Orientation distribution definition

As a prelude to the discussion it is necessary to consider the definition of orientation in terms of the Euler angles, and the definition ofan orientation distribution function in terms ofan expansion ofLegendre functions. These definitions set the scene for examining the information which can be obtained from different spectroscopic techniques. In this review, infra-red and Raman spectroscopy and nuclear magnetic resonance, will be considered. 

Fig. 3.2.2 The orientational distribution function, (a) Definition of the orientation angle 0.

The orientational distribution fimction P (cos 0) enters the shape of the wideline spectrum 5(f2) in a slightly hidden way. The angular dependence of the resonance frequency is given by (3.1.23) via the orientation of the magnetic field in the principal axes system XYZ of the coupling tensor (cf. Fig. 3.1.2), while the orientational distribution function specifies the distribution of the preferential direction n in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). Figure 3.2.3 shows the relationship between the different coordinate frames and the definition of the relative orientation angles. 

Since xj/ is an even function, the average of products of odd number of components of p is zero. Also, since the distribution function is normalized (Eq. 5.2) and p is a unit vector, one has akk= 1- From the definition, one can also find the symmetry = Uji. Therefore, the second-order orientation Uy has only five independent components. Figure 5.1 shows some extreme cases of fiber orientation distributions and corresponding values of the second-order orientation tensor components. 

Figure 4. The three main substrate classes (a) smooth surfaces on which surface molecules have a definite orientational distribution (represented surface obtained on a rubbed polyimide film [52]) (b) interpenetrable surfaces of dangling chains (c) topographies (represented grooved surface) with a favorable (left) and unfavorable director field R. In all cases, a is the macroscopic anchoring direction.

This follows from the definition for the orientational distribution function f(0) ... 

For a one-component chiral nematic of D synmietry, a local D2 symmetry, and an effective molecular symmetry of D2, the four local order parameters S, D, A and B of the slices of the chiral nematic phase need to be considered [8]. Because the numerical factors In the definitions are different in the literature, the functions of the Eulerian angles are given also in EQNS (4) to (7) where now the orientational distribution function is normalised ... 

In turn, the coefficients of the expanded series JV, and Q can be obtained from equations (8) and (9) respectively. Clearly, from equations (8) and (9), the coefficients and Qj are the averages of the distribution functions with respect to the Imnth and Imth orders of spherical harmonics respectively. In other words, the coefficients are the Imnth and /mth averages of the orientation distribution functions, and are a sort of general definition of the orientation factors as will be discussed later. 

This definition of the quadrupole is obviously dependent upon the orientation of the chargi distribution within the coordinate frame. Transformation of the axes can lead to alternativi definitions that may be more informative. Thus the quadrupole moment is commonl defined as follows ... 

In words, s describes the interaction of the solute charge distribution component p, with the arbitrary solvent orientational polarization mediated by the cavity surface. The arbitrary weights p,, previously defined by (2.11), enter accordingly the definition of the solvent coordinates, and reduce, in the equilibrium solvation regime, to the weights tv,, such that the solvent coordinates are no longer arbitrary, but instead depend on the solute nuclear geometry and assume the form se<> = lor. weq. In equilibrium, the solvent coordinates are correlated to the actual electronic structure of the solute, while out of equilibrium they are not. 

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