Rigid dipoles

Note 1. The Free Energy Lost by a Polar Dielectric in an Electrostatic Field. Let Fig. 76 depict a permanent rigid dipole whose axis makes an angle 0 with a uniform field of intensity E. If y is the dipole moment, the potential energy of the dipole in the field is — Ey cos 0. If the dipole is held in this fixed position, any increment dE in the intensity of the field will clearly mean a change in the potential energy of the dipole, equal to — y cos 0 dE. 

In practice a polar molecule is not a rigid dipole but suffers some distortion in the field. In this case we shall assume that in the step (a) not only is the axis of every dipole held in a fixed position, but also the distortion of the molecule is held at a fixed value. During this step (a) the total polarization P of the dielectric will remain constant. 

The total mutual potential energy of the rigid dipole and the charge q... 

The first step in studying the orientation ordering of two-dimensional dipole systems consists in the analysis of the ground state. If the orientation of rigid dipoles is described by two-dimensional unit vectors er lying in the lattice plane, then the ground state corresponds to the minimum of the system Hamiltonian... 

Second, an alternative hat-curved-cosine-squared potential (HC-CS) model is also considered, which, as it seems, is more adeuate than the HC-HO model. The CS potential is assumed to govern angular deflections of H-bonded rigid dipole from equilibrium H-bond direction. The HC-CS model agrees very well with the experimental spectra of water. 

This expression is obtained by using only the Maxwell equations and the equations of classical mechanics and the condition (26) for q(t). Because of this, Eq. (27b) is valid irrespectively of the type of the projection = p g. This projection may correspond to reorientation of a rigid dipole or to vibration of bound charges of a nonrigid dipole. 

This section presents the continuation of Section V. In the latter a new model [10] termed the hat-curved model was described, where a rigid dipole reorients in a hat-like intermolecular potential well having a rounded bottom. This well differs considerably from the rectangular one, which is extensively applied to polar fluids. Now the theory of the hat-curved model will be generalized, taking into account the non-rigidity of a dipole that is, a simplified polarization model of water is described here. 

Liquid T (K) A. Fitted Parameters Parameters Relevant to a Rigid Dipole Moment Parameters Relevant to a Nonrigid Dipole Moment ... 

Figure 35. Frequency dependence in the submillimeter wavelength region of the real (a, b) and imaginary (c, d) parts of the complex permittivity. Solid lines Calculation for the composite HC-HO model. Dashed lines Experimental data [51]. Dashed-and-dotted lines show the contributions to the calculated quantities due to stretching vibrations of an effective non-rigid dipole. The vertical lines are pertinent to the estimated frequency v b of the second stochastic process. Parts (a) and (c) refer to ordinary water, and parts (b) and (d) refer to heavy water. Temperature 22.2°C.

Reorientation of a rigid dipole l in a narrow hat-curved potential... 

We shall remove an important drawback of the polarization model described in Section VI by considering another variant of a composite model than that described in previous Section VILA. We use again a linear-response theory to find the contribution of a vibrating dipole to the total permittivity . We split the total concentration N of polar molecules into the sum Nm and Nv b, where each term refers to rotation of a like rigid dipole (viz. with the same electric moment p) but characterized by different law of motion ... 

Change of g influences insignificantly the resonance line pertinent to the ensemble of rigid dipoles reorienting in a hat-like potential well, since the resonance absorption/loss peaks are usually placed at much higher frequencies than the Debye loss peak. 

All the SB in crystals are thus of some kind of order-disorder transitions with JT or PJT origin of the ordering distortion units. Order-disorder transitions in crystals with rigid dipole molecules may be considered as an extreme case of such SB when two or more possible positions of the molecule in the lattice are regarded as due to corresponding PJT distortions from their averaged high-symmetry hypothetical formation (similar to enantiomers, see below). 

B. Higher Harmonics Generation in Field-Induced Birefringence by a Suspension of Rigid Dipoles Zero Bias Field... 

These formulas coincide with those derived in Ref. 137 and explain the intermediate asymptotics, 40 = const(o>) confirmed there experimentally. The limiting value of the phase is less than half that for rigid dipoles /(( xB —> 00) = 90°. 

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