Incoherent signal

Neutron cross sections themselves follow no trend. Moreover, as was shown above, quite different values are foimd for different isotopes of the same element. This is extensively exploited in dif action studies of liquid solutions [4]. Occasionally, when isotopes of the same element have scattering lengths of opposite sign, the average coherent cross section can be reduced to zero by making a sample with the correct proportions of each isotope. Such null-scattering samples produce only incoherent signals. 

As it was discussed in O Sect. 29.2, incoherent (inelastic) neutron scattering accounts for the individual motions of atoms. Thus, in the crystalline solid state, in the absence of diffusion, most of the incoherent signals originate from molecular vibrations. Most chemical applications have something to do with the role of hydrogen to which neutrons are more sensitive than any other probe. The reason for this can be understood by looking at the incoherent inelastic cross section, which can be related to the (vibrational, one-phonon approximated) density of states distribution function Z co) as (see, e.g., Marshall and Lovesey 1971 Bacon 1977)... 

The relative magnitude of the coherent and incoherent structure factors mainly depends on the nature of the nuclei. For almost all the atoms bcoh inc of the principal exceptions is the H atom for which bcoh = -3.74 fm, bine = 25.22 fm. Because of that the incoherent signal of protons is 50 times larger than the coherent one. For structural analysis, i.e. for the determination of the equilibrium positions of the sample atoms, it is preferable to work with deuterated samples. In contrast dynamical measurements of individual guests require a highly incoherent scatterer and hydrogen is the best candidate. 

In many signals, noise is mainly distinguished by incoherence, that is, the error at one measurement is not correlated with the error at any other. This property can be exploited to reduce the average amount of error, since a wavelet analysis of an incoherent signal consists of many small components. By neglecting all components in a wavelet wries that fall below a threshold, the noise components will be preferentially suppressed. The threshold must be chosen based on an estimate of the noise energy in the signal. 

Under the assumption that the sensors both have weU-known responses, the instrument corrected output, from the two sensors, should only differ in instrumental self-noise. By estimating the coherent signal between the two records, the coherence signal is removed resulting in the incoherent signal which is attributed to the self-noise. It can be seen from this estimate of the self-noise that it is critical to have well-described transfer functions for both instruments i and j. Since the transfer functions are used in the calculation, errors in the transfer function will produce errors in the self-noise estimates of the instrument. This method was originally proposed by Holcomb (1989), who later characterized error sources (Holcomb 1990). In the latter work, Holcomb also suggested alternative two-sensor methods, under the assumptions that the two sensors... 

Figure 1 Schematic representation of the 13C (or 15N) spin-lattice relaxation times (7"i), spin-spin relaxation (T2), and H spin-lattice relaxation time in the rotating frame (Tlp) for the liquid-like and solid-like domains, as a function of the correlation times of local motions. 13C (or 15N) NMR signals from the solid-like domains undergoing incoherent fluctuation motions with the correlation times of 10 4-10 5 s (indicated by the grey colour) could be lost due to failure of attempted peak-narrowing due to interference of frequency with proton decoupling or magic angle spinning.

Each FID signal is accompanied by noise however, the noise is incoherent— sometimes positive, sometimes negative—so that it increases more slowly than the desired nuclear signal. A series of N FID s has a signal-to-noise ratio //V times better than a single FID, allowing spectroscopists to obtain useful chemical information from otherwise unreceptive nuclei or from dilute solution samples having few of the nuclei of interest. 

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