Electric dipole line strength

Another important quantity in the context of El transitions is the electric dipole line strength, defined as [7]... 

Judd (29), in his classic paper of 1962, used the odd parity terms of the ligand field to accomplish this admixture. After applying second order perturbation theory and several simplifying assumptions, he showed that the electric dipole line strength between J-manifolds may be expressed as the sum of three terms, each being the product of an intensity parameter and a reduced matrix element of the tensor operator U of rank X. The electric dipole line strength, Se(j, can be written in the form... 

Judd (1962) and Ofelt (1962) have shown that calculations of f-f intensities in rare-earth spectra can be made tractable by treating the excited configuration as completely degenerate. The electric-dipole line strength can then be expressed as a sum of products of phenomenological intensity parameters il and matrix... 

The intensity of an electric dipole transition in absorption or emission depends, on one hand, on factors particular to the experiment measuring the intensity, e.g., the number density of molecules in the initial state of the transition and, for absorption experiments, the absorption path length and the intensity of the incident light. On the other hand, the intensity involves a factor independent of the experimental parameters. This factor, the line strength 5(f <— i), determines the probability that a molecule in the initial state i of the transition f <— i will end up in the final state f within unit time. 

If we assume that the initial state i and the final state f are both non-degenerate, then the line strength of the electric dipole transition between them [3] is given by... 

As in the case of LS coupling, when there are N equivalent electrons in the outer shell, both the line strength and the oscillator strength should be multiplied by N as well as by the corresponding CFP [ 10,12]. As in the LS scheme the two forms of the electric dipole length transition operator have been employed here in the calculation of the radial transition integral, I nl, n l ). 

The crystal field model may also provide a calciflation scheme for the transition probabilities between levels perturbed by the crystal field. It is so called weak crystal field approximation. In this case the crystal field has little effect on the total Hamiltonian and it is regarded as a perturbation of the energy levels of the free ion. Judd and Ofelt, who showed that the odd terms in the crystal field expansion might connect the 4/ configuration with the 5d and 5g configurations, made such calculations. The result of the calculation for the oscillator strength, due to a forced electric dipole transition between the two states makes it possible to calculate the intensities of the lines due to forced electric dipole transitions. 

The characteristic absorption and emission spectra of lanthanide compounds in the visible, near ultra-violet and infra-red is attributed to transitions between 4/ levels due to the fact that they present a sharp line with oscillators strengths typically of the order of 10 . These transitions are electric dipole forbidden but became allowed as forced electric dipole transitions. 

Fro. IX-1.—Values of the ratio of polarisation P to field strength E for hydrogen chloride gas, as a function of the reciprocal of the absolute temperature. The slope of the line is a measure of the permanent electric dipole moment of the molecules, and the intercept of the line is a measure of the temperature-independent polarizability of the molecules. 

Though not strictly functioning as resistors/conductors, carbon nanotubes have just been reported in an extraordinarily vapor-sensitive capacitive device [19]. The electric field lines emanating from the nanotubes are responsible for a localized dielectric response that can be modulated by minute quantities of adsorbate on the nanotube surface. A layer of hydrogen-bonding polymer, or even a mono-layer terminated in mildly acidic groups, increased sensitivity to parts per billion levels. Response strength was correlated with the dipole moment of the analytes. 

The oscillator strength, P, of a spectral line for an electric dipole transition from the component A) of the ground state to the component B) of the excited state may be written... 

In addition to the transition probability A, the atomic oscillator strength/and the line strength S are often used in the literature. The conversion factors between these quantities are (for electric-dipole transitions) ... 

The Judd-Ofelt theory was applied to study the spectral properties of 3at.%Nd Lu20j ceramic in this work. For Nd ions, the magnetic dipole (md) transitions are much weaker than the electric dipole (ed) transitions and they can be neglected in the J-O calculations. The experimental line strength for electric dipole (ed) transitions Sediexp) can be obtained from the absorption spectrum according to the following equation (2)... 

The aforementioned matrix elements of electric dipole operators cannot be calculated analytically. Fortunately, the realistic limit for large N leads to closed forms for the matrix elements of exp(-an ). It is customary to introduce the standard Herman-Wallis notation for denoting the line strength ... 

Formulas for the line strengths for magnetic-dipole and electric-quadrupole radiation were included by Condon and Shortley (1935) in their classic text. Only minor modifications in the standard theory were necessary to allow for the presence of a lanthanide ion in a crystal. For electric-dipole radiation, on the other hand, the sequence of representations (69) has to be replaced by the product of matrix elements... 

The great bulk of the spectra of the free lanthanides consists of lines that correspond to electric-dipole transitions. The contribution to the Hamiltonian is thus proportional to , for each electron i, and the whole problem turns on evaluating the matrix elements of Condon and Shortley (1935) listed the relative strengths of the lines of several transition arrays. The methods of Racah (1942b, 1943) can be readily applied to S,r,. This operator possesses the tensorial characteristics of and thus leads to the selection rules AL, AJ = 0, +1 ... 

An example of the resolution of this type of Stark spectroscopy is given in Fig. 6 at the fairly low electric field strength of 150 V/cm. Three Mp-components of the N = 3 - 2, J = 2.5 - 1.5, F = 5 - A transition in the X Z", y = 0 state of Cal near 12.3 GHz are well resolved in a 800 kHz mw scan. The electric field shifts were followed up to 900 V/cm and are depicted in Fig. 7. The measured data points lie well on the solid lines which represent the line positions calculated with the fitted dipole moment. The high accuracy of this method allowed the determination of the vibrational dependence of the electric dipole moment e.g. in the case of SrF. ... 

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