Inhomogeneously broadened line

In a review of the 1988 International Conference on Luminescence, McClure (97) commented that No longer can one tolerate the loss of information in an inhomogeneously broadened line . 

Fig. 2.7 Dependence of the experimental line width Cexp on the effective absorber thickness t for Lorentzian lines and inhomogenously broadened lines with quasi-Gaussian shape (from [9])...

Here <( t ) f(t")> is the autocorrelation function of the electromagnetic field. For the case of excitation by a conventional light source, where the amplitudes and the phases of the field are subject to random fluctuations, the field autocorrelation function differs from zero for time intervals shorter than the reciprocal width of the exciting source. In the limit 8v A, that is when the spectral width, 8v, of the source exceeds the inhomogenously broadened line width, the field autocorrelation function can be represented as a delta function... 

Now n = pA is just the average number of levels within the halfwidth of the inhomogenously broadened line, so that... 

A distinction arises in the description of the lineshape when the dense, optically inactive manifold is replaced by a sparse one, corresponding to the intermediate case 15>. This may apply to small molecules such as SO2 and NO2, and also to large molecules characterized by a small energy gap between the two interacting electronic states, such as naphthalene 16> V) and 3,4-benzpyrene 18>. According to Jortner, Rice and Hochstrasser 8>, separate, inhomogenously broadened lines are predicted for the intermediate case. 

Fig. 18.4 The spectrum of a single molecule (upper panel) is superimposed on similar spectra of other molecules (middle panel) to yield the inhomogeneously broadened line shape (lower panel).

Fig. 3.4.1 Homogeneously and inhomogeneously broadened lines, (a) Echo train generated by repeated refocussing of the FID (CPMG method, cf. Fig 2.2.10(b)). (b) The Fourier transform of the slowly decaying echo envelope is the homogeneously broadened line, (c) The Fourier transform of the fast decaying echo is the inhomogeneously broadened line.

Fig. 14. Schematic of selective excitation and ID exchange spectroscopy, (a) Typical pulse sequence with a soft selective pulse centered at pulsation a>s with a frequency dispersion AcoP <3C Aoj much smaller than the typical linewidth. After an evolution time te smaller or of the order of the spin-lattice relaxation time, a reading sequence of hard pulses that covers uniformly the whole broad line is applied, (b) Effect of a selective excitation on a homogeneously broaden line, (c) Selective frequency labeling of an inhomogeneously broaden line at the irradiation pulsation cos of the first soft pulse. For a soft n pulse, the magnetizations of all the spins that can exchange energy at this pulsation are reversed. By following the difference spectra between the spectra acquired at different evolution times te and the fully relaxed spectrum AS(te) — S(t -> oo) — S(te), limits or evaluation of the correlation time tc of the motion can be achieved.

A simple example of an inhomogeneously broadened line is a resonance broadened by inhomogeneity of the applied dc magnetic field. If the applied field has a spatial inhomogeneity, otherwise identical spins in different parts of the sample will have slightly different resonance frequencies. As a result, the FID will be much shorter than that determined by... 

The H dipolar powder pattern has features which affect other NMR experiments. First, this is an inhomogeneously broadened line composed of many, narrow but homogenously broadened peaks. Second, the two subspectra generated by ms( H) = +1/2 are mirror images of each other. Third, powder-pattern averages of axially-symmetric units, such as H, give the characteristic Pake... 

The method to obtain relaxation times from CW microwave saturation measurements for an inhomogenously broadened line is based on assumptions given in the literature [77, 78], The ESR line shape is then expressed as a convolution of a Gauss and a Lorentz function [81] ... 

The line-shape given by (9.6) is a Voigt function that can be evaluated numerically by a standard procedure [81]. For a single inhomogenously broadened line, the transition probability is set to 1 as in a simple two-level system. The relaxation times are given by ... 

Figure 8. Fluorescence excitation spectra for pentacene in/j-terphenyl at 1.5 K measured with a tunable dye laser of linewidth 3 MHz. The laser detuning frequency is referenced to the line center at 592.321 nm. (a) Broad scan of the inhomogeneously broadened line all the sharp features are repeatable structure, (b) Expansion of 2 GHz spectral range showing several single molecules, (c) Low-power scan of a single molecule at 592.407 nm showing the lifetime-limited width of 7.8 MHz and a Lorentzian fit. After Ref. 7.

For inhomogeneously broadened line shapes it necessarily follows that no information about time-dependent fluctuations of the chromophore s transition frequency (which I will call spectral dynamics) can be obtained from the line shape itself. This does not mean that such dynamic fluctuations do not occur it simply means that either their amplitude is much smaller than the inhomogeneous line width or that their time scale is much longer than the inverse of the inhomogeneous line width. In either case these dynamic fluctuations are of great interest because they result from time-dependent changes in the local environments of chromophores, and hence can provide information about solid-state dynamics. 

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