Lineshape Lorentzian, natural

Doppler broadening has a Gaussian lineshape, and its convolution with the Lorentzian natural lineshape yields a Voigt profile. In typical experiments, this effect can be neglected since the Doppler width is usually much smaller than the resolution of the apparatus. Collisional line broadening is also Lorentzian, and the Lorentzian component of measured lines must be carefully extrapolated to zero pressure. 

Suppose that a pulse Fourier transform proton NMR experiment is carried out on a sample containing acetone and ethanol. If the instrument is correctly operated and the Bq field perfectly uniform, then the result will he a spectrum in which each of the lines has a Lorentzian shape, with a width given hy the natural limit 1/(7tT2). Unfortunately such a result is an unattainable ideal the most that any experimenter can hope for is to shim the field sufficiently well that the sample experiences only a narrow distribution of Bq fields. The effect of the Bq inhomogeneity is to superimpose an instrumental lineshape on the natural lineshapes of the different resonances the true spectrum is convoluted by the instrumental lineshape. 

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 amplitude of the oscillation is a maximum when the driving frequency equals the natural resonance frequency and the width of the resonance line depends on the degree of damping of the system. If the liquid in which the system is immersed is not viscous, the line will be narrow. On the other hand, the line will be quite broad for a viscous liquid. The line-shape produced by such a damped harmonic oscillator is a Lorentzian lineshape. The full width at half height of such a... 

Tj and Tg are usually different in solids, although this can happen quite often in liquids as well. (This should be considered as a blessing rather than a nuisance since nature is trying to give us additional information. ) In IV. A. 2. we say that solids usually have a lineshape close to a Gaussian while most liquids have Lorentzian lineshapes. Since we have so far treated T as a time constant for an exponential decay, there is an inconsistency here. How should we define a T2 for a Gaussian or some other non-Lorentzian line ... 

The lineshape factor g(v) for the 1s-2s case has to be chosen with some care since the natural width, which is about 1 Hz, is clearly not the dominant contribution. However, I will assume here for simplicity that the dominant lineshape is Lorentzian in form, caused by pressure... 

The present treatment of solvent broadening is of course an oversimplification. It does not take into account the dynamic nature of the interaction or the contributions of solvent excitations to the resonant state. On a more elementary level, the representation of A by a Lorentzian is probably unrealistic since solvent-broadened lines often approximate a Gaussian shape. Our choice of a Lorentzian is made to obtain analytical expressions for the cross section a Gaussian lineshape requires numerical procedures, but presents no fundamental difficulties. It turns out (O. Sonnich Mortensen, unpublished) that, for a given halfwidth A, a Gaussian tends to distort REPs and polarization dispersion curves more strongly than a Lorentzian. 

The energy distribution of the emitted radiation and that of the resonant absorption cross section have identical lineshapes. Both are Lorentzian lines centered at the nuclear transition energy E, with fullwidth at half height given by the uncertainty principle energy width of the nuclear excited state. This width, the natural linewidth, is defined by... 

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