A Möbius strip of single crystals Satoshi Tanda*¹, Taku Tsuneta*¹, Yoshitoshi Okajima*¹, Katsuhiko Inagaki*¹, Kazuhiko Yamaya*', Noriyuki Hatakenaka*2 *1 Department of Applied Physics, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan e-mail: tanda@eng.hokudai.ac.jp *² NTT Basic Research Laboratories, Atsugi, Kanagawa 243-0198, Japan Reprinted from Nature, Vol. 417, No. 6887, pp. 397-398, 23 May 2002 © Nature Publishing Group, 2002
A crystalline ribbon of niobium and selenium can be coaxed into a novel topology. Möbius strip is produced by twisting a ribbon of material through 180° and joining its two ends, resulting in a distinct, one-sided topology¹-⁴. Here we describe a Möbius structure formed by crystals of a compound of niobium and selenium, NbSe 3 . It is surprising that a crystalline ribbon should adopt this exotic topology in view of its inherent rigidity, which would be expected to prevent it from either bending or twisting. NbSe3 is a typical low-dimensional inorganic conductor, which shows charge- density-wave  (CDW)  phase  transitions  at 58 K and 145 K. Its crystals are synthesized by chemical-vapour transport and assume fibrocrystalline ribbon and whisker shapes. We have been able to create a Möbius strip of single NbSe 3 crystals by modifying the conventional growth conditions. We soaked a mixture of Se and Nb powder at 740 °C in an evacuated quartz tube for a period varying from a few hours to a few days. The growth conditions differed from those that are typically used, in that they relied on a furnace that had a large temperature gradient; this creates a crucial non-equilibrium state inside the quartz tube in which selenium circulates through vapour, mist and liquid (droplet) phases. Figure 1 shows scanning electron micro- scopic (SEM) images of different topologies that are assumed by NbSe3 crystals produced under these conditions. Ring diameter and width are typically 100 μm and 1 μm, respectively (Fig. 1a); Möbius crystals are typically around 50 μm in diameter and less than 1 μm in width (Fig. 1b); and ‘figure-of- eight’ crystal strips (Fig. 1c), with a double twist, have a circumference of about 200 μ m and a width of 1 μm. We confirmed by X-ray diffraction and transmission electron microscope diffraction that the ring and Möbius structures are equivalent to a conventional crystalline ribbon with single- phase and monoclinic properties. The mechanism of ring formation by the ribbon-shaped NbSe 3 crystals depends on their being bent by the surface tension of the viscous selenium droplet on which they have grown. The ribbon grows along the equator of the droplet to minimize bending energy, finally meeting its own tail to form a perfectly seamless ring (Fig. 1d). Growth of the Möbius strip is difficult by comparison because of the twisting involved. Bending of a bar or beam is accompanied by twisting, and the low symmetry in a monoclinic crystal also promotes bending with twisting 5, as shown their twisting (shown schematically beneath images; white ribbons represent NbSe, crystals, red spheres represent selenium droplets). a, Ring structure (0 π twist). b, Mobius strip (1 π twist). c, Figure-of-eight strip (2 π twist). d, NbSe, fibres (white streaks) bend into rings around a selenium drop of diameter about 50 μ m. The NbSe, ribbon is spooled onto a selenium droplet by surface tension until its ends bind together. e, High-magnification image showing a twist in a ribbon of NbSe, crystals; spooling of the ribbon can produce a twist, as in the formation of a Möbius crystal, owing to its anisotropic elastic properties. Scale bars, 10 μ m. in Fig. 1e. Droplet rotation, which we observed frequently, could also help to produce the twist. The figure-of-eight crystal strips arise as a result of either double encircling or dou- ble twisting. According to White’s theorem 6, a doubly encircling loop is topologically equivalent to a 2π-twisted loop, which is analogous to a ring conformer of DNA . These different topological arrange- ments of NbSe  all show the characteristic CDW phase transitions, indicating that the samples are crystalline and relatively ordered, and that the CDW displacements are identical to those in conventional, untwisted crystals 8,9 . This was demonstrated by satellites in the electron-diffraction pat- terns (CDW wave vector Q = 0.24 ± 0.01), by anomalies due to CDW phase transitions in the temperature dependence of the resistivity at 145 K and 54 K, and by non- linear conduction due to CDW sliding. We have developed a new growth tech- nique for topological crystals by using a