Distribution, uniaxial, symmetric

In order to evaluate quantitatively the orientation of vibrational modes from the dichroic ratio in molecular films, we assume a uniaxial distribution of transition dipole moments in respect to the surface normal, (z-axis in Figure 1). This assumption is reasonable for a crystalline-like, regularly ordered monolayer assembly. An alternative, although more complex model is to assume uniaxial symmetry of transition dipole moments about the molecular axis, which itself is tilted (and uniaxially symmetric) with respect to the z-axis. As monolayers become more liquid-like, this may become a progressively more valid model (8,9). We define < > as the angle between the transition dipole moment M and the surface normal (note that 0° electric field of the evenescent wave (2,10), in the ATR experiment are given by equations 3-5 (8). 

The above general analysis is readily extended to the case of biaxially oriented samples, as shown by Nomura et al If the orientation distribution is symmetrical with respect to the X1X2, XjJ and X2 3 planes, then biaxial orientation fimctions of the form sin cos2 may be obtained from measurements of A2—Aj)l A3+A2+Ai). Assuming that the distributions over 6 and are independent, then functions such as cos2 = 2cos —1 can subsequently be obtained, as shown by Stein. Owing to experimental difficulties, however, no applications of these analyses seem to have yet been made. For both uniaxial and biaxial orientation. 

In the case of a uniaxially symmetric distribution of the molecular axes M, the intensities corresponding to the P and A directions lying along the fixed-frame axes (Z corresponds to the symmetry axis) can be conveniently expressed through the following quantities ... 

Before considering the fluorescence polarization technique which leads to a measurement of the orientation distribution of characteristic vectors, we introduce some convenient quantities to describe the orientation of uniaxially symmetric systems. 

If the orientation distribution function is uniaxial, and symmetric about the x axis, the averages in equation (5.44) take on the following simple forms,... 

In studying uniaxially deformed polymers, we can confine ourselves to systems with cylindrically symmetric correlation functions. It should be recognized that this does not impose any symmetry on the electron density distribution Q (X). Introducing cylindrical coordinates (x, X3, x ) and (s S3, s,) and integrating over the azimutal angles x, and s. Equations (27) and (28) become... 

Molecular order is descnhedhy the orientational distribution function P(0) [Mcbl]. This is the probability density of finding a preferential direction n in the sample under an angle 0 in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). For simplicity, macroscopically uniaxial samples with cylindrically symmetric molecules are considered. Then, one angle is sufficient to characterize the orientational distribution function. In practice, not the angle 0 itself but its cosine is used as the variable and for weak order the distribution function is expanded into Legendre polynomials P/(cos 0),... 

Uniaxial drawing of polymer samples introduces a uniaxial distribution of molecular orientations which often can be approximated by a Gaussian. The NMR spectrum then depends on the angle between the drawing direction and the magnetic field. For example, the spectrum of unoriented FIFE is well approximated by a symmetric powder pattern (Fig. 10.3.1(a), cf. Fig. 3.1.3), while the spectrum of a uniaxially oriented sample exhibits two peaks when the direction of molecular order is parallel to the magnetic field (Fig. 10.3.1(b)) [Hepl]. 

This step introduces the statistical distribution of units into the analysis. In the most general case, there is a distribution of orientations, P(a, /3, y), describing the sample anisotropy. When the orientation is uniaxial, the distribution is cylindrically symmetrical about the draw axis, and P(a, )S, y) reduces to a function of one variable, P(i8) [3, 5]... 

The second type of explanation for finding values of R less than 3 involves the assumption that the emission and absorption axes of the fluorescent molecule are not coincident. Kimura et al. have considered a model in which the absorption and emission axes each have, independently, a cylindrically symmetric distribution of orientations around a third unique axis in the molecule, and Nobbs et al. have considered a model which includes both this and the possibility that there is a fixed angle between the emission and absorption axes which are otherwise uniformly distributed around a third unique axis. In the more general model at least three parameters are required to specify the relationships between the directions of the emission and absorption axes and that of the unique axis of a fluorescent molecule and these are not generally known. For orthotropic symmetry, five v, are required to characterise the distribution of orientations of the unique axes and if the constant NqIo is included, there is a total of at least nine unknown quantities. No attempt has so far been made to evaluate these from intensity measurements on an orthotropic sample. For a uniaxial sample only two parameters, cos O and cos O, are required for characterising the distribution of orientations and by making various approximations the total number of unknown quantities can be reduced to six. Their evaluation then becomes a practical possibility. 

And what about optically uniaxial phases fti the case of a nematic, the molecular distribution is independent of the precession angle (O = const) but may depend on angle 4. For a smectic A, in the first approximation, the orientational distribution function is the same as for the nematic. However, there is some interaction between the translational and orientational degrees of freedom that can be taken into account as a correction to/(i9,0). At first, cmisider a distribution function for a uniaxial phase consisting of axially symmetric molecules [12]. 

In the rest state, the distribution of chain end-to-end vectors is spherically symmetric. In an ideal uniaxial deformation, the distribution of x-, y-, and z-components is cylindrically symmetric. The normalized distribution for the x-component (the stretch direction) is ... 

S and D refer to the optical axis of the uniaxial phase which is chosen as the space-fixed Xj-axis and to the molecule-fixed x coordinate system. Whereas S characterizes the order of the orientation of the molecule-fixed Xj-axis with respect to the optical axis, the parameter D is a measure for the deviation from a rotationally symmetrical distribution of a molecules about... 

In a colloidal solution of anisotropic particles an electric or a magnetic field can also cause orientation and thereby double refraction. In this case the particles are oriented by the field with their longest axis parallel or perpendicular to the field direction while the BROwriian movement again disturbs this orientation. There is thus produced a Boltzmann distribution symmetrical about one axis whereby the colloid behaves in this case as a uniaxial doubly refractive body. 

Figure 29 Uniaxially oriented system. The distribution of molecular axis is symmetric around the reference axis. A dipole-transition moment is aligned symmetrically around the molecular axis at an angle (. ...

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