The Study Of Smectic-A Liquid Crystal Based On Quantum Theory

When liquid crystal (LC) is in smectic-A (SA) phase, its molecules disperse in layers (sandwich) and the molecular long axis tropism is in some order and the director is perpendicularto the layers. Three order parameters that character SA are η,τ ,σ. Considering collectiveeffects of reciprocity systemic, elementary excitation of condensed physics method is adopted. Quantum theory is used to study SA- New microcosmic theory of SA is given.Firstly, quantum theory is introduced to liquid crystal microcosmic theory. LC molecule has two degrees of freedom, molecular long axis tropism Ω and center of mass of molecule r . Considering that the direction of Ω is the same as of rotational angular momentum L aboutmolecular long axis, Ω can be expressed as L divided by its length. New operator τ can be constructed by using L . And then the part relative to Ω in molecules among potential can beexpressed as operator τ. Considering that r limit to a small area we adopt lattice model. In the model, cell center form space lattice. Quantum state of center of mass of molecule is corresponding to the particle in potential trap.Secondly, Hamiltonian operator is taken into account. McMillan form of the singlemolecule potential function is adopted, H is express as τ(i)(i = 1,2,3...),pi and Pi .pi, is position direction of center of cell and Pi.is corresponding momentum. Boson operators ai and ai+ called anti-orientatianon are introduced by exerting Holstein-Primakoff transformation tooperator τ(i). As center of mass of molecule r is concerned its Hamiltonian state is harmonicoscillator state and phonons bi and bi+ are introduced in. In the paper, we express H as thistwo kinds of operators. In some approximation, H can be express as the summation of constantterm and perfect "anti-orientatianon" gas Hamiltonian term and perfect "phonons" gas term.Thirdly, order parameters η,τ ,σ of smectic-A phase can be expressed as this two kindsof elementary excitations and its expression is accomplished. By using quantum statistical theory, we give anti-orientatianon numbers and phonons numbers. We calculate order parametersη,τ ,σ by using numerical value method, and compare it with experiment value.