Fourier transforms absorption lineshape

In real experiments after Fourier transformed the lineshapes are mixtures of absorptive and dispersive signals and are related to the delayed FID acquisition (first-order phase error). The delayed acquisition is a consequence of the minimum time required to change the spectrometer from transmit to receive mode, during this delay the magnetization vectors process according to their chemical shift frequencies. The zero-order phase error arises because of the phase difference between the magnetization vectors and the receiver. In NMR-SIM the delayed acquisition is not necessary because the ideal spectrometer approach does not require any switching time and the first order phase correction is normally zero if no other sources of phase deviations are present. 

For all of the eases eonsidered earlier, a C(t) funetion is subjeeted to Fourier transformation to obtain a speetral lineshape funetion I(co), whieh then provides the essential ingredient for eomputing the net rate of photon absorption. In this Fourier transform proeess, the variable co is assumed to be the frequeney of the eleetromagnetie field experienced by the molecules. The above considerations of Doppler shifting then leads one to realize that the correct functional form to use in converting C(t) to I(co) is ... 

In the case of resonance absorption of synchrotron radiation by an Fe nucleus in a polycrystalline sample, the frequency dependence of the electric field of the forward scattered radiation, R(oj), takes a Lorentzian lineshape. In order to gain information about the time dependence of the transmitted radiation, the expression for R(oj) has to be Fourier-transformed into R(t) [6]. 

The basic processing of ID and 2D data requires obligatory processing steps for transforming the raw data (FID) into a "readable spectrum, i.e. Fourier transformation and phase correction to produce a spectrum with absorptive lineshapes. Finally, a few additional step.s (calibration, peak picking, integration) as discussed in chapter 4 are required before the spectrum is eventually plotted. 

The lineshape function which describes the absorption and dispersion modes of an unsaturated, steady-state NMR spectrum is proportional to the Fourier transform of the function MxID(t) (24, 25, 99)... 

Here T2 is transverse relaxation time of an electron spin. The EPR absorption lineshape is related to free induction decay G(t) through the Fourier transformation [16]. After averaging over all angles between the surface and the external magnetic field H the following equation was obtained for D = 2 systems [138] ... 

After you Fourier transform your FID, you get a frequency-domain spectrum with peaks, but the shape of the peaks may not be what you expected. Some peaks may be upside down, whereas others may have a dispersive (half up-half down) lineshape (Fig. 3.36). The shape of the peak in the spectrum (+ or — absorptive, + or — dispersive) depends on the starting point of the sine function in the time-domain FID (0° or 180°, 90° or —90°). The starting point of a sinusoidal function is called its phase. Phase errors come in all possible angles, including those intermediate between absorptive and dispersive (Fig. 3.37). The spectrum has to be phase corrected ( phased ) after the Fourier transform to obtain the... 

The Fourier transform converts the FID into a Lorentzian peak with absorptive lineshape (after phase correction). The full width of this peak at one half of the peak s height (the linewidth ) is inversely related to the decay time constant of the FID, ... 

An exponentially decaying FID gives a Lorentzian lineshape upon Fourier transformation. The general form of the absorptive Lorentzian line is IabS = 1/(1 + v2), whereas the dispersive line has the form Idisp = v/(l + v2), where I is the intensity at each point in the spectmm. Far from the peak maximum (v2 >> 1), we have Iabs 1/v2 and Idisp l/y- This is the reason that the dispersive lineshape extends much further from the peak maximum. 

In die Fourier transform of a real time series, die peakshapes in the real and imaginary halves of die spectrum differ. Ideally, the real spectrum corresponds to an absorption lineshape, and die imaginary spectrum to a dispersion lineshape, as illustrated in Figure 3.20. The absorption lineshape is equivalent to a pure peakshape such as a Lorentzian or Gaussian, whereas die dispersion lineshape is a little like a derivative. 

