Lorentzian emission

The emission of the surface exciton, coupled to the bulk polaritons, may also be calculated using the above scheme, which, here also, coincides exactly with the quantuum-mechanical results. Instead of the lorentzian emission (3.20), we obtain an emission proportional to... 

The exponential decay of the A population corresponds to a Lorentzian line shape for the absorption (or emission) cross section, a, as a fiinction of energy E. The lineshape is centred around its maximum at E. The fiill-width at half-maximum (F) is proportional to... 

As a result, the energy E of photons emitted by an ensemble of identical nuclei, rigidly fixed in space, upon transition from their excited states (e) to their ground states (g), scatters around the mean energy Eq = E. Eg. The intensity distribution of the radiation as a function of the energy E, the emission line, is a Lorentzian curve as given by the Breit-Wigner equation [1] ... 

In the following, we consider the shape and the width of the Mdssbauer velocity spectrum in more detail. We assume that the source is moving with velocity u, and the emission line is an unsplit Lorentzian according to (2.2) with natural width E. If we denote the total number of y-quanta emitted by the source per time unit toward the detector by Nq, the number N E)AE of recoU-free emitted y-rays with energy y in the range to -f dE is given by ([1] in Chap. 1)... 

In a Mdssbauer transmission experiment, the absorber containing the stable Mdssbauer isotope is placed between the source and the detector (cf. Fig. 2.6). For the absorber, we assume the same mean energy q between nuclear excited and ground states as for the source, but with an additional intrinsic shift A due to chemical influence. The absorption Une, or resonant absorption cross-section cr( ), has the same Lorentzian shape as the emission line and if we assume also the same half width , cr( ) can be expressed as ([1] in Chap. 1)... 

Thus, the experimental Mossbauer spectmm of a thin single-line absorber is a Lorentzian line, with full-width at half maximum twice the natural Une width of the separate emission and absorption lines Texp = 2E. 

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 Lorentzian shape of x-ray emission lines is well founded in quantum theory and has been substantiated experimentally (Hoyt, 1932). Siegbahn et al. (1967) discuss the aluminum anode x-ray source as applied to ESCA. Beatham and Orchard (1976) list doublet separations and half-widths derived from the literature and optimized by computer simulation. Kallne and Aberg (1975) and Senemaud (1968) also provide values. 

Spin and velocity of photons from pair production and bremsstrahlung17 may help decide between emission theories and (Lorentzian or Einsteinian) relativistic theories. 

Figure 2. Powder emission spectrum of tr-[Rh py ClJ Cl and a lorentzian band analysis yielding three superimposed progressions. Reproduced with permission from Ref. 9. Copyright 1981, VCH-Verlag.

Figure k. Powder emission spectra of CsgSeClg and a band resolution in terms of lorentzian functions. 

For a correct analysis of photoionization processes studied by electron spectrometry, convolution procedures are essential because of the combined influence of several distinct energy distribution functions which enter the response signal of the electron spectrometer. In the following such a convolution procedure will be formulated for the general case of photon-induced two-electron emission needed for electron-electron coincidence measurements. As a special application, the convolution results for the non-coincident observation of photoelectrons or Auger electrons, and for photoelectrons in coincidence with subsequent Auger electrons are worked out. Finally, the convolutions of two Gaussian and of two Lorentzian functions are treated. 

For an optical transition to the middle of the excitonic band, the low-temperature limit width is dominated by phonon spontaneous emission with a lorentzian lineshape, more or less distorted by the density of excitonic states at the final energy — h 2s. With increasing temperature, the line broadens and reaches the high-temperature limit (2.104). 

The Franck-Condon factors of polarizable chromophores in Eq. [153] can be used to generate the complete vibrational/solvent optical envelopes according to Eqs. [132] and [134]. The solvent-induced line shapes as given by Eq. [153] are close to Gaussian functions in the vicinity of the band maximum and switch to a Lorentzian form on their wings. A finite parameter ai leads to asymmetric bands with differing absorption and emission widths. The functions in Eq. [153] can thus be used either for a band shape analysis of polarizable optical chromophores or as probe functions for a general band shape analysis of asymmetric optical lines. 

