Linewidth, natural

An excited atom can emit its excitation energy as spontaneous radiation (Sect. 2.7). In order to investigate the spectral distribution of this spontaneous emission on a transition Ei Ek, v/t shall describe the excited atomic electron by the classical model of a damped harmonic oscillator with frequency co, mass m, and restoring force constant k. The radiative energy loss results in a damping of the oscillation described by the damping constant y. We shall see, however, that for real atoms the damping is extremely small, which means that y 0).  

The amplitude x(t) of the oscillation can be obtained by solving the differential equation of motion 

The frequency co = co — y /4) / of the damped oscillation is slightly lower than the frequency coq of the undamped case. However, for small damping (y coq) we can set co — coo and also may neglect the second term in (3.4). With this approximation, which is still very accurate for real atoms, we obtain the solution of (3.3) as 

The frequency coq = Invo of the oscillator corresponds to the central frequency coik = (Ei — Ek)/h of an atomic transition Ei - Ek  

An excited atom can emit its excitation energy as spontaneous radiation (Sect. 2.8). In order to investigate the spectral distribution of this spontaneous emission on a transition Et shall describe the excited atomic electron by the classical 

Because the amplitude x(t) of the oscillation decreases gradually, the frequency of the emitted radiation is no longer monochromatic as it would be for an oscillation with constant amplitude, but shows a frequency distribution related to the function x(t) in (3.5) by a Fourier transformation (see Fig.3.2). 

The lower integration limit is taken to be zero because x(t) = 0 for t 0. Equation (3.7) can be easily integrated to give the complex amplitudes A(u.)  

For comparison of different line profiles it is useful to define a normalized intensity profile g(u) - wq) = CI(u)) such that 

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The most ftmdamental limitation on sharpness of spectral lines is the so-called natural linewidth. Because an... 

In aes, the resolution is largely independent of the characteristics of the analy2er or source and is dictated by the natural linewidth of the Auger line (usually several eV). Therefore, in using a CMA for aes, the analyst is more concerned with transmission (and hence, sensitivity) than with resolution. In contrast to xps, the optimhation of variables is achieved for aes in the CRR mode of operation. The large transmission of the CMA relative to the CHA make it the more desirable analy2er for aes. 

Natural linewidths are broadened by several mechanisms. Those effective in the gas phase include collisional and Doppler broadening. Collisional broadening results when an optically active system experiences perturbations by other species. Collisions effectively reduce the natural lifetime, so the broadening depends on a characteristic impact time, that is typically 1 ps at atmospheric pressure ... 

Sodium-molecule collision time 150 /LIS Peak D2 cross section for natural linewidth 1.1 IQ- m ... 

The Mossbauer effect, discovered by Rudolf L. Mossbauer in 1957, can in short be described as the recoil-free emission and resonant absorption of gamma radiation by nuclei. In the case of iron, the source consists of Co, which decays with a half-life of 270 days to an excited state of Fe (natural abundance in iron 2%). The latter, in turn, decays rapidly to the first excited state of this isotope. The final decay generates a 14.4 keV photon and a very narrow natural linewidth of the order of nano eV. 

It is worth mentioning that >NH protons may often appear somewhat broader than their -OH counterparts, for another reason >NH protons have another relaxation mechanism available to them (quadrupole relaxation) because the 14N nucleus has an electric quadrupole moment. This extra relaxation capability can lead to a shorter relaxation time for >NH protons, and since the natural linewidth of a peak is inversely proportional to the relaxation time of the proton(s) giving rise to it, a shorter relaxation time will give rise to a broader peak. 

Transitions have a natural linewidth associated with their lifetime (via the uncertainty principle) but this is usually small. 

The natural linewidth comes from the lifetime, r, of the upper state of a spontaneous transition, which is related to the Einstein A coefficient so that r = A l faster transitions have shorter lifetimes and vice versa, and similarly an allowed transition will have a short lifetime for the upper state whereas forbidden transitions will have a long lifetime. The lifetime consideration is very important in the laboratory where transitions have to occur on the timescale of the experiment, otherwise they are not observed. Hence in the laboratory allowed transitions are observed and in general (but not specifically) forbidden transitions are not seen. For astronomy this does not matter. So what if a forbidden transition has a lifetime of 30 million years - the Universe is 15 billion years old - if you wait long enough it will happen. The rules of spectroscopy need to be understood but in space anything goes ... 

The natural linewidth is the smallest contribution to the line profile of a transition and is only rarely seen as limiting within the laboratory. For an electronic transition with a lifetime of 10000 ps the linewidth is of order 0.00053 cm-1 but for a rotational transition the lifetime linewidth 5.3 x 10-15 cm-1. The best microwave spectra recorded in the laboratory have a linewidth of a few Hz or 10-12 cm-1, which is close (but not very) to the natural linewidth limit. 

Linewidth The spread in wavelengths or frequencies associated with a transition in an atom or molecule. There are three contributions natural linewidth associated with the lifetime of the transition pressure broadening associated with the presence with the other molecules nearby Doppler broadening associated with relative motion of the molecule and light source. 

This is so when absorbers are thin and when the lines are broader than the natural linewidth. For heavier samples, saturation effects come into play. In case of a sextet, the factor b, in (5-7) forms the reason that the outer peaks are more affected than the inner peaks, with the result that the line intensity ratios become lower than the expected ratio of 3 1. 

IJt) is the measured linewidth of a Mossbauer peak 71 is the linewidth of the source ra is the linewidth of the infinitely thin absorber rml is the natural linewidth... 

