Singular lines and singular points of ferromagnetic spin systems and of nematic liquid crystals

2014 The spins in a ferromagnet define a vector field. Their simplest line singularities are therefore disclinations of unit strength, and their point singularities are of the types analysed by Poincaré. A nematic liquid crystal is normally unpolarised. As a result, a vector placed along the nematic axis at a point may be carried continuously through « good » liquid crystal and return inverted. The simplest line singularities are therefore disclinations of half unit strength. A double circuit restores the vector to its original orientation. The director field thus has the character of the wave function of a spinor. The extension of this idea to three dimensions allows a preliminary description of the point singularities of nematic structures. LE JOURNAL DE PHYSIQUE TOME 33, NOVEMBRE-DÉCEMBRE 1972, Classification Physics Abstracts 02.00, 14.82, 16.40, 17 64 1. The singularities of ferromagnetic spin systems. We begin with an outline of Poincaré’s analysis of the singularities of vector fields in two and three dimensions. We then use this analysis and the theory of disclinations to consider the Bloch wall separating two ferromagnetic domains magnetised in opposite directions, the Néel line which separates two portions of the Bloch wall which have opposite helicities, and the Bloch point which separates two Néel lines which have opposite disclination strengths. A similar analysis is possible for Néel walls, Bloch lines and their singular points. 1.1 POINCARÉ’S ANALYSIS IN TWO DIMENSIONS. We consider a vector field (X, Y) in the plane (x, y). In Poincaré’s analysis [1], (X, Y) is the velocity of a particle situated at (x, y) ; in our analysis it is an unnormalised indicator of the direction of magnetisation at (x, y). We suppose that, in the neighbourhood of the origin, where all the coefficients are real. Unless Xo and Yo both vanish, the direction (X, Y) is well determined at the origin, which is then an ordinary point. The simplest singularities are those for which Xo = Yo = 0, while ai , bl, a2, b2 are not all zero. They are in general screw disclinations of strength ± 1. Disclinations of larger integral strengths may be produced by the confluence of these unit disclinations. We look for the regions in which the vector field (X, Y) is parallel or antiparallel to the radius vector (x, Y). This requires which has non-zero solutions only if Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019720033011-120108900