Dipole oscillator

Many methods for the evaluation of from equation ( Al.5.20) use moments of the dipole oscillator strength distribution (DOSD) defined, for molecule A, by... 

Kumar A and Meath W J 1992 Dipole oscillator strength properties and dispersion energies for acetylene and benzene Mol. Phys. 75 311... 

Meath W J and Kumar A 1990 Reliable isotropic and anisotropic dipole dispersion energies, evaluated using constrained dipole oscillator strength techniques, with application to interactions involving H2, N2 and the rare gases Int. J. Quantum Chem. Symp. 24 501... 

Fig. 1. An incident electromagnetic field of intensity, having an associated electric field, U, induces dipole oscillation in the absorbers. The transmitted...

RAIRS spectra contain absorption band structures related to electronic transitions and vibrations of the bulk, the surface, or adsorbed molecules. In reflectance spectroscopy the ahsorhance is usually determined hy calculating -log(Rs/Ro), where Rs represents the reflectance from the adsorhate-covered substrate and Rq is the reflectance from the bare substrate. For thin films with strong dipole oscillators, the Berre-man effect, which can lead to an additional feature in the reflectance spectrum, must also be considered (Sect. 4.9 Ellipsometry). The frequencies, intensities, full widths at half maximum, and band line-shapes in the absorption spectrum yield information about adsorption states, chemical environment, ordering effects, and vibrational coupling. 

The first term in the brackets represents a static electric field in the material and the second term represents a dipole oscillating at 2co, tvdce the frequency of the incident light. This is a process knovm as SHG. 

If the interaction Hamiltonian in the Coulomb term is expanded in a series about the separation vector, the first term of the expansion is a dipole-dipole interaction, the second a dipole-quadrupole interaction, etc.<4> Again reverting to a classical analog (dipole oscillators), the energy of interaction between the two dipoles is inversely proportional to the third power of the... 

The BEB model was developed to overcome this problem. The dipole oscillator strength is assumed to have a simple form based on the approximate shape of the function for ionization of ground-state hydrogen ... 

A fourth possibility is electrodynamic bonding. This arises because atoms and molecules are not static, but are dynamically polarizable into dipoles. Each dipole oscillates, sending out an electromagnetic field which interacts with other nearby dipoles causing them to oscillate. As the dipoles exchange electro-magnetic energy (photons), they attract one another (London, 1937). 

To obtain a clear understanding of electrodynamic bonding, start with the field of a static electric dipole. Then, let the dipole oscillate so it emits electromagnetic waves (photons). Consider what happens when the emitted field envelopes another dipole (London, 1937). Finally, determine the factors that convert neutral molecules into dipoles (that is, their polarizabilities). 

Here/(q) is the dipole oscillator strength distribution at q and e is the base of natural logarithm. The lowest excitation potential may be taken for qmin, whereas qmax = (E + EB)/2 with EB a defined mean electron binding energy (Mozumder and La Verne, 1984). 

Not much is known about these processes, but they must be included to give a total picture. Emissions of Lyman and Balmer spectra of the H atom upon e-impact on hydrocarbons, H2, and H20, discussed in Sect. 4.3.2, fall in this category. Similarly, many of the excited states observed in dissociated radicals via electron impact on stable molecules (Polak and Slovetsky, 1976) also belong to this category. It is known from the dipole oscillator spectrum of H20 (Platzman, 1967) that most ionizations are accompanied by considerable excitation. Excitation transfer to the neighboring neutral molecule followed by fast dissociation cannot be ruled out. 

To further reduce of the cross section formula (4.11), we note that it is proportional to the area of the curve of Fn(K)/en plotted against In (Kag)2 between the maximum and minimum momentum transfers. Since T is large and the generalized oscillator strength falls rapidly with the momentum transfer, the upper limit may be extended to infinity. In addition, the minimum momentum transfer decreases with T in such a manner that the limit Fn(K) may be replaced by /, the dipole oscillator strength for the same energy loss. This implies that a mean momentum transfer can be defined independently of T such that the relevant area of the curve of Fn(K)/ n is equal to (// ) [ (In Kag)2 - (In Ka0)2]. Thus, by definition (Bethe, 1930 Inokuti, 1971),... 

The dipole oscillator strength is the dominant factor in dipole-allowed transitions, as in photoabsorption. Bethe (1930) showed that for charged-particle impact, the transition probability is proportional to the matrix elements of the operator exp(ik r), where ftk is the momentum transfer. Thus, in collision with fast charged particles where k r is small, the process is again controlled by dipole oscillator strength (see Sects. 2.3.4 and 4.5). 

It is not easy to calculate oscillator strengths from first principles except in some very simple cases. On the other hand, the oscillator strength distribution must fulfill certain sum rules, which in some cases help to unravel their character. Referring the (dipole) oscillator strength for the transition from the ground state with excitation energy n to state n as fn, a sum may be defined by... 

Various aspects of fluorophore emission at surfaces have been investigated, particularly within the past two decades. For nondissipative surfaces (e.g., bare glass), the lifetime(14) and the inversely related total radiated power 15 for a single emission dipole, modeled as a continuous classical oscillator, have been calculated as functions of orientation and distance from the surface. The radiated intensity emitted from a continuous dipole oscillator has been calculated as a function of observation angle, dipole orientation, and distance.(16 21)... 

Oddershede s earlier results [3-5] calculate the directional values of the dipole oscillator strength distribution for use in the Bethe theory [9], which is valid for high-energy projectiles. Our approach, since we have not implemented the possibility of treating unbound electrons, is restricted to calculating stopping cross... 

Instead of one resonance frequency per individual electron, Bethe recovered the spectrum of resonance frequencies for the atom, weighted by dipole oscillator strengths satisfying the sum rule... 

Dipole oscillator strengths form important input into all stopping models based on Bethe or Bohr theory. Emphasis has frequently been on total /-values which show only little sensitivity to the specific input. More important are differential oscillator-strength spectra, in particular at projectile speeds where inner-shell excitation channels are closed. Spectra bundled into principal or subshells [60] are sufficient for many purposes, but the best available tabulations are based on analysis of optical data rather than on theory, and such data are unavailable for numerous elements and compounds [61]. 

From classical electromagnetic theory (Jackson, 1975) for an oscillating dipole, the power radiated from the dipole oscillator is given by... 

The first term in eq. [21] is the contribution of the intrinsic rotational strengths if oscillators a and/or b are themselves chiral. The second term is the coupled oscillator contribution due to the intrinsic moments and the third term is the coupled oscillator contribution due to the geometric arrangement of the two electric dipole oscillators. The latter two contributions give rise to a conservative bisignate couplet in the observed spectrum, if the coupled modes are sufficiently separated in frequency such that the positive and negative contributions do not cancel. 

In the classical theory of scattering (Cohen-Tannoudji et al. 1977, James 1982), atoms are considered to scatter as dipole oscillators with definite natural frequencies. They undergo harmonic vibrations in the electromagnetic field, and emit radiation as a result of the oscillations. 

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