Fourier broadening

In electron-spin-echo-detected EPR spectroscopy, spectral infomiation may, in principle, be obtained from a Fourier transfomiation of the second half of the echo shape, since it represents the FID of the refocused magnetizations, however, now recorded with much reduced deadtime problems. For the inhomogeneously broadened EPR lines considered here, however, the FID and therefore also the spin echo, show little structure. For this reason, the amplitude of tire echo is used as the main source of infomiation in ESE experiments. Recording the intensity of the two-pulse or tliree-pulse echo amplitude as a function of the external magnetic field defines electron-spm-echo- (ESE-)... 

Once the basic work has been done, the observed spectrum can be calculated in several different ways. If the problem is solved in tlie time domain, then the solution provides a list of transitions. Each transition is defined by four quantities the mtegrated intensity, the frequency at which it appears, the linewidth (or decay rate in the time domain) and the phase. From this list of parameters, either a spectrum or a time-domain FID can be calculated easily. The spectrum has the advantage that it can be directly compared to the experimental result. An FID can be subjected to some sort of apodization before Fourier transfomiation to the spectrum this allows additional line broadening to be added to the spectrum independent of the sumilation. 

As with pressure broadening, this exponential time dependence, when subjected to Fourier transformation, yields ... 

In order to find the effect of broadening of the surface on the structure parameters H and K, we first study the ordered phases with the diffusive interfaces. The ordered phases can be described by the periodic surfaces (0(r)) = 0 and we can compare and with H and K. The numerators in the definitions (76) and (77) in the Fourier representation assume the forms [68]... 

The isotope has a nuclear spin quantum number I and so is potentially useful in nmr experiments (receptivity to nmr detection 17 X 10 that of the proton). The resonance was first observed in 1951 but the low natural abundance i>i S(0.75%) and the quadrupolar broadening of many of the signals has so far restricted the amount of chemically significant work appearing on this rcsonance, However, more results are expected now that pulsed fourier-transform techniques have become generally available. 

When applied to the XRD patterns of Fig. 4.5, average diameters of 4.2 and 2.5 nm are found for the catalysts with 2.4 and 1.1 wt% Pd, respectively. X-ray line broadening provides a quick but not always reliable estimate of the particle size. Better procedures to determine particle sizes from X-ray diffraction are based on line-profile analysis with Fourier transform methods. 

Several peaks of interest (ideally higher order reflections of the same type hkl, 2h, 2k, 21, 3h, 3k, 31,. .., nh, nk, nl) are fitted by Fourier series the same procedure is applied to the diffraction lines of a reference sample, in which size and strain effects are negligible, in order to determine the instrumental line broadening. Such information is used in order to deconvolute instrumental broadening from sample effects (Stokes-Fourier deconvolution [36]). 

Information concerning size distribution and strain profile can be obtained from the cosine Fourier coefficients, which describe the symmetric peak broadening. 

It is a known property of Fourier transforms that given a convolution product in the reciprocal space, it becomes a simple product of the Fourier transforms of each term in the real space. Then, as the peak broadening is due to the convolution of size and strains (and instrumental) effects, the Fourier transform A 1) of the peak profile I s) is [36] ... 

In this short summary of peak profile analysis, we only considered the broadening due to the dimension and the strain, and we have considered only the Fourier-cosines transform (i.e. the symmetric part of the peak) that is the most frequent case. 

Fig. 6a-e OCH2 signal of compound 1 (200 MHz) a Only Fourier transformation b Fourier transformation preceded by multiplication of FID by a negative line broadening function (-0.3 Hz) c Fourier transformation preceded by multiplication of FID by a shaped sine bell function (SSB = 1) d Fourier transformation preceded by multiplication of FID by a positive line broadening function (0.8 Hz) e Fourier transformation preceded by multiplication of FID by a positive line broadening function (1.9 Hz)... 

Elimination of Instrumental Broadening and Crystal Size Effect. Fourier transform of Eq. (8.13) turns the convolutions into multiplications (Sect. 2.7.8)... 

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

Fig. 8. The 195Pt-NMR spectra of a DMF solution of [Pt2(en)3(PRI)2(N02) (N03)](N03)2 0.5 H20 (11) at 5°C, acquired on a Bruker WM-250 spectrometer operating at 53.6 MHz. (a) Power spectrum of the Fourier transform of a 1 K FID accumulated with a 5-jjls pulse width, 100-kHz spectral width, and 2000 K transients, (b and c) Normal Fourier transforms of 1 K FIDs accumulated with 10-fis pulsewidths, 42-kHz spectral width, and 64 K transients per spectrum. All FIDs were treated with 400-Hz line broadening functions to suppress noise (58).

Figure 2.10 (a) 2D H- N HETCOR correlation spectrum of fully N-labeled complex [(=SiO)2Ta(=NH) (NHj)] and [=Si- NH2] and comparison with 2D double quantum (b) and triple quantum (c) correlation spectra. An exponential line broadening of 100 Hz was applied to all the proton dimensions before Fourier transform. The dotted gray lines correspond to the resonances of the tantalum NH, NH2 and NH3 protons. The dotted circles underline the absence of auto-correlation peaks for the imido proton in the double quantum spectrum (b), and for the amido proton in the triple quantum, spectrum (c) (from Reference [9]). 

Fig. 6 Stacked Fourier transforms of the NMR spin echoes in D-RADP-25 versus echo delay time for various temperatures. In contrast to D-RADP-20 (Fig. 5) the PE rim at vl does not disappear at low temperatures but continuously broadens to become the glass rim...

A review of deconvolution methods applied to ESCA (Carley and Joyner, 1979) shows that Van Cittert s method has played a big role. Because the Lorentzian nature of the broadening does not completely obliterate the high Fourier frequencies as does the sine-squared spreading encountered in optical spectroscopy (its transform is the band-limiting rect function), useful restorations are indeed possible through use of such linear methods. Rendina and Larson (1975), for example, have used a multiple filter approach. Additional detail is given in Section IV.E of Chapter 3. 

The previous sections have dealt primarily with infrared absorption spectra, although the conclusions can in general be applied to other types of spectra. Here additional uses of deconvolution will be demonstrated. In the first example, a Fourier transform spectrum is simulated and several attempts to deconvolve this spectrum show limited success. In the second example, pressure-broadening effects in an infrared absorption spectrum and a Raman spectrum are simulated. An attempt at removing these effects by deconvolution shows some promise. 

Deconvolution, the inverse operation of recovering the original function o from the convolution model as given in Eq. (1), employs procedures that almost always result in an increase in resolution of the various components of interest in the data. However, there are many broadening and degrading effects that cannot be explicitly expressed as a convolution integral. To consider resolution improvement alone, it is instructive to consider other viewpoints. The uncertainty principle of Fourier analysis provides an interesting perspective on this question. 

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