Gaussian line broadening

The composite filter 7(g)) may either be the true inverse filter, truncated for oo large if necessary, or any of the variations described in Section IV. In their original work, Rendina and Larson chose 7(g)) = (co)/t(co), where //(co) is a Gaussian line-broadening function that limits the ultimate resolution obtainable but yields a manageable 7(g)). For their studies Rendina and Larson used Ns = 4. 

Figure 1 Simulated MAS (rows Ml and M2) and QCPMG (rows Q1 and Q2) spectra corresponding to the static limit (fc=10-9 Hz) of a two-site jump process corresponding to parameter sets P1-P5 in Table 1. In rows Q1 and Ml, the Hamiltonian includes the only HQ(1) whereas both HQ(1) and Hq(2) are included for the spectra in rows Q2 and M2. Gaussian line broadenings of 30 (A), 50 (B) or 75 (C-E) Hz were applied prior to Fourier transformation.

Figure 4 Simulated 14N (43.34 MHz) rotor-synchronized MAS spectra corresponding to a two-site jump process and detection of either singlequantum (columns Al, Bl, Cl) or double-quantum coherence (columns A2, B2, C2). The spectra in columns A (A1.A2) are calculated using parameter set P6, whereas parameter sets P7 and P8 are used for columns B (Bl, B2) and C (Cl, C2), respectively. Gaussian line broadenings of 75 (Al, Bl, Cl), 50 (A2, B2) or 100 (C2) Hz were applied prior to Fourier transformation. The logarithm of the rate constant k is indicated at each row of spectra.

Figure 6 Simulated 2H (73.58 MHz) QCPMG spectra corresponding to a 3-by-2-site jump process using parameter set P2b in Table 1 and the single-frame set of Euler angles were69 (0.0,124.0, 0.0), (57.594, 55.006, 91.716), (302.406, 55.006, 268.206), (0.0, 124.0,180.0), (57.594, 55.006, 271.716), (302.406, 55.006, 88.206). The logarithm of the rate constants and k2 are indicated at each row and column of the spectra. All spectra were apodized by Gaussian line broadening of 50 Hz prior to Fourier transformation.

Figure 7 Simulated 2H (73.58 MHz) MAS spectra corresponding to a 3-by-2-site jump process the same parameter set as in Figure 6. The logarithm of the rate constants and k2 are indicated at each row and column of the spectra. All spectra were apodized by Gaussian line broadening of 50 Hz prior to Fourier transformation.

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

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]... 

A computer simulation was performed to observe the effect of variable line width on the calculated ESR first derivative spectrum. Since dipolar interaction is the major contribution to line broadening in the ESR spectrum of PVC radicals, a Gaussian line shape is expected for each of the ten absorptions. Therefore each line was assigned a Gaussian shape, the variables being relative amplitude position in the spectrum and line... 

Fig. 1 Top Behavior of the electronic linear chiroptical response in the vicinity of an excitation frequency. Re = real part (e.g., molar rotation [< ]), Im = imaginary part (e.g., molar ellipticity [0]). Without absorption line broadening, the imaginary part is a line-spectrum (5-functions) with corresponding singularities in the real part at coex. A broadened imaginary part is accompanied by a nonsingular anomalous OR dispersion (real part). A Gaussian broadening was used for this figure [37]. Bottom Several excitations. Electronic absorptions shown as a circular dichroism spectrum with well separated bands. The molar rotation exhibits regions of anomalous dispersion in the vicinity of the excitations [34, 36, 37]. See text for further details...

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