Dipoles randomly oriented

Theoretical Results for a Distribution of Dipoles Random Orientations... 

An aqueous electrolyte solution consists of a variety of charged and uncharged species, e.g. cations, anions, water dipoles, organic molecules, trace impurities, etc. which under equilibrium conditions are randomly oriented so that within the solution there is no net preferentially directed field. However, under the influence of a potential difference, the charge will be transported through the solution by cations and anions that migrate to... 

Figure 5-2. In the absence of an applied magnetic field a), the molecular magnetic dipoles are randomly oriented on application of an external field b), the dipoles tend to orientate parallel to the field.

Steady-State Fluorescence Depolarization Spectroscopy. For steady state depolarization measurements, the sample is excited with linearly polarized lig t of constant intensity. Observed values of P depend on the angle between the absorption and emission dipole moment vectors. In equation 2 (9), Po is the limiting value of polarization for a dilute solution of fluorophores randomly oriented in a rigid medium that permits no rotation and no energy transfer to other fluorophores ... 

It may be worthwhile to compare briefly the PECD phenomenon discussed here, which relates to randomly oriented chiral molecular targets, with the likely more familiar Circular Dichroism in the Angular Distribution (CDAD) that is observed with oriented, achiral species [44 7]. Both approaches measure a photoemission circular dichroism brought about by an asymmetry in the lab frame electron angular distribution. Both phenomena arise in the electric dipole approximation and so create exceptionally large asymmetries, but these similarities are perhaps a little superficial. 

We consider the expression of the lab frame photoelectron angular distribution for a randomly oriented molecular sample. The frozen core, electric dipole approximation for the differential cross-section for electron emission into a solid angle about a direction k can be written as... 

A paraelectric substance is not polarized macroscopically because the dipoles are oriented randomly. However, they can be oriented by an external electric field (orientation polarization). The orientation is counteracted by thermal motion, i.e. the degree of polarization decreases with increasing temperature. 

Figure 12. Angular distribution of luminescence radiated by a randomly oriented dipole located at a glass surface.

Considering that J 2 I 2 and that the ionization rates at R = Rq have little 9 dependence [35], the dominant difference should be due to the electron and nuclear dynamics in the steps 2 and 3. The observed single molecule responses are obtained by superposing the radiation from all the molecules with random orientation coherently. For linearly polarized laser field, whose direction is defined as x axis, the observed dipole moment is given by... 

CD spectra are usually measured in solution, and these spectra result from the interaction of the individual chromophores of a single molecule with the electromagnetic field of light. The interaction with neighboring molecules is often negligible. Moreover, because molecules in solution are tumbling and randomly oriented, the mutual interaction between two molecules, which is approximated by a dipole-dipole interaction, is negligible. 

Given randomly oriented dipoles, the absorption probability rate is proportional to the intensity 7, = E,tJ 2. At z = 0, the evanescent intensities are... 

The collected fluorescence 3F [from Eq. (7.39)] clearly depends on the orientation distribution of the dipoles and the incident polarization through the dependences on 0 and E. We will assume a special but common case here randomly oriented dipoles with a z-dependent concentration near the surface, excited by a p-polarized evanescent wave. 

Figure 7.7. The weightings wj / versus i/(z) for the special case of p-polarized incident light and randomly oriented dipoles. In this case, k -1 = w]) 1.

In a microscope, standard polarized epi-illumination cannot distinguish order from disorder in the polar direction (defined as the optical axis) because epi-illumination is polarized transverse to the optical axis and observation is along the optical axis at 180°. However, microscope TIR illumination can be partially polarized in the optical axis direction (the z-direction of Section 7.2) and can thereby detect order in the polar angle direction. Timbs and Thompson(102) used this feature to confirm that the popular lipid probe 3,3 -dioctadecylindocarbocyanine (dil) resides in a supported lipid monolayer with its dipoles parallel to the membrane surface, but labeled antibodies bound to the membrane exhibit totally random orientations. 

The scattering matrix elements StJ corresponding to (5.47) can be obtained from (3.16). However, of possibly greater interest than the most general scattering matrix is that for a collection of identical, but randomly oriented, anisotropic dipoles this scattering matrix is proportional to where %... 

All the essential ingredients for calculating the average scattering matrix for a randomly oriented anisotropic dipole are now at hand. From the relations (5.50), after a good bit of algebra, we obtain... 

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