Polarization moments transversal components

Let us now imagine that the function p(0,axial symmetry with respect to the z-axis, as, e.g., in Fig. 4.1(6), and that it is created by pulsed excitation at t = 0. In this case its expansion over multipoles includes transversal polarization moments pq (these are complex quantities) where Q 0. If the precession frequency uj< is much larger than the relaxation rate, then the components pQ will depend on time, in the time scale wj,1, following (4.9), according to the simple harmonic law ... 

We thus see that the purely magnetic evolution (4.11) of the polarization moment is, in essence, a linear change in time of its phase ip according to (4.12), with conservation of the module Mod pq (circle in Fig. 4.1(c)). The factor e1 means that the dependence pq (t) is periodic with a period Tq = 2ir/Qu)ji, i.e. that each transversal component of a polarization moment passes into itself with its own frequency Quj>. This is in full agreement with what has been said before in Section 2.3 on the connection between the coherence and symmetry of p(6,ip). The model presented affords the conservation of the shape of the angular momenta distribution p(0,ip) in the course of precession (see Fig. 4.1(6)). Incidentally, it may not seem quite appropriate in this context to maintain the statement that the magnetic field itself destroys coherency , as described by the transversal components pq, Q 0. Indeed, it follows from (4.11) that at... 

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... 

In the molecules are upright, and since there is no head-to-tail ordering (the director being apolar) there is no polarization normal to the layers. Moreover, even if the molecules themselves are chiral, there is equal probability of their assuming any orientation about their long axes. Hence the transverse component of the dipole moment is averaged out and there is no net polarization parallel to the layers. 

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. 

Similar to the treatments for Isotropic phases we have to compute the sum of the induced and orientation polarization Following Maier and Meier, we consider a molecule with a permanent dipole moment /x that makes an angle )3 with the long molecular axis The anisotropy of the polarizability is accounted for by two principal elements a, and a, along and transverse to The components of fx (in the molecule-fixed coordinate system) are ix, = fi cos /3, fi, = fi sin /3. The components a, fi, and (in the laboratory system) depend on the orientation of the molecular axis, which makes an angle with the z axis, and thus are connected with the order parameter S ... 

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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