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of our analysis. The zig-zag pattern is comprised of Bi that are all in the same plane (denoted with arrows). This plane is defined by the displacement of any Bi atom in the chain, so the central Bi atom can only partici- pate in zig-zag ordering in one direction at a time. The two inline nearest neighbors of the central atom should have a tendency to displace opposite from the central 

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Bi displacement vector. We can therefore quantify the correlation by examining the angle φ between the dis- placement vector of the central Bi versus the displace- ment vectors of the two relevant nearest neighbors. Each double-tetrahedral shape in the supercell is examined in this manner. A schematic showing selected orientations of neighboring Bi displacements and their corresponding values of φ is given in Fig. 10. 

In Fig. 11(a), the relative angle φ histogram is plotted for the RMC simulations. It does not have the same six- fold modulation as the averaged Bi angles θ in Fig. 9(b) because φ is defined relative to its neighbors , while θ is defined relative to the crystal axes . We investigate the tendency for ordering by shuffling Bi displacements: the set of all individual Bi displacements from their ideal sites is preserved, but redistributed randomly among the Bi atoms. As a result, their orientations with respect to each other are disrupted. Note that, as a result of the removal of this local correlation, the shuffled φ curve is simply a six-fold cosine curve with modulation analo- gous to θ in Fig. 9(b). The difference between the RMC and shuffled φ , plotted in Fig. 11(b), shows a peak at φ = 180 $^{◦}$, indicating a preference for zig-zag ordering. 

