Radiative width

In most optical excitations the resolution is determined by the Doppler effect or the finite linewidth of the light source. The Doppler effect gives a typical frequency width of 1 GHz, and the width of the light source can be anywhere from 1 kHz to 30 GHz. We assume that these widths are larger than the radiative width. The photoionization cross sections from the ground states of H, alkali, and the alkaline earth atoms are given in Table 3.3. 20... 

Generally, when studying autoionizing levels, we have to take into consideration both (radiative and radiationless) channels of their decay. The total natural width of the autoionizing level will be the sum of its autoionizing and radiative widths. 

Effective nuclear charge grows with increase of ionization and excitation of an atom, therefore, the fluorescence yield in ions and, to a less extent, in excited atoms, tends to increase. Radiative and Auger linewidths, as well as fluorescence yields, depend on relativistic and correlation effects to less extent in comparison with the probabilities of separate transitions, because in sums the individual corrections partly compensate each other. Therefore, calculations of radiative widths in the Hartree-Fock-Pauli approach lead to reasonably accurate results. 

A possibility to extend this set comes from the use of an Electron-Cyclotron-Resonance-Ion-Trap (ECRIT), which will be realized using the cyclotron trap itself [22]. Here, hydrogen-like electronic atoms will be produced to obtain narrow calibration lines independent of an accelerator s pion beam. The radiative widths of light elements with Z k. 15 are of the order of a few 10 meV because of the absence of non-radiative inner-shell transitions. 

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... 

Here r,(co) shows the effect of the surface reflectivity, which appears as a lorentzian line, centered at the surface resonance, if we neglect the variation of rv(co) with co around the surface resonance ( lOcnU1). The surface excitations are renormalized relative to the bulk-free surface, leading for coupled surface excitons to a frequency shift ds and to a new radiative width rs, both quantities simply related to the complex amplitude of the bulk reflectivity ... 

Therefore, as a general trend, Ts decreases when the energy gap between surface and bulk states is made weaker Figs. 3.1-3 provide a perfect illustration of the expression (3.26) for the bulk effect on the surface emission. A more detailed analysis of the bulk effect will be given below. However, this reduction of the surface radiative width may be interpreted classically as the destructive interference between the emission of the surface and that of its electrostatic image in the bulk.140 The bulk reflectivity amplitude rv(to) is quasi-metallic near resonance and at low temperatures. 

This section has been devoted to the study of the surface excitons of the (001) face of the anthracene crystal, which behave as 2D perturbed excitons. They have been analyzed in reflectivity and transmission spectra, as well as in excitation spectra bf the first surface fluorescence. The theoretical study in Section III.A of a perfect isolated layer of dipoles explains one of the most important characteristics of the 2D surface excitons their abnormally strong radiative width of about 15 cm -1, corresponding to an emission power 10s to 106 times stronger than that of the isolated molecule. Also, the dominant excitonic coherence means that the intrinsic properties of the crystal can be used readily in the analysis of the spectroscopy of high-quality crystals any nonradiative phenomena of the crystal imperfections are residual or can be treated validly as perturbations. The main phenomena are accounted for by the excitons and phonons of the perfect crystal, their mutual interactions, and their coupling to the internal and external radiation induced by the crystal symmetry. No ad hoc parameters are necessary to account for the observed structures. 

Figure 4.2. Variation of the radiative width y, or coherent emission rate, of a 2D disordered exciton as a function of the disorder strength A. For A > g,r, the emission becomes incoherent. For all distributions, including the gaussian distribution, there is threshold behavior with a sudden takeoff of the coherent emission.

To conclude, we can draw an analogy between our transition and Anderson s transition to localization the role of extended states is played here by our coherent radiant states. A major difference of our model is that we have long-range interactions (retarded interactions), which make a mean-field theory well suited for the study of coherent radiant states, while for short-range 2D Coulombic interactions mean-field theory has many drawbacks, as will be discussed in Section IV.B. Another point concerns the geometry of our model. The very same analysis applies to ID systems however, the radiative width (A/a)y0 of a ID lattice is too small to be observed in practical experiments. In a 3D lattice no emission can take place, since the photon is always reabsorbed. The 3D polariton picture has then to be used to calculate the dielectric permittivity of the disordered crystal see Section IV.B. 

In the definition of line broadening it is necessary to exercise some discrimination. On the one hand spectral linewidths of less them 0.17 cm-1 are observed for some of the vibronic bands of the lowest singlet system of benzene 1f 2 - -1diff in the vapor phase W, while on the other hand many electronic spectra have been encountered, in particular in higher excited singlet and triplet systems, for which few or no vibrational features are apparent. In crystal spectra at 4 K, linewidths as sharp as 0.5 cm-1 are often obtained for the lowest excited state of any multiplicity, despite coupling with the lattice modes, which may be expected to lead to considerable broadening. Nevertheless, these crystal linewidths are considerably more than the linewidths observed in the vapor phase and certainly more than the natural radiative widths. 

Let us first study Eq. (6) in co space. Clearly cs(ai) is the amplitude of s> as prepared by the laser jr(ai - cou), which of course will have a certain coherence width around the frequency cou. Further note that the denominator of Eq. (6) has poles at the frequencies a>kJ, these poles being topped off by the radiative width y of s>. cs(co) has poles whenever co = again topped off by the radiative width y. At these poles c,(ai) is maximal and we expect strong absorption while at the poles of the denominator, c,(ai) is minimal and we expect no, or at least very little, absorption. 

The fluorescence decays with the radiative width, a result we would have expected. 

The fluorescence oscillates in time with frequency 2t>. Of course, since we neglected the radiative width, it seems to oscillate forever. If we reintroduce this width, realizing that each molecular eigenstate in this symmetric case gets y/2, we have... 

We have one further piece of information, the radiative lifetime of the singlet, which has been determined from the oscillator strength of the lB3u electronic transition as determined by its absorption in solution or gas phase.10-12 The best value appears to be 290 nsec or a radiative width of 3.3 MHz... 

If the retarded interaction is ignored and the operator Hmt is removed from the total Hamiltonian (4.2), then the operator H becomes a sum of two independent Hamiltonians, one of them (Hi) describing the crystal elementary excitations - those which occur when the retardation effects are ignored and the second (H2) giving the elementary excitations - transverse photons in vacuum. The presence of the operator H3 leads to the interaction between carriers with the transverse electromagnetic field. In the case of an atomic gas this interaction causes, in particular, the so-called radiative width of energetic levels of excited states. In the case of an infinite crystal possessing translational symmetry the radiative width of excitonic states vanishes.29... 

Giant radiative width of small wavevector polaritons in one-and two-dimensional structures ( polariton superradiance )... 

The radiative width of an excited isolated molecule is proportional to (see (1))... 

Hence the above calculated radiative width in one-dimensional crystals with k < ko, as follows from (4.93), is greater than the corresponding quantity in an isolated molecule by l/k0d times. In the optical spectrum region l/k0d = X/(2ttd) 102-103. In consequence, the lifetime of exciton states in one-dimensional ideal molecular crystals may be several orders of magnitude smaller than the corresponding lifetime in isolated molecules. 

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