Polarization moments longitudinal components

Assuming an axially symmetric potential, the anisotropy energy of n) will be an even function of the longitudinal component of the magnetic moment s n. The averages we need to calculate are aU products of the form = (n =i (cn ))a> where the c are arbitrary constant vectors. Introducing the polar and azimuthal angles of the spin d, tp), we can write as... 

It follows from (4.9) and (4.10) that the magnetic field does not affect the longitudinal component of the polarization moments bPo, aPo (they are real quantities) where Q = q = 0, since the magnetic terms become equal to zero. This can easily be understood from the following. In accordance with Section 2.3, an ensemble of angular momenta possessing only... 

Since longitudinal orientation /d can arise only from diagonal matrix elements /mm, which are not affected by external perturbation in the form of anisotropic collisions or in the form of an external field, at linearly polarized excitation we have /d = 0 irrespective of the type of perturbation. This means that the orientation which may emerge must be transversal, i.e. the corresponding components / of the polarization moment must appear. According to (5.40) we can write... 

The question arises why cis-polyisoprene, different from poly(vinyl-acetate), shows in its dielectric spectrum the chain reorientation. The reason becomes clear when we look at the chemical constitution of polyisoprene, and focus in particular on the associated dipole moments. Figure 5.22 displays the chemical structure. The main point is that isoprene monomers are polar units which possess a longitudinal component p of the dipole moment, which always points in the same direction along the chain. As a consequence, the longitudinal components of the dipoles of all monomers become added up along the contour, giving a sum which is proportional to the end-to-end distance vector R. In the dielectric spectrum the kinetics of this total dipole of the chain is observable, hence also the chain reorientation as described by the time dependence R t). 

For a Rouse-chain built up of Nr polar sequences, each one carrying a dipole moment with a longitudinal component pjj, the total dipole moment Pp is given by... 

It is of interest to compare these results with those for the field dependencies of the relaxation times and for T for the longitudinal and for the transverse polarization components of a polar fluid in a constant electric field Eq. As shown in [52, 55] the relaxation times and T are also given by Eqs. (5.55) and (5.56), where = nEJkT, p. is the dipole moment of a polar molecule and is the Debye rotational diffusion time with = 0. Thus, Eqs. (5.55) and (5.56) predict the same field dependencies of the relaxation times Tj and T for both a ferrofluid and a polar fluid. This is not unexpected because from a physical point of view the behavior of a suspension of fine ferromagnetic particles in a constant magnetic field Hg is similar to that of a system of electric dipoles (polar molecules) in a constant electric field Eg. 

Consider first an anisometric molecule with the longitudinal p, and transversal p, permanent dipole moments in an isotropic phase. There are two relaxation modes mode 1, rotations of p, around the long axis, and mode 2, reorientation of p,. Figure 10-1. The mode 1 has a smaller relaxation time, Tj < Tj, because of the smaller moments of inertia involved. When this isotropic fluid is cooled down into the NEC phase, the dynamics is affected by the appearance of the nematic potential associated with the orientational order along the director n. The mode 1 remains almost the same as in the isotropic phase, and contributes to both the parallel and perpendicular components of dielectric polarization (determined with respect to n). Mode 2 is associated with small changes of the angle between p, and n it contributes to the parallel component of dielectric polarization. Mode 3 is associated with conical rotations of p, around the director (as the axis of the cone) it is effective when the applied electric field is perpendicular to n and contributes... 

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