Liquid Crystals - Iam-Choon Khoo

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href="#fb3_img_img_10a945b2-c05c-5bb5-bf80-26c39d959147.png" alt="StartLayout 1st Row ModifyingAbove normal upper Gamma With right harpoon with barb up Subscript m Baseline equals ModifyingAbove upper M With right harpoon with barb up times ModifyingAbove upper H With right harpoon with barb up 2nd Row equals normal upper Delta chi Superscript m Baseline left-parenthesis n dot ModifyingAbove upper H With right harpoon with barb up right-parenthesis left-parenthesis n dot ModifyingAbove upper H With right harpoon with barb up right-parenthesis period EndLayout"/>

      These electric and magnetic torques play a central role in various field‐induced effects in liquid crystals.

      3.4.1. Linear Susceptibility and Local Field Effect

Image described by caption.

      The optical dielectric constants originate from the linear polarization ModifyingAbove upper P With right harpoon with barb up generated by the incident optical field ModifyingAbove upper E With right harpoon with barb up Subscript o p on the nematic liquid crystal:

      From the defining equation

      (3.30b)ModifyingAbove upper D With right harpoon with barb up equals epsilon 0 ModifyingAbove upper E With right harpoon with barb up plus ModifyingAbove upper P With right harpoon with barb up equals ModifyingAbove Above ModifyingAbove epsilon With right harpoon with barb up With right harpoon with barb up colon ModifyingAbove upper E With right harpoon with barb up comma

      Here ModifyingAbove Above ModifyingAbove chi With right harpoon with barb up With right harpoon with barb up Superscript left-parenthesis 1 right-parenthesis is the linear (sometimes termed “first order”) susceptibility tensor of the nematics. ModifyingAbove Above ModifyingAbove chi With right harpoon with barb up With right harpoon with barb up Superscript left-parenthesis 1 right-parenthesis is a macroscopic parameter and is related to the microscopic (molecular) parameter, the molecular polarizabilities tensor αij, in the following way:

d Subscript i Baseline equals alpha Subscript italic i j Baseline upper E Subscript j Superscript l o c Baseline comma

      (3.31a)ModifyingAbove d With right harpoon with barb up equals ModifyingAbove Above ModifyingAbove alpha With right harpoon with barb up With right harpoon with barb up colon ModifyingAbove upper E With right harpoon with barb up Superscript l o c Baseline comma

      (3.31b)ModifyingAbove upper P With right harpoon with barb up equals upper N ModifyingAbove d With right harpoon with barb up comma

      where di is the ith component of the induced dipole ModifyingAbove d With right harpoon with barb up and N is the number density. In Chapter 8, a rigorous quantum mechanical derivation of α in terms of the dipole matrix elements or oscillator strengths and the energy levels and level populations will be presented. The connection between the microscopic parameter αij and the macroscopic parameter χij is the local field correction factor (i.e. the difference between the externally applied field and the actual field as experienced by the molecules). Several theoretical formalisms have been developed to evaluate the field correction factor, ranging from simplified to complex and sophisticated ones.

      Most of the approaches used to obtain the local field correction factor are based on the Lorentz results [5], which state that the internal field (i.e. the local field as experienced by a molecule ModifyingAbove upper E With right harpoon with barb up Superscript l o c in a solid) is related to the applied field ModifyingAbove upper E With right harpoon with barb up Superscript a p p by

      (3.32a)ModifyingAbove upper E With right harpoon with barb up Superscript l o c Baseline equals StartFraction n squared plus 2 Over 3 EndFraction ModifyingAbove upper E With right harpoon with barb up Superscript a p p Baseline period

      In particular, Vuks [10] analyzed experimental data and proposed that the local field in an anisotropic crystal may be taken as isotropic and expressed in the form

      (3.32b)ModifyingAbove upper E With right harpoon with barb up Superscript l o c Baseline equals StartFraction left pointing angle n squared right pointing angle plus 2 Over 3 EndFraction left pointing angle ModifyingAbove upper E With right harpoon with barb up right pointing angle comma

      where left pointing angle n squared right pointing angle equals one third left-parenthesis n Subscript x Superscript 2 Baseline plus n Subscript y Superscript 2 Baseline plus n Subscript z Superscript 2 Baseline right-parenthesis and nx, ny, and nz are the principal refractive indices of the crystal. This approach has been employed in the study of liquid crystals [11]. A more generalized expression for anisotropic crystals is given in

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