We evaluate numerically the probability of linking, i.e. the probability of a given pair of self-avoiding polygons (SAPs) being entangled and forming a nontrivial link type L. In the simulation we generate pairs of SAPs of N spherical segments of radius rd such that they have no overlaps among the segments and each of the SAPs has the trivial knot type. We evaluate the probability of a self-avoiding pair of SAPs forming a given link type L for various link types with fixed distance R between the centers of mass of the two SAPs. We define normalized distance r by where denotes the square root of the mean square radius of gyration of SAP of the trivial knot 01. We introduce formulae expressing the linking probability as a function of normalized distance r, which gives good fitting curves with respect to χ2 values. We also investigate the dependence of linking probabilities on the excluded-volume parameter rd and the number of segments, N. Quite interestingly, the graph of linking probability versus normalized distance r shows no N-dependence at a particular value of the excluded volume parameter, rd = 0.2.

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We theoretically investigate the fluorine doping in LaOBiS 2-type quaternary compounds (LaOBiS 2 , NdOBiS 2 , LaOBiSe 2 , and LaOSbSe 2), which are promising candidates of thermoelectric and superconducting materials. These compounds possess a layered structure comprising blocking LnO (Ln = La, Nd, etc.) layers and conducting PnCh 2 (Pn = Bi, Sb; Ch = S, Se) layers. Their carrier concentration is generally tuned via substitutional doping of F atoms in the O site for improving the thermoelectric performance or the superconductivity; however, the tunability of the electrical properties via F doping strongly depends on constituent elements. In order to elucidate the difference, we theoretically examine the electronic and structural properties of these F-doped systems using first-principles calculation. Our results show that the monoclinic distortion of the mother compound, which is closely related to the Pn element, can drastically decrease the capability of F doping. Replacement of the Ln atom from La to Nd in LnOBiS 2 makes F doping difficult, which is consistent with experimental observation. We also find that the tetragonal structure is gradually stabilized by F-doping for all the systems investigated in this study. Our results will be important knowledge for controlling the electrical properties of LaOBiS 2-type compounds both as thermoelectric and superconducting materials.