Linewidth intensity dependence

NMR spectroscopy, while containing a high information-density, is quite an unsensitive method. Therefore, the main challenge is to obtain well-resolved spectra of high signal-to-noise ratio. Apart from sample concentration, the signal intensity depends mainly on the linewidth. Linewidth, in turn, depends mainly on relaxation. 

The Bloch equations (Eq. 5) can be solved under different conditions. The transient solution yields an expression for 0-22 (0> time-dependent population of the excited singlet state S. It will be discussed in detail in Section 1.2.4.3 in connection with the fluorescence intensity autocorrelation function. Here we are interested in the steady state solution (an = 0-22 = < 33 = di2 = 0) which allows to compute the line-shape and saturation effects. A detailed description of the steady state solution for a three level system can be found in [35]. From those the appropriate equations for the intensity dependence of the excitation linewidth Avfwhm (FWHM full width at half maximum) and the fluorescence emission rate R for a single absorber can be easily derived [10] ... 

The absolute measurement of areas is not usually usefiil, because tlie sensitivity of the spectrometer depends on factors such as temperature, pulse length, amplifier settings and the exact tuning of the coil used to detect resonance. Peak intensities are also less usefiil, because linewidths vary, and because the resonance from a given chemical type of atom will often be split into a pattern called a multiplet. However, the relative overall areas of the peaks or multiplets still obey the simple rule given above, if appropriate conditions are met. Most samples have several chemically distinct types of (for example) hydrogen atoms within the molecules under study, so that a simple inspection of the number of peaks/multiplets and of their relative areas can help to identify the molecules, even in cases where no usefid infonnation is available from shifts or couplings. 

The sinc fiinction describes the best possible case, with often a much stronger frequency dependence of power output delivered at the probe-head. (It should be noted here that other excitation schemes are possible such as adiabatic passage [9] and stochastic excitation [fO] but these are only infrequently applied.) The excitation/recording of the NMR signal is further complicated as the pulse is then fed into the probe circuit which itself has a frequency response. As a result, a broad line will not only experience non-unifonn irradiation but also the intensity detected per spin at different frequency offsets will depend on this probe response, which depends on the quality factor (0. The quality factor is a measure of the sharpness of the resonance of the probe circuit and one definition is the resonance frequency/haltwidth of the resonance response of the circuit (also = a L/R where L is the inductance and R is the probe resistance). Flence, the width of the frequency response decreases as Q increases so that, typically, for a 2 of 100, the haltwidth of the frequency response at 100 MFIz is about 1 MFIz. Flence, direct FT-piilse observation of broad spectral lines becomes impractical with pulse teclmiques for linewidths greater than 200 kFIz. For a great majority of... 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

The intensity of a Mossbauer spectrum depends not only on the recoil-free fractions of the source and the absorber and on the number of absorbing nuclei, but also on the linewidth of the absorption lines and on whether or not saturation effects occur. The following approximate expression is valid for relatively thin absorbers [17] ... 

The computation of far-field radiation from a collection of incoherently radiating dipoles is in general quite a complicated problem. To calculate the angular dependence of the far-field intensity, the volume distribution of excited states must first be obtained, which, as we have seen, depends on the volume distribution of the absorbers and the electromagnetic field which stimulates them. The fields in turn depend on the frequency and linewidth of the exciting light source. Then the emission problem for the excited-state distribution (both spatial and frequency) must be solved including reorientation and depolarization effects. 

Fortunately in a number of cases there is experimental information on these points from broad band pump/probe experiments when the anharmonicity A is larger than the linewidth but much smaller than the bandwidth 8(o of the laser. Then the 0-1 transition is seen as a bleaching signal and the 1-2 (66,67,71) as well as the 2-3 and often higher quantum number transitions (68,95) appear as new absorptions to an extent that depends on the pump intensity. A direct comparison of the total linewidths (1/T2) of these transitions, and the population relaxation times for the v = 1, v = 2 and perhaps higher levels can be obtained from such data. For N3 we found that ratio of the state to state relaxation from v = 2 to v = 1 was 1.8 times that for v = 1 to v = 0, not far from the harmonic value of 2 (50,95). However, the bandwidth of both transitions was roughly the same. 

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