In this type of experiment, the echo and antiecho are linearly combined with the same amplitude to yield an amplitude-modulated signal in Pure absorption lineshapes may then be obtained in the frequency domain spectrum after a two-dimensional Fourier transform is performed. The disadvantage of this method is that it is not possible to discriminate the sign of the MQ coher-... 

First, either a Lorentzian or Gaussian filter is applied to the FID to reduce the amount of noise. The choice of lineshape will depend on the shape of the frequency domain spectrum, the lineshape is related to how the fluorine spins interact with their environment. The filter linewidth is generally similar to or slightly less than the T2 value (T2 can be estimated from the spectral linewidth). After application of the time domain filter, a fast Fourier transform (FFT) is performed. The resultant frequency domain spectrum will then need to undergo phase adjustment to obtain a pure absorption spectrum. The amount of receiver dead time (time lost between the end of the excitation pulse and the first useful detection time point) will determine the presence and extent of baseline artifact present as well as how difficult phase adjustment will be to accomplish. 

A 2D NMR experiment can lead to a data set that is either phase modulated or amplitude modulated as a function of fj, depending on the particular experiment and coherence pathways selected. A regular ID spectrum consists of absorption A(p) and dispersion peaks corresponding to the real and imaginary parts of the spectral lines, respectively. In 2D experiments, phase modulation in fj results in twisted 2D real lineshapes as a result of the Fourier transformation of bi-exponential time domain... 

In Section 6.2,3 we have seen that a simple quantum mechanical theory based on the golden rule yields an expression for the absorption lineshape that is given essentially by the Fourier transform of the relevant dipole correlation function (/ii(0)/ii(Z)). Assuming again that ix is proportional to the displacement x of the oscillator from its equilibrium position we have... 

Suppose that we record a spectrum with the simple pulse-acquire sequence using a 90° pulse applied along the x axis. The resulting FID is Fourier transformed and the spectrum is phased to give an absorption mode lineshape. 

Fourier transformation with respect to q gives peaks with an absorption lineshape, but this time in theiq dimension an absorption mode signal at 12, in Fx is denoted Af K The time domain signal becomes, after Fourier transformation in each dimension... 

A data set from an experiment to which TPPI has been applied is simply amplitude modulated in tx and so can be processed according to the method described for cosine modulated data so as to obtain absorption mode lineshapes. As the spectrum is symmetrical about Fx = 0 it is usual to use a modified Fourier transform routine which saves effort and space by only calculating the positive frequency part of the spectrum. 

With amplitude modulation the cosine and sine components may be handled in two ways to achieve quadrature detection in fl. They may be acquired in subsequent scans by either incrementing the pulse or receiver phase and the data co-added in the computer memory or they may be acquired sequentially and stored separately. With the first approach direct Fourier transformation yields frequency discrimination in fl but no absorptive lineshapes whilst with the second approach additional processing steps are necessary to achieve both, frequency discrimination and absorptive lineshapes. 

What makes the neutral final state or complete screening condition approximate rather than exact is the time required for the system to respond to the core excitation (the various time scales are discussed by Gadzuk (1978)). In this work we are concerned only with the excitation energy, the threshold for absorption. Since the threshold corresponds to a long-time process (in the sense of a Fourier transform of a correlation function), the actual dynamics of complete screening will have negligible effect on the threshold energy but will affect the lineshape. 

If the off-diagonal matrix elements that describe the coherence between a ground state and an excited electronic state decay exponentially with time, the homogeneous absorption line should have a Lorentzian shape (Figs. 10.6 and 10.7 Eqs. (2.70) and (10.35)). More generally, as we discussed in Sect. 10.6, the spectral lineshape is the Fourier transform of the relaxation function ... 

Fig. 10.10 Absorption spectral lineshapes calculated as the Fourier transform of the Kubo relaxation function <)> (). (A) (indicated in arbitrary time units) is varied, while a is fixed at 1 reciprocal time unit (B) is fixed at 1 time unit and a is varied as indicated. To use the full Fourier transform (Eq. 10.70), < (r) is treated as an even function of time (Fig. 10.1 lA)...

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