures we are discussing in this treatment, the thermal motion probability density is either constrained by symmetry to be isotropic or assumed to be isotropic and in any case has little effect on critical net details. Thus we omit smearing functions from the discussion other than to say they are isotropic, Gaussian, and mildly overlapping.

**Body-Centered Cubic Symmetry Breaking Family**

In order to point out some additional properties about orbifolds and
critical nets on orbifolds, we examine a series of related cubic space
group orbifolds that accommodate the body-centered cubic critical net.
The series of cubic space group orbifolds that are related by
group/normal-subgroup relationships starting
with Imm is shown in
the linearized critical nets of Fig.
4.3, which includes the cesium chloride and body-centered cubic
critical net crystal structure types. The symbols within the
nodes are lattice complex symbols, which will be defined and discussed
in Sect. 5. All we need to know about them for now is that I, P, and F
represent the
body-centered cubic, primitive cubic, and face-centered cubic
configurations of points, respectively, and P_{2}
is a
primitive cubic array with doubling of periodicity along each axis.
Group/subgroup relations for the
cubic space groups are discussed in Sect. 6. and Appendix A.

Notes on the above illustration:

- A straight arrow between graphs points toward a normal subgroup, a straight arrow within a graph points toward a site of "lower density", an arrow between adjacent levels within a graph indicates a critical net Morse function separatrix, and a curved arrow between nonadjacent levels within a graph indicates a symmetry axis of the space group orbifold that is not embedded into the critical net Morse function.
- A number greater than 1 labeling a line of a graph indicates a 2-, 3-, 4-, or 6-fold crystallographic rotation axis while 1 indicates a path within a general position zone.
- A primed number indicates the axis lies in a mirror.
- A thick circle indicates a projective plane suspension point arising from an inversion point not in a mirror.
- For a group/subgroup pair, each axis within the parent graph is either split into two identical axes or reduced in group order by one half (e.g., 4' -> 4' + 4', 4' -> 4, or 4' -> 2') in the subgroup graph.
- A superscript number on a lattice complex symbol denotes the degree of positional freedom at that site.
- Mult, the sum of Wyckoff multiplicities for a row of elements in a
graph, is the same for all groups in the illustration except #228, which
has 8 times that number because of its multiple cell (e.g., I -> I
_{2}) lattice complexes. - Integer ratios of adjacent multiplicities provide the coordination vector, Coord, for the bcc critical graph (8,2,6,4,2,4) denoting 8 passes around a peak, 2 peaks around a pass, etc. as illustrated at the top of Fig. 4.1.
- By adding the shortest peak to pit path (4' for #229) to the graph, we also obtain the number of peaks around pits (2) and pits around peaks (6) as coordination numbers.
- The extended coordination vector [e.g., (6)(8,2,6,4,2,4)(2) for bcc] can be used as a local topological description of critical net coordination topology for simple critical nets. However, local topology may not be adequate to differentiate two closely related structures, such as fcc and hcp. The global information provided by the lattice complex symbols resolves this superstructure problem.

**Alternative Illustration Techniques for Critical Nets on Orbifolds**

The component graphs in Fig. 4.4 were derived from space group specific fundamental domain drawings similar to that shown in Fig. 4.1. Fig. 4.4 also shows the I lattice complex critical net for each orbifold illustrated. An alternate approach, which seems more amenable to automation, is discussed in Appendix A. Making such sketches for the more complex situations requires distortion of a crystallographer's normal geometric intuition and learning some basic cut and paste tricks that all geometric topologists seem to know but never document (i.e., the tricks of the trade). Lacking any formal training in topology, our bootstrap approach involved lots of reading, short apprenticeship periods with a professional topologist consultant, and employment of topology graduate students from the University of Tennessee part time.

In our experience, one of the more difficult tasks is describing the
underlying topological space for a space group orbifold. A simplifying
feature of the linearized critical net on orbifold drawings
shown in Fig. 4.3 is that they can be used
without fully specifying the underlying topological space details.
The drawing in Fig. 4.3 gives the connectivity of the singular set
elements within the orbifold, but antipodal gluing pattern information
is omitted from topological spaces containing RP^{2} or
RP^{3} projective planes since those elements no longer have a
conventional shape in the linearized critical net on orbifold drawing.

There are several different underlying spaces for the orbifolds presented
in Fig. 4.3 (i.e., silvered 3-ball: 229, 221, 224, 223; 3-sphere: 211,
208; real projective 3-space (RP^{3}): 197;
double suspended projective 2-space (RP^{2}):
222, 201, 218, 228; and singly suspended projective 2-space adjoined to
silvered 3-ball: 204, 217).

The orbifold for the parent group, Imm, which is pictured in Fig 4.1, is also shown in the middle of the second row of Fig. 4.4. The red path shown in Fig. 4.4 is the critical net while symmetry axes not in the critical net are shown in blue. Two-fold axes are not labeled. Mirrors are indicated by blue stippling. There are two mirrors in Imm orbifold separated at the equator by 4- and 2-fold axes. On either side of that drawing are topologically equivalent drawings with one hemisphere flattened.

**Orbifold Transformations**

In Fig. 4.4, removing a mirror and dividing the order of
the equatorial axes by two
causes the opposite hemisphere to reproduce a mirror image of
itself as shown for the left
and right figures in the top row. They each now have a single mirror
boundary since the equator is no longer divided by even-order axes, which again
produces silvered 3-ball underlying spaces. When both mirrors are removed
simultaneously (top row middle), the entire bottom hemisphere must wrap
around to the top hemisphere carrying the interior 2-axis into a complete
loop and producing a 3-sphere underlying space, which is the normal space
for knots and links. Many polar (orientable) space groups yield orbifolds
with a 3-sphere (S^{3}) underlying space.

To form the two outside orbifolds in the third row from the corresponding ones on the second, we retain the mirrors shown as planes, removing the mirrors shown as hemispheres and again reduce the order of the equatorial axes by half. But now we have to double everything that was lying on the hemisphere surface where the mirror was located so we now have a cone with an antipodal identification operation on the cone's surface, which means the boundary cone is now a projective plane. The order of the interior 2-fold axis is halved, and the critical net along that path shown in orange to indicate it is now along a 1-axis (general position) rather than a 2-axis. The resulting underlying space is half bounded by a mirror and half by a projective plane with a suspension point at the cone apex; thus it may be called a singly suspended silvered ball. The middle orbifold in the third row has both mirrors removed producing two antipodal projective plane cones and that underlying space may be called a double suspension.

The final orbifold in the bottom row right requires a more complex set
of cut and paste operations in the transition from the third row left to
the bottom row left in which the 3-axis is moved from the cone surface
to the cone interior. This is related to the sliding gluing edge
phenomenon for projective planes described in Sects. 2.1 and 2.2. The rest of
the operations are simple repetitions of those used previously. The
underlying space for I23 turns out to be real projective 3-space
RP^{3}. It
has an antipodal relationship relative to the center of the ball for all
points on the surface of the ball. Care must be taken not to confuse
this with a crystallographic inversion center which holds throughout
space. All antipodal relations operate only on the gluing edge, not the
interior.

5. Lattice Complexes on Critical Nets on Orbifolds

3. Introduction to Critical Nets

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*Page last revised: June 12, 1996*