Oscillator strength formula

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),... 

From the data of Hoogschagen and Gorter (104), the oscillator strength of the 5D4-+7F6 transition was obtained. By means of the Ladenburg formula, the spontaneous coefficient A46 was calculated. Using the relative-emission intensities, the rest of the A4J spontaneous-emission coefficients could be calculated. From these and a measured lifetime of 5.5 x 10 4 sec at 15°C, he calculated a quantum efficiency of 0.8 per cent. Kondrat eva concluded that the probability of radiationless transition for the trivalent terbium ion in aqueous solution is approximately two orders of magnitude greater than for the radiation transition. 

Oscillator strengths of non-relativistic Ek- and Mfc-transitions may be calculated by the general formula (24.14). Transition probabilities may be found for /c-radiation using formula (4.10) or (4.11) and for Mk-... 

Thus, by employing general formulas for oscillator strengths, results can be obtained which are useful both in practical calculations and in studying the role of correlation effects on transition probabilities. 

Thus, as was the case for wavelengths (formulas (31.2) and (31.3)), the oscillator strengths and transition probabilities behave differently for the transitions with An = 0 and An f 0. However, these regularities are valid only approximately. Especially large deviations may occur for the cases where correlation or relativistic effects are large. 

Here p is the frequency of plasmon oscillations in a system of free electrons (3.7). The oscillator strengths ft introduced previously differ from the usual fm (see Section IV) in their normalization (Efl, / = 1). A method for calculating the thus defined oscillator strengths from experimental values of e2 is presented in Ref. 89. Since the energy range essential for collective oscillations is ho> < 30 eV, the electrons of inner atomic shells can be disregarded. Thus, the value of ne is determined by the density of valence electrons only, and only the transitions of these electrons should be taken into account in the sum over i in formula (3.15). A convenient formula for calculating the frequencies molecular liquids is presented in ref. 89 ... 

Thus, in order to calculate the differential cross sections it is enough to know the generalized oscillator strengths. On the other hand, if the cross sections are found experimentally, formulas (4.13) and (4.18) enable us to find the experimental values of the oscillator strengths.117 We will briefly dwell on the properties of the generalized oscillator strengths.113,118... 

Keeping in mind the characteristic properties of generalized oscillator strengths, the authors of Ref. 120 have proposed the following semiem-pirical formula for /(to, q) ... 

Although the actual form of f(w, q) is different from formula (4.27), the latter leads to reasonable results when we use it to calculate the cross sections of inelastic collisions and the ionization losses.120 As one of the reasons for using approximation (4.27), one can consider the fact that the data concerning the Bethe surfaces for molecules are very scant, while there is extensive information about the optical oscillator strengths of molecules both in the discrete and in the continuous regions of the spectrum (see Refs. 119 and 121). 

Both theoretical122-124 and experimental125 studies of the behavior of f0n versus q show that at qa0 < 1 the oscillator strengths for optically allowed transitions rapidly fall with increase of q, which is in agreement with formula (4.27). However, at qa0 > 1 the behavior of/0n( ) depends on the type of a transition. In the case of Rydberg transitions, f0n(q) has characteristic maxima and minima that are absent in the case of excitation of valence electrons. According to Ref. 123, their appearance is due to the existence of nodes in molecular orbitals. 

Apparently, Bragg s rule is only approximate. When atoms combine into a molecule, their oscillator strengths and the energy levels of valence electrons change essentially. Consequently, the true value of IM must also be different from the one given by formula (5.5). An obvious example is the difference between the value 7H2 = 19eV calculated by Platzman158 and the value = 15 eV predicted by formula (5.5). 

In the local response model each electron density volume element is separately characterized by a two-parameter formula giving electric dipole oscillator strength as a function of frequency [12]. One of the two parameters is fixed by the oscillator strength sum rule, while the other is an effective mean excitation energy, taken to be the plasma energy huip by Andersson et al [9]. This model requires introduction of a low-density cutoff of the dipole response, because a... 

If the excited level contains e2 states hv above the groundstate (representing e( mutually orthogonal states), Einstein s formula from 191733 relates the radiative life-time rrad to the oscillator strength P of absorption ... 

For many simple gas molecules [e.g. the rare gases, Hg, Nj, Og, CH4), the empirical dispersion curve has been found to be representable, in a large frequency interval, by a dispersion formula of the type (14) consisting of one single term only. That means that for these molecules the oscillator strength for frequencies of a small interval so far exceed the others that the latter can entirely be neglected. In this case, and for the limiting case v -> O (polarisability in a static field) the formula (14) can simply be written ... 

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