9. Equations 4.3 and 4.4 relate the scalar orientational order parameter S for a liquid crystal to the molecular orientational distribution function P(,). (a) Show by an explicit calculation that in the isotropic state, where P(,) is uniform over 0 and 02,S is zero. (b) Approximate the distribution P(,) in the nematic state by assuming P(,) is independent of (uniaxial), and P()={cos00/2otherwise and calculate the orientational parameter S. (c) A material undergoes the following sequence of structural transitions: crystal to smectic at 110C, smectic to nematic at 130C, and nematic to isotropic liquid at 170C. Sketch representative plots that illustrate the orientational parameter S(T) and, for each of the four states, the angular-dependent pair-distribution function g(r,). (d) A material transforms from the crystal to plastic-crystal state at 125C and from the plastic crystal to isotropic liquid at 165C. Sketch representative plots that illustrate S(T) and, for each structural state, g(r,). 9. Equations 4.3 and 4.4 relate the scalar orientational order parameter S for a liquid crystal to the molecular orientational distribution function P(,). (a) Show by an explicit calculation that in the isotropic state, where P(,) is uniform over 0 and 02,S is zero. (b) Approximate the distribution P(,) in the nematic state by assuming P(,) is independent of (uniaxial), and P()={cos00/2otherwise and calculate the orientational parameter S. (c) A material undergoes the following sequence of structural transitions: crystal to smectic at 110C, smectic to nematic at 130C, and nematic to isotropic liquid at 170C. Sketch representative plots that illustrate the orientational parameter S(T) and, for each of the four states, the angular-dependent pair-distribution function g(r,). (d) A material transforms from the crystal to plastic-crystal state at 125C and from the plastic crystal to isotropic liquid at 165C. Sketch representative plots that illustrate S(T) and, for each structural state, g(r,)