Spectrum envelope Fourier transformation

ESE envelope modulation. In the context of the present paper the nuclear modulation effect in ESE is of particular interest110, mi. Rowan et al.1 1) have shown that the amplitude of the two- and three-pulse echoes1081 does not always decay smoothly as a function of the pulse time interval r. Instead, an oscillation in the envelope of the echo associated with the hf frequencies of nuclei near the unpaired electron is observed. In systems with a large number of interacting nuclei the analysis of this modulated envelope by computer simulation has proved to be difficult in the time domain. However, it has been shown by Mims1121 that the Fourier transform of the modulation data of a three-pulse echo into the frequency domain yields a spectrum similar to that of an ENDOR spectrum. Merks and de Beer1131 have demonstrated that the display in the frequency domain has many advantages over the parameter estimation procedure in the time domain. 

Fig. 27a-c. Electron spin echo envelope modulation of Co(acacen), temperature 4K. a) Nuclear modulation pattern of Co(acacen) diluted into a Ni(acacen) 1/2 H20 single crystal. Crystal setting rotation axis I,

Fourier transform of the nuclear modulation pattern (From R. de Beer1 4)) c) Stick spectrum ENDOR frequencies (AmN = 1, 2) calculated from the hfs and quadruple tensors in Ref. 59 dashed lines ms = - 1/2, full lines ms = 1/2... 

ESEEM is a pulsed EPR technique which is complementary to both conventional EPR and ENDOR spectroscopy(74.75). In the ESEEM experiment, one selects a field (effective g value) in the EPR spectrum and through a sequence of microwave pulses generates a spin echo whose intensity is monitored as a function of the delay time between the pulses. This resulting echo envelope decay pattern is amplitude modulated due to the magnetic interaction of nuclear spins that are coupled to the electron spin. Cosine Fourier transformation of this envelope yields an ENDOR-like spectrum from which nuclear hyperfine and quadrupole splittings can be determined. 

It is also possible to Fourier transform the echo envelope modulation to obtain a frequency spectrum showing the various nuclear hyperfine frequencies. In practice there are some experimental difficulties and the intensities of the Fourier transformed spectrum which are related to the number of Interacting nuclei are not well defined. However, the frequency spectrum does help to Identify frequency components associated with specific nuclei and the presence of weak isotropic hyperfine interactions. In the... 

In order to test the predictions of the model, plates were made up from selected formulations and tested in air. The thicknesses of the various layers were as stated previously, while the length was one meter and the width was one-third meter. Damping measurements were made by two different methods. In both cases, the plates were suspended in a shock chord and accelerometers placed at different locations on the plate. The first used a reverberation meter and the half-power method (16). In the second, an impact hammer was used to tap the structure and the outputs of the accelerometers fed into a Fourier transform based spectrum analyzer to examine the envelope of vibration (17). The results presented here are based upon the second method. 

FIG. 11. J-spectrum obtained by Fourier transformation of the spin-echo envelope from the protons of 1,1,2-trichloroethane. The interval between echoes is 2r = 40 milliseconds and echoes have been sampled for 30 seconds. Responses are observed at 0 Hz, 2-97 Hz, and 5-94 Hz. From ref. 145. 

MAS spectrum, consisting of narrow lines separated by the rotor firequency cur. The maxima of the rotational echoes in (b) follow the evolution of the isotropic mean. If the MAS FID is sampled at the echo maxima only, no sidebands will appear in the MAS spectrum. A similar situation is encountered for fast spinning, when the rotational echoes are no longer resolved. The envelope of the sideband spectrum is the Fourier transform of the rotational echo decay (c) [Mar2]. For slow spinning speeds, it approaches the shape of the wideline spectrum (a) of the nonspinning sample. 

The spectrum obtained by Fourier transform of figure A 1.6.12 trightl tal is shown in figure Al.6.12 (right) (bl. Qualitatively, it has all the features of the spectrum in figure A1.6.11 a broad envelope with resolved vibrational structure underneath, but with an ultimate, unresolvable linewidth. Note that the shortest time decay, 5, determines the overall envelope in frequency, 1/5 the recurrence time, T, determines the vibrational frequency spacing, InlT, the overall decay time determines the width of the vibrational features. Moreover, note that decays in time correspond to widths in frequency, while recurrences in time correspond to spacings in frequency. 

The short time limit is Gaussian and determines the envelope of the spectrum, shown in Fig. 6. Since the Fourier transform of a Gaussian in time is a Gaussian in frequency and since a phase factor corresponds to a shift in the frequency, the frequency mQ corresponds to the center of the Franck-Condon band (it is the frequency of the transition turning... 

Then the envelope of the spectrum is specified by the first My constraints and the additional, M — A/, constraints determine the finer structure seen only at resolution better than Ath. This is seen by coarse graining the spectrum (Eq. (62)) over At t. As discussed in Sec. II, the convolution theorem implies that the corresponding Fourier transform satisfies C(tr) = 0, r > My. This can also be seen directly from Eq. (67). Hence the Lagrange multipliers, Eq. (76) satisfy... 

The rate of the initial decay of P (t) is related to the width of the overall absorption envelope for the 8727 to 8927 cm > frequency range of the experimental spectrum. If a Fourier transform were taken of a Lorentzian /(w) for only the principal peak in the spectrum, the decay rate would be significantly smaller. 

Power spectrum obtained by multiplying the envelope in Figure 13 by an arbitrary term, cos(Fourier transforming. The peak separation is 20.46 KHz.8... 

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