Fig. 28. Excitation spectra of Pt(2-thpy)2 dissolved in n-octane (a) at zero magnetic field and T = 4.2 K and (b) for different magnetic fields at T = 1.5 K. Concentration = 10 mol/1. For detection, the energy of 16,444 cm with a band width of = 5 cm was used, in order to monitor simultaneously the 713 cm and 718 cm vibrational satellites that correspond to the emissions of the triplet substates I and II, respectively. Under magnetic fields (b), the detection energy is red-shifted according to the size of the field-induced red shift of these satellites. The total excitation spectrum is composed of different spectra. The spectral resolution of the equipment is = 5 cm and = 160 cm for energies below and above the vertical line near 22,300 cm respectively. Note, the halfwidth given in (b) refers to the fwhm of a Lorentzian line shape function which was fit to the red flank of the corresponding peak. (Compare also to the Refs. [74] and [95])...

Fig. 5 Measured x-ray emission spectrum of Ceo (heavy solid line) compared with carbon 2p DOS calculated by the DV-Xa method (thin solid Hne). Vertical line net 2p atomic orbital population calculated by the DV-Aa method. The thin solid line is the result of the convolution of 0.5-eV FWHM Lorentzian function and the net atomic orbital population. The peak at 270 eV is a substrate signal. (Taken from Kawai and Motoyama [20].)...

Fig. 15 Effect of cluster choice on the calculation of oxygen 2p DOS (oxygen Ka x-ray emission spectrum) of CU2O. The 2p net atomic orbital population is broadened by 2.0-eV-FWHM Lorentzian function. (Taken from Sakai [29].)...

In 1996, Nemet and Kozma showed the emission spectrometry of gold laser-produced plasma to be of interest for analytical purposes a delay time of 800-1000 ns was found to ensure nni/-thermal equilibrium and thorough atomization in the plasma. The line profiles obtained under such conditions (both resonant and Stark-broadened) were fitted to a symmetric Lorentzian curve [170]. Recently, LIBS was used in combination with effective chemometric tools to develop a determination method for gold in homogeneous samples that allows the characterization of jewellery products. The results confirmed the LIBS technique as an effective alternative to the hallmark official methods [143,144,171]. 

Our discussions so far have been limited to assuming a normal, Gaussian distribution to describe the spread of observed data. Before proceeding to extend this analysis to multivariate measurements, it is worthwhile pointing out that other continuous distributions are important in spectroscopy. One distribution which is similar, but unrelated, to the Gaussian function is the Lorentzian distribution. Sometimes called the Cauchy function, the Lorentzian distribution is appropriate when describing resonance behaviour, and it is commonly encountered in emission and absorption spectroscopies. This distribution for a single variable, x, is defined by... 

Deep centres are often present in photoconductors and they can trap the photo-generated carriers. The statistical trapping (recombination or capture) and subsequent release (generation or emission) of these carriers leads to an extra source of noise called generation-recombination (g-r) noise. The presence of this noise depends on the purity of the material used as a photoconductor, but in some cases, it is inherent to the deliberate technological process as recombination centres can be added to reduce the time constant of the detector for specific applications. The time constant r of a single trap is related to its capture and emission time constants rc and re by r 1 = r 1 + t"1, and when the g-r noise arises from a trap with a definite value of r, the observed noise spectrum has a Lorentzian dependence on the modulation frequency /, peaking at /o = l/2nr. 

The total width, T, is the sum of partial widths, which can be calculated but not observed separately. Only the total width can be observed experimentally. This width does not depend on whether the line is observed in an absorption, photoionization, photodissociation, or emission spectrum because the width (or the lifetime) is characteristic of a given state (or resonance). In contrast, the peak profile can have different line shapes in different channels the line profile, q, is dependent on the excitation and decay mode (see Sections 7.9 and 8.9). For predissociation into H+CT, the transition moment from the X1E+ state to the 3n (or 3E+) predissociating state is zero, consequently q = oo and the lineshape is Lorentzian. In contrast, the ratio of the two transition moments for transitions to the XE+ continuum of the X2n state and to the (A2E+)1E+ Rydberg states leads to q 0 for the autoionized peaks (see Fig. 8.26) (Lefebvre-Brion and... 

---