Line strength

Both line positions and line strengths must be understood in order to interpret the structure of observed spectra and derive abundances and temperatures of contributing molecules. Following Planck (1901) and Einstein (1906b), we treat radiation as composed of photons with energy E — hcv, where h is the Planck constant and cv is the frequency of the radiation in hertz. The intensity is proportional to the arrival rate of photons. If these photons originate from molecules in a small volume, the intensity is proportional to the number of molecules in the optical path undergoing transitions at that frequency. 

The electric dipole transition rate between two energy levels of a molecule, E and Em, depends on the probability per second that a molecule in E will make the transition to Em, and on the number of molecules in the initial state. The strengths of emission or absorption lines between an upper level, E , and a lower level, Em, are given by 

Anm and Bmn are the Einstein coefficients for spontaneous emission and for absorption and stimulated emission, respectively. The Einstein coefficients are defined here in terms of the square of the electric dipole moment matrix, 

The number of molecules Ni in the initial state depends on the total number of molecules, the distribution of energy levels, the degeneracy of individual levels (number of levels of identical energy), and the temperature of the gas. In equilibrium the initial state population is 

We illustrate the main characteristics of line strength by considering diatomic molecules. The vibrational states (without rotation) are all nondegenerate, so dn = 1. Ignoring anharmonicities, vibrational energies are given by Eq. (3.3.6), 

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In the ideal case for REMPI, the efficiency of ion production is proportional to the line strength factors for 2-photon excitation [M], since the ionization step can be taken to have a wavelength- and state-mdependent efficiency. In actual practice, fragment ions can be produced upon absorption of a fouitli photon, or the ionization efficiency can be reduced tinough predissociation of the electronically excited state. It is advisable to employ experimentally measured ionization efficiency line strengdi factors to calibrate the detection sensitivity. With sufficient knowledge of the excited molecular electronic states, it is possible to understand the state dependence of these intensity factors [65]. 

These results provide so-called "selection rules" because they limit the L and M values of the final rotational state, given the L, M values of the initial rotational state. In the figure shown below, the L = L + 1 absorption spectrum of NO at 120 °K is given. The intensities of the various peaks are related to the populations of the lower-energy rotational states which are, in turn, proportional to (2 L + 1) exp(- L (L +1) h /STi IkT). Also included in the intensities are so-called line strength factors that are proportional to the squares of the quantities ... 

The peak absorption (scattering) cross sections are thus useful comparative measures of detectivity because the latter is a product of the line strength and the practical line resolution. 

By measuring the relative intensities of satellite and main lines, the population ratio is obtained, if it can be assumed that the dipole moment and line strength is not appreciably different in the two cases. From the population ratio R, the energy interval AE is obtained from the Boltzmann law i.e.,... 

This case is shown schematically in Fig. 5c. In Eq. (50), qj. are generalized y-photon asymmetry parameters, defined, by analogy to the single-photon q parameter of Fano s formalism [68], in terms of the ratio of the resonance-mediated and direct transition matrix elements [31], j. is a reduced energy variable, and <7/ y, is proportional to the line strength of the spectroscopic transition. The structure predicted by Eq. (50) was observed in studies of HI and DI ionization in the vicinity of the 5<78 resonance [30, 33], In the case of a... 

The temperature of a molecule within any astronomical environment may vary from the intense cold of the interstellar medium with a temperature of 10 K to the temperature within a sun spot 4000 K close to the temperature at which a molecule would fall apart. The relative intensity of transitions along the progression is given by a line strength factor ... 

A library of stellar spectra or absorption-line strengths, taking into account differences in a-element iron and possibly other element abundance ratios. The spectra may be either observational or synthetic, i.e. theoretically computed. 

Tiff, from continuum slope, Baimer jump in hotter stars and hydrogen-line strengths in F-G stars. 

Luminosity or gravity from H-line strengths in hotter stars, Balmer jump and molecular features in F-K stars. 

Fig. 3.25. Trends of nebular line strengths in H n regions with oxygen abundance. This figure shows oxygen abundance in H n regions of the Milky Way and spiral and irregular galaxies (determined using measured electron temperatures) vs. log R23, after Pilyugin (2003) the p parameter is the line ratio [O iii]/([0 11] + [O hi]).

Colour and line-strength gradients are also observed across elliptical galaxies, as is to be expected from dissipative effects (see Appendix 5). However, the detailed... 

As in the previous case of infrared transitions, one wants to calculate the line strengths S(v,J —> v, J ) defined in Eq. (2.127). For Raman transitions there are two contributions, as discussed in Chapter 1. The so-called trace scattering is induced by the monopole operator... 

It is convenient to write the line strength for trace scattering in general as... 

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... 

The energy density function p v) is defined so that dE—p v)dv is the amount of available radiation energy per unit volume originating in radiation with frequency in the infinitesimal interval [v,v + dv]. Thus, p v) is expressed in the SI units J/(m Hz) = J s/m, so that Bg and Bg have the SI units m /(J s ). Ag is expressed in s The Einstein coefficients defined in this manner are related to the line strength by... 

In the present section, we obtain an expression for the line strength in equation (4) in a form suitable for numerical calculation. This derivation closely follows the theory developed in Refs. [18,19] and in Chapter 14 of Ref. [3], and so we give only an outline here. 

With the phase choices made for the basis functions in the present work, we obtain the real, positive line strength in equation (21) in the form Iwl, where s is real, that is, as the module square of a purely imaginary number. 

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