Line shape equation

Like in the past, most of the data on the QEC effect involve this class of the metal hydrides. The NMR formalism used in the description of the coherent and stochastic dynamics in the trihydrides is a phenomenological generalization of that describing the two-particle case. In consistence with the spirit of the semiclassical AB theory, the dynamics in the two two-particle subsystems (of which either engages the nucleus in the centre) are assumed to be independent. In the trihydrides, the central hydride atom lies in the molecular symmetry plane such that the three hydride nuclei form an A2B system. Therefore, the pertinent line shape equation has the form... 

Fig. 19. Experimental variable-temperature spectra of a single crystal of ammonium persulfate-J4. The outermost doublet in the spectrum at 32 K which appears practically unchanged in the spectra at the remaining temperatures comes from the static deuteron situated at the reorientation axis. The intense multiplet in the centre is the resolved (3 doublet. The remaining, weak lines that disappear above 28 K are the a lines. Right column Theoretical spectra simulated using the AB line shape equation [i.e., Eq. (8) in which the second dissipative term is dropped]. The presence of the static deuteron was disregarded in the calculations. The quantities Vt and Ajyn are equivalents of A and Aciass, respectively. (Reproduced with permission from Ref. 77. Copyright by the American Institute of Physics, 2002).

Spectral lines are fiirther broadened by collisions. To a first approximation, collisions can be drought of as just reducing the lifetime of the excited state. For example, collisions of molecules will connnonly change the rotational state. That will reduce the lifetime of a given state. Even if die state is not changed, the collision will cause a phase shift in the light wave being absorbed or emitted and that will have a similar effect. The line shapes of collisionally broadened lines are similar to the natural line shape of equation (B1.1.20) with a lifetime related to the mean time between collisions. The details will depend on the nature of the intemrolecular forces. We will not pursue the subject fiirther here. 

Figure B2.5.12 shows the energy-level scheme of the fine structure and hyperfme structure levels of iodine. The corresponding absorption spectrum shows six sharp hyperfme structure transitions. The experimental resolution is sufficient to detennine the Doppler line shape associated with the velocity distribution of the I atoms produced in the reaction. In this way, one can detennine either the temperature in an oven—as shown in Figure B2.5.12 —or the primary translational energy distribution of I atoms produced in photolysis, equation B2.5.35.

Equation (2.27) illustrates what is called the natural line broadening. Since each atom or molecule behaves identically in this respect it is an example of homogeneous line broadening, which results in a characteristic lorentzian line shape. 

At low temperatures the electron and hole created by the probe light beam can form a bound state (called an exdtori) because of the Coulomb interaction between them. In this case the exponent m in Equation (1) becomes 2 and the line shape is only a first derivative. ... 

These modulation methods do not accelerate the electron-hole pairs and hence produce only a first-derivative Modulation Spectroscopy. Their line shapes are given by Equation (1), with m = 2. 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

The relationship between the line shape of an NMR spectrum and the lifetime of chemical processes is provided by the Bloch equations. Let us imagine that there is a chemical equilibrium... 

Several other points are worth noticing Well-defined motions lead to well-defined spectra. Thus chain motion can be studied, where the analysis of 2H line shapes yields directly the number of conformations accessible for a given segment. Moreover, even more complicated motions not considered explicitely here lead to spectra, the angular dependence of which can be described by Equation (1 a). The corresponding line shapes can easily be calculated facilitating the analysis of the data. In glassy... 

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

A plot of v vs. T2(a>o co) is shown in Figure 5.1. Equation (5.14) corresponds to the classical Lorentzian line shape function and the absorption curve of Figure 5.1 is a Lorentzian line . The half-width at half-height is easily found to be ... 

Compute line shape F(Bres, D with Equation 4.8 ADD intensity F to absorption-array END STEP in dB END STEP in cp END STEP in cos0... 

Line shape analysis was performed for the binding of some dihydroxycholate ions to /1-cyclodextrin.205 The dihydroxycholates show different 18-CH3 signals for the complexed and free dihydroxycholate ions. To extract exchange rate constants from the NMR spectra, a complete line-shape simulation was performed. The simulation requires input of the chemical shift difference between the two sites, the line width in the absence of exchange, the residence time in each site (thg and Tg), and the relative population (fHG and fG) of each site (Equation (11)). The values were varied until the simulated and experimental spectra could be superimposed. The dissociation rate... 

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