Figure 34 shows spectra of TA and SOL from a 24-year-old male volunteer recorded at 1.5 T (a) and 3.0 T (b). Besides the clearly improved SNR, which is elevated by a factor of 1.7 to 1.8 at 3.0 T, distinct differences can be observed for the two magnetic field strengths In both muscles, IMCL and EMCL are clearly better separated at 3.0 T as the methylene resonance of IMCL shows smaller natural linewidths (in ppm). However, EMCL signals remain with a broad lineshape in TA as well as in SOL, since the lineshape is dominated by susceptibility induced static field inhomogeneities. Crs and TMA signals are... 

The spectral linewidths of fluorescence lines are determined in most spectral lamps by Doppler effect and pressure broadening and are therefore normally much broader than the natural linewidth, which is approached only by low-pressure hollow cathode lamps 23) operated at liquid helium temperatures. 

With this technique the Doppler width could be reduced by two orders of magnitude below the natural linewidth, and spectral structures within the Doppler width could be resolved. Examples are the resolution of hyperfine structure components in an 12-beam using a single-mode argon laser (tunable within a few gigahertz) or the investigation of the upper state hfs-splitting in the atomic... 

Another method for lifetime determination uses the level-crossing technique 12 t) which measures the natural linewidth (and with it the lifetime) from the change of the spatial fluorescence intensity distri-... 

Line broadening due to inhomogeneity in the static magnetic field. Ho, as well as in the rf pulse Hj, can contribute to the observed resonance. However, studies of standard sairples, of known natural linewidths, enable the contributions from this source to be determined. In the present case these causes contribute only a few percent, i.e., a few Hz, to the total linewidth and are thus inconsequential to the present problem. Before discussing the different motional contributions to the linewidth. 

Fig. 2. (a) Raw 300 MHz proton spectrum of a mixture of acetone and ethanol in deuteri-ochloroform (b) after reference deconvolution using the acetone signal as reference and an ideal lineshape of a 1 Hz wide Lorentzian and (c) after reference deconvolution with an ideal lineshape characterized by a negative Lorentzian width of 0.1 Hz and a Gaussian width of 0.4 Hz. The 0.1 Hz Lorentzian term represents the approximate difference in natural linewidth between the ethanol and acetone signals, and is responsible for the wings on... 

The stability of the Au, cluster compound is of course supplied at least in part by the ligand shell, and this complicates a direct comparison of the above theoretical predictions for imaginary bare Au,. But experimentally, the presence of an MES spectrum consisting of a superposition of distinct lines of the natural linewidth for gold from four different structural sites also constitutes proof that gold core of the Au, cluster is, at least on the time scale of the MES measurements (i.e. 0.1 ps < t < 10 ps) [112,113], a solid up to temperatures of at least 30 K. Surface melting on this time scale can also be refuted for Au, for the same reason. The same has also been observed by MES on the water soluble compound Au, [46],... 

Fig. 19, an unapodized spectrum [response function (sin nx)/nx = sinc(x)] is shown in trace (b). For such a spectrum there will be sidelobes and negative absorption if the natural linewidths are narrower than the full width of the sine-shaped response function. These are seen in Fig. 19, where the linewidth is three points and the response function width eight points. Here the phrase instrument response function may have a slightly different definition, but the meaning is clear. For such a response function, the direct deconvolution methods fall short. 

Figure 9.11 Suppression of the 2P-1S spontaneous emission in the hydrogen atom, for which the natural linewidth is 1.66 X 10 cm". The solid lines display the decay of the optimized superposition of the Autler-Townes split levels with no interruptions. The dot-dashed lines are the decay curves of the same superposition states in the presence of interruptions. The dashed lines display the average decay of the two Autler-Townes split components. The optimization time t (marked by a triangle) is 0.2/T(= 0.65 ns), and the total time range displayed is up to 3/r(= 10 ns). A is the Autler-Townes splitting induced by the CW laser and T denotes the natural linewidth. Reprinted figure by permission from Ref. [38]. Copyright 2003 by the American Physical Society.

The uncertainty of this result is 46 Hz, which is only about ten times larger than the natural linewidth 4 Hz. The experimentalists envisage further improvement of the accuracy of this measurement with the perspective to achieve an accuracy better than 1 part in 10 [36, 41]. 

The natural linewidth of the 2P states in muonic hydrogen and respectively of the 2P — 2S transition is determined by the linewidth of the 2P — IS transition, which is equal hP = 0.077 meV. It is planned [64] to measure 2P — 2S Lamb shift with an accuracy at the level of 10% of the natural linewidth, or with an error about 0.008 meV, which means measuring the 2P — 2S transition with relative error about 4 x 10 . 

The new value has an experimental error which corresponds to measuring the hyperfine energy splitting at the level of Z z/ea p/(T)t//i) 7 x 10 of the natural linewidth. This is a remarkable experimental achievement. 

In the above expressions, iA is the lifetime of nuclei in site A, and tb is the lifetime of nuclei in site B cuA and coB are the chemical shifts of sites A and B, respectively pA and pB are the populations for each site and T2 is the natural linewidth. In fitting data to the equation, the rate constant is varied until the linewidth in the calculated spectrum matches that in the experimental spectrum. 

The width of the spectral line equals the sum of the widths of initial and final levels. Due to the short lifetime of highly excited states with an inner vacancy, their widths, conditioned by spontaneous transitions, are very broad. The other reasons for broadening of X-ray and electronic lines (apparatus distortions, Doppler and collisional broadenings) usually lead to small corrections to natural linewidth. 

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