spherical droplet as a spool. This technique could be extended to a wide range of mat- erials, and we have already obtained topo- logical variants of tantalum compounds with selenium and sulphur (TaSe3 and TaS 3 ). It may also be possible to grow crys- tals to a desired size by controlling the non- equilibrium conditions inside the furnace and thus the size of the droplets. Our crystal forms offer a new route to exploring topo- logical effects in quantum mechanics 10-13, as well as to the construction of new devices. Satoshi Tanda*, Taku Tsuneta*, Yoshitoshi Okajima*, Katsuhiko Inagaki*, Kazuhiko Yamaya*, Noriyuki Hatakenaka† * Department of Applied Physics, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan e-mail: tanda@eng.hokudai.ac.jp † NTT Basic Research Laboratories, Atsugi, Kanagawa 243-0198, Japan 1. Walba, D. M., Richards, R. M. & Haltiwanger, R. C. J. Am. Chem. Soc. 104, 3219–3221 (1982). 2. Mila, F., Stafford, C. & Capponi, S. Phys. Rev. B 57, 1457-1460 (1998). 3. Hayashi, M. & Ebisawa, H. J. Phys. 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Competing financial interests: declared none. fitness of randomly mutated strains of Escherichia coli. a , The fitness values for the ancestral strains of mutators and non- mutators are the same (left); 'mutated', fitness los after 3,240 generations of mutation accumulation;  qroE c , fitness recovery of strains in which groE is overexpressed. Culture conditions, fitness assays and phenotypic markers are described elsewhere 11 , as is the mutation-accumulation protocol12. b, The regulated E. coli groE operon was changed to a constitutive form (groE ) by putting the structural genes under the control of the β -lactamase gene promoter (P bla). Recombinant genotypes were obtained by transduction using P1 wi (ref. 13). The groE  construct contains part of the flanking yieH gene, the inducible tetracycline- resistance operon ( tetA/R ) for transductant selection, the Pbla promoter, and the groE coding regions (S and L). Restriction- enzyme cutting sites are indicated; P, promoter. Further details are available from the authors. mutated strains lost fitness (Fig. 1a) as a result of the accumulation of deleterious mutations (W= 0.5168 ± 0.0159 for muta- tors; W=0.5208 ± 0.0087 for non-mutators; paired t-tests, P< 0.0001). We then replaced the regulated groE operon of the mutated strains with a consti- tutive allele (groEc ; Fig. 1b). To estimate the fitness of the groEc strains, we ran competi- tion experiments against the corresponding ancestors on DM25. Surprisingly, none of the groEc strains reached the expected cell density during growth overnight. A simple explanation could be that an overproduc- tion of GroEL of about 86 ± 16-fold is deleterious because it diverts amino acids away from other cellular functions. To test this, we grew each groEc strain and its ancestor in DM25 supplemented with increasing concentrations of tryptone (a mixture of peptides and amino acids). We found that groE overexpression was deleteri- ous in the presence of small amounts of tryptone, but that there were no significant differences in cell density between groEc strains and their non-mutated ancestors at tryptone concentrations of 0.1% or higher. In endosymbionts, amino acids are not limiting because these are abundant in their intracellular environment". To determine whether fitness estimates depend on the environment, we estimated fitness on DM25 and on DM25+0.1% tryptone. Both fitness estimates were corre- lated (partial correlation test, P< 0.0001). Figure 1a shows that the average fitness of the groE  strains derived from non- mutator strains (W=0.8801 ± 0.0214) was 75.9% greater than that of the mutated strains (paired t-test, P< 0.0001), but was 12% less than that of the ancestors (P = 0.0002); the average fitness of the groEc strains derived from evolved mutators was W=0.8152 ± 0.0167,  which is 61.6% greater than that of the mutated strains (paired t-test, P<0.0001) but 18.48% less than that of the ancestral strains (P < 0.0001). Is fitness recovery a result of the buffer- ing of deleterious effects by GroEL, or is it simply a general benefit associated with increased concentrations of GroEL? In favour of the first possibility, the advantage of GroEL overproduction is evident only in an  amino-acid-rich  environment;  also,  if mutation compensation is occurring, we would expect a positive correlation between the extents of fitness loss and recovery, as evidenced by their partial correlation (1-tailed test, P=0.0089). 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