Dipole, field moment

This simple derivation omits the angular dependence of the field which varies as the cosine of the angle between the dipole axis (the moment vector) and the distance expressed as a vector, r. Therefore, the field is a maximum along the axis of the dipole. Equation 3.18 makes the point that the dipole field decreases rapidly with distance. The units here are electrostatic (CGS) for simplicity. 

For polarizable charge distributions, additional classical-type interactions arise from the induced dipole, quadrupole, and higher moments on each monomer, which are proportional to the fields created by the asymmetric charge distribution on the other monomer. The proportionality constants for each multipole field are the monomer polarizabilities aa and ah (a111 for dipole fields, a(Q) for quadrupole fields, etc.). The leading two induction interactions are ... 

Here p is the stellar dipole magnetic moment and M is the mass accretion rate through the disk. For this formula we assume a purely dipolar field higher multipoles weaken the dependence on M because the field is effectively stiffen... 

Physically, this formula describes the power dissipated by a harmonic oscillator (the emission dipole with moment t) as it is driven by the force felt at its own location from its own emitted and reflected electric field. PT is calculable given all the refractive indices and Fresnel coefficients of the layered model(12 33)... 

In Equation 12.11, v is the frequency of incident radiation (cm-1), v o is the frequency corresponding to the energy difference between ground and excited electronic state, the sum is over all excited states, and d0n and d o are the dipole transition moments between the ground and excited state (dno = (n d 0) = / I nd I o dx, the T s are wave functions and d is the dipole moment operator). At low frequency (v -> 0) Equation 12.11 reduces to the static field expression... 

To put the problem in proper perspective, let us consider an instructive example an individual magnetic dipole of moment p subject to a magnetic field H and embedded in a thermal bath. The dipole can switch between the up and down... 

The energy of interaction of a fixed dipole and a polarisable spherically symmetrical molecule depends on the angle 0 between the dipole axis and the line joining the centre of the molecule and the dipole. For a dipole of moment u, the field at a distance r and at an angle 0 is given by... 

We have used the following method for calculating electric fields. Suppose the magnitude of electric field due to the charge distribution of a molecule at a point in its vicinity is E. If a point dipole having moment p is placed at that point, its potential energy of interaction with the electric field would be given by... 

The observed frequency dependent behavior of biopolymer solutions often resembles that of polar liquids and their solutions in nonpolar solvents, the most conspicuous differences being the very large effects of small concentrations of solute and the much lower frequencies. The classic theory of Debye (6) for polar liquids and solutions assumes N independent permanent dipoles of moment p reorienting in an applied field E(t) =... 

Polarizability goes as the cube of radius The electrostatic potential set up by a dipole of moment /xclipoie has the form (/dipole A2) cos (), where () is the angle between the dipole direction and the line to the position where the dipole potential is being sensed.5 For example, a metallic sphere of radius a placed in a constant external electric field Eo... 

Vibrational product state distributions have been obtained for reactions studied in crossed molecular beams using the technique of beam electric resonance spectroscopy [109]. This method uses the focusing action of electric quadrupole and dipole fields to measure the radio frequency Stark spectrum of the reaction products, which must possess a dipole moment. This has restricted this technique to reactions producing alkali halides. 

The origin of the electric dipole intensity for the AMj = 1 transitions studied merits further consideration. If the static magnetic field is 5 kG, the motional electric field has a magnitude of approximately 3 V cm-1 and is perpendicular to the applied magnetic field. This electric field mixes a state [./, Mj) with the states. J 1, Mj 1) and in order to obtain non-zero electric dipole transition moments for the transitions. /. Mj) o IJ, Mj 1), the oscillating electric field must be applied parallel to the static magnetic field. 

Here r, and a>oj are the damping constants and resonance frequencies of the oscillators, and E ocj is the local field at location xqj, given by the external held and the fields of the induced dipoles with moment / / = e/x/ of all other oscillators l j. A Fourier decomposition yields for the spectral components of the displacements... 

In the complexes [Ln(H20)y]3+, [Ln(oda)3]3, the dynamic polarization first-order electric dipole transition moment is minimized by negative interference due to the out-of-phase relation between the contributions of the [ML3] and [ML6] ligand sets [109,110]. For [Ln(oda)3]3 and other D3 complexes, only the anisotropic polarizability contributions are non-zero for AMj = 1 transitions in the [Eu(H20) ]3+ and [Eu(oda)3]3 complexes the contribution of the cross-term to the dipole strength of the 7Fo —> 5D2 and5 Do — 7F2 transitions has a magnitude comparable with that of the dominant crystal field or dynamic polarization contribution [111]. 

VCD on achiral molecules becomes feasible when an achiral sample is put into a strong magnetic field. The magnetic field vector can in this case be parallel or antiparallel to a component of the dipole transition moment and this way positive and negative signals become possible. The method is consequently termed magnetic VCD, MVCD. 

The effective magnetic field, from the orbital part onto the conduction electrons, can be approximated from the spin orbit splitting but it should also be remembered that the effective field could be larger because the forces between the f-electron spin and the conduction electron spins, will add to the dipole field of the orbital moment. To illustrate the effect of this dynamic effective mass m+ we present the change in the f-band width of Ce in YCe as a function of the effective mass. Figure 2. Other calculations supporting this analysis will be published elsewhere. 

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