Radiative line shape

This distribution is called the natural or radiative line-shape and is illustrated in Fig.8.1. It is a bell-shaped curve known as the Lorentzian distribution, (w-Wq,y), whose full width at half maximum intensity, (FWHM), is given by... 

Specifically, the collision-induced absorption and emission coefficients for electric-dipole forbidden atomic transitions were calculated for weak radiation fields and photon energies Ha> near the atomic transition frequencies, utilizing the concepts and methods of the traditional theory of line shapes for dipole-allowed transitions. The example of the S-D transition induced by a spherically symmetric perturber (e.g., a rare gas atom) is treated in detail and compared with measurements. The case of the radiative collision, i.e., a collision in which both colliding atoms change their state, was also considered. 

A. Ben-Reuven. Radiatively damped collisions of ultracold atoms. In L. Frommhold and J. W. Keto, eds., Spectral Line Shapes 6, p. 206, Am. 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

The most time consuming parts of the forward model are the calculation of the absorption coefficients and the calculation of the radiative transfer. A spectral resolution of Av = 0.0005 cm 1 is considered necessary in order to resolve the shape of Doppler-broadened lines. To avoid repeated line-shape and radiative transfer calculations at this high resolution, two optimizations have been implemented ... 

The radiative correction in the lowest order to the emission line shape of metastable hydrogen atom can be written in the form of additional contribution to a numerator in (10)... 

Fig. 5. tyNe(6 — 5) transitions. The line shape is identical with the spectrometer response function because of the small radiative width of 10 meV. A line width of 26 or 550 meV is achieved for a silicon crystal of 95 mm in diameter... 

The absorption edge shifts to the blue (Fig. 1). The photoluminescence has a broad band peaking at 640 nm. The luminescence line shape is not Lorentzian and has a strong Stokes shift. Photoluminescence excitation (PLE) spectra have revealed a fine substructure of the band at its short-wave wing whose origin is attributed to the intrinsic luminescence contribution and to radiative recombination on defects. 

Figure 7.10 LIF lifetime measurements, following an excitation laser pulse of duration At = 4.5 ns FWHM. If the lifetime of the excited levelis longerthan the excitation pulse, then the lifetime can be extracted from theslopeof the semi-logarithmic plot (trace a) if the radiative lifetime signal is detected with electronics of similar time constants, then / C-response deconvolution needs to be applied (trace b) and if the lifetime is of similar length or slightly shorter than the laser pulse, full line shape function deconvolution procedures are required (trace c). Data shown in trace (b) are adapted from Verdasco et al Laser Chem., 1990, 10 239, with permission of Taylor Francis Group...

The origin of each element composing the nuclear-ensemble approach can be traced back to decades ago, first with the works of Heller, Wilson and others in the 1980s, where absorption bands were computed based on molecular dynamics [9]. It is also influenced by the works of Skinner [10], which provided a useful link between Kubo s stochastic theory of the line shape [11] and molecular dynamics, and by the reflection principle [12], which approaches bound to continuum transitions from the nuclear-ensemble perspective. The intuitive character of the nuclear-ensemble approach has created a situation where although the method is frequently employed, there is no clear derivation of its formalism. This information gap makes difficult to understand the reasons for its limitations and to propose ways to improve the method. In this contribution, we derive equations for absorption cross sections and radiative decay rates based on the nuclear-ensemble method. The main approximations are made explicit, and improvements on the method are proposed, in particular ways to get rid of arbitrary parameters. 

The broadening Fj is proportional to the probability of the excited state k) decaying into any of the other states, and it is related to the lifetime of the excited state as r. = l/Fj . For Fjt = 0, the lifetime is infinite and O Eq. 5.14 is recovered from O Eq. 5.20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (O Eq. 5.4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes caimot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, aU decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions (1) all broadening parameters are assumed to be identical, Fi = F2 = = r, and (2) the value of F is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in O Eq. 5.4 with the substitution to to+iTjl, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. 

In the following sections on the interaction of radiation with gas molecules we begin with an overview of the physical principles of radiative transitions in molecules in Sections 3.1 and 3.2, proceed to discussions of the properties of diatomic and polyatomic molecules in Sections 3.3 and 3.4, and, finally, examine line strengths in Section 3.5 and line shapes in Section 3.6. Interactions of radiation with solid and liquid surfaces, as well as cloud particles, are the subject of Sections 3.7 and 3.8. For further information on molecular spectroscopy we refer the reader to text books, such as Pauling Wilson (1935), Herzberg (1939,1945,1950), Townes Schawlow (1955), or Steinfeld (1974). The book by Murcray Goldman (1981) is... 

The Line Shape Function for Radiative Transitions 165 TaWe 7.2 The MNDO-optimized molecular structural parameters of p-terphenyl. 

Cr + ions in aluminum oxide (the ruby laser) show a sharp emission (the so-called Ri emission line) at 694.3 nm. To a good approximation, the shape of this emission is Lorentzian, with Av = 330 GHz at room temperature, (a) Provided that the measured peak transition cross section is c = 2.5 x 10 ° cm and the refractive index is = 1.76, use the formula demonstrated in the previous exercise to estimate the radiative lifetime, (b) Since the measured room temperature fluorescence lifetime is 3 ms, determine the quantum efficiency for this laser material. 

The temperature dependence of the observed emission is equally well interpreted in the scheme of Section II.D.3.b The lines emitted from the main distribution D, are asymmetric, with a width of the order of kT for the high-energy part. At very low temperature (0.4 K)86 the main distribution approaches a width of about 4 cm - which indicates that the thermalization regime is slower than the radiative relaxation rate, of the order of 1 ns. In addition, the shape of the second distribution D2, at 25 082 cm1, sharpens as the thermal barrier,85 which inhibits the relaxation very near the middle of the zone, weakens at very low temperature. 

---