Natural Linewidth of Absorbing Transitions

In a similar way the spectral profile of an absorption line can be derived for atoms at rest. The intensity I of a plane wave passing in z-direction through an absorbing sample decreases along the distance dz by 

The absorption coefficient [cm ] for a transition i) — k) depends on the population densities Nj, N]j of lower and upper level, and on the optical absorption cross section a- [cm ] of each absorbing atom, see (2.21)  

The absorption profile a(w) can be obtained from our classical model of a damped oscillator with charge q under the influence of a driving force qE caused by the incident wave with amplitude E = Eoe . The corresponding differential equation 

In a sample with N oscillators per unit volume the macroscopic polarization P which is the sum of all dipole moments per unit volume is therefore 

On the other hand, the polarization can be derived in classical electrodynamics from Maxwell s equations using the dielectric constant Cq or the susceptibility X, i-C-. 

The absorption profile a co) is Lorentzian with a FWHM of Acu = y, which equals for free atoms at rest the natural linewidth. The difference n — hq = n — I between the refractive indices in a gas and in vacuum yields a dispersion profile. 

Note The relations derived in this section are only valid for oscillators at rest in the observer s coordinate system. The thermal motion of real atoms in a gas introduces an additional broadening of the line profile, the Doppler broadening, which will be discussed in Sect. 3.2. The profiles (3.24a, 3.24b) can therefore be observed only with Doppler-free techniques (Vol. 2, Chaps. 2 and 4). 

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The transmission throngh an absorber of thickness teff as a ftmction of the relative velocity V between source and absorber is given by the evalnation of the transmission integral discussed, for example, in Ref 2. In case of Fe Mdssbaner spectroscopy for almost all samples which are not enriched in the Mossbauer isotope Fe (the natural abundance of Fe is 2%), the evaluation of the transmission integral leads to Lorentzian line shape with a minimum linewidth of 0.19mms , which is twice the natural linewidth of the Fe Mossbauer transition. The absorption cross section as a ftmction of source velocity is then given as (with... 

The 64-keV level of Gd has a lifetime of 460 ns, giving it a very narrow natural linewidth of only 0-0093 mm s the resonance was first reported in 1966 [84]. The Eu parent has a short lifetime of 15-4 h and the i Gd(y,p) Eu preparation requires separation of the required source activity from other contaminants. The combined quadrupole splitting of the GdaOj source and absorber from the f, f El transition gave line separations an order of magnitude greater than the linewidth. 

The 37T5-keV transition is from the / = f + excited state to the / = f-1-ground state, the multipolarity being pure Ml. The first measurements used a source of chemically separated "Sn electroplated as 8-tin onto copper, and absorbers of Sb metal and SbjOs [24]. Isotopic enrichment of absorbers was not required to obtain a significant absorption with both source and absorber at 80 K. The tin source matrix was used in most of the early work, although "Sn/SnOa has also been used with success [26]. A third source is Ca SnOs, which by analogy with the same sources used in Sn work (see Chapter 14.1) is likely to have the narrowest line and highest recoil-free fraction of the three [27, 28]. Detailed comparative data are not available, but experiments are usually made with the source and absorber at 80 K in all cases. Linewidths close to the natural width (2-1 mm s ) can be obtained with an InSb absorber and a "Sn/jS-Sn source [29]. The recoil-free fraction of a "Sn/SnOj source is 0-32 at 80 K and 0-16 at room temperature [26]. The internal conversion coefficient derived from the same measurements is Or 10. 

The 49-8-keV resonance of thorium-232 was reported in 1968 following experiments using Coulomb excitation of a thorium metal target by 4-5-MeV a-particles at 80 K [1]. The transition is highly internally converted (aj = 260), which reduces the effective excitation rate considerably. The linewidth obtained from an absorber of Th02 at 110 K was 16-7 mm s"S which is close to the natural width of 15-7 mm s thereby implying negligible quadru-pole interaction, consistent with the cubic lattices of the metal and ThOa. No hyperfine effects have yet been reported. 

Only those molecules with velocity components Vx in the interval Avx = hco jk around Vx = (co - coo)/k essentially contribute to the absorption of the monochromatic laser wave, because these molecules are shifted into resonance with the laser frequency co within the natural linewidth Scon of the absorbing transition. 

The expanded beam from a HeNe laser at A = 3.39 p,m with lOmW power is sent through a methane cell T = 300 K, / = 0.1 mbar, beam diameter 1 cm). The absorbing CH4 transition is from the vibrational ground state (r 00) to an excited vibrational level with r 20 p,s. Give the ratios of Doppler width to transit-time width to natural width to pressure-broadened linewidth for a collision cross section CTj = 10 cm. ... 

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