Fj frequency domain

During the detection period denoted t2 (not the relaxation time T2 ) the NMR signal is captured electronically and stored in a computer for subsequent workup. Although detection occurs after evolution, the first Fourier transformation is applied to the time domain data detected during the t2 detection period to generate the f2 frequency axis. That is, the t2 time domain is converted using the Fourier transformation into the f2 frequency domain before the tj time domain is converted to the fj frequency domain. This ordering may seem counterintuitive, but recall that q and t2 get their names from the order in which they occur in the pulse sequence, and not from the order in which the data set is processed. 

Interferogram. A 2-D data matrix that has only undergone Fourier transformation along one axis to convert the time domain to the fj frequency domain. An interferogram will therefore show the frequency domain on one axis and the t, time domain on the other axis. 

Fi axis, fi axis. Syn. fj frequency axis. The reference scale applied to the fj frequency domain. The fi axis may be labeled with either ppm or Elz. 

An algorithm developed by Cooley and Tukey simplified this extremely time-consuming calculation, bringing it within the capability of modern microcomputers. Today, the transformation takes only a few seconds, after which the frequency-domain spectrum Fj can be plotted. (The frequency-domain spectrum corresponding to Figure 3.16 will be discussed in Chapter 5.)... 

The term two-dimensional (2D) NMR spectrum refers to a data set where signal intensity is a function of two frequency domains (Fj and F2). The corresponding FID data are collected as a function of two time domains (detection and evolution/mixing) and then Fourier transformed in each dimension. The resulting data are most commonly displayed in a contour format. 

The term 2D NMR, which stands for two-dimensional NMR, is something of a misnomer. All the NMR spectra we have discussed so far in this book are two dimensional in the sense that they are plots of signal intensity versus frequency (or its Fourier equivalent, signal intensity versus time). By contrast, 2D NMR refers to spectroscopic data that are collected as a function of two time scales, t (evolution and mixing) and 2 (detection). The resulting data set is then subjected to separate Fourier transformations of each time domain to give a frequency-domain 2D NMR spectrum of signal intensity versus two frequencies, F, (the Fourier transform of the r, time domain) and Fj (the Fourier transform of the F time domain). Thus, a 2D NMR spectrum is actually a three-dimensional data set ... 

Following conversion of t2 to f2, we have a half-processed NMR data matrix called an interferogram. The interferogram is not a particularly useful thing in and of itself, but performing a Fourier transformation to convert the q time domain to the q frequency domain renders a data matrix with two frequency axes (fj and f2) that will (hopefully) allow the extraction of meaningful data pertaining to our sample. 

A 2D J experiment, or any 2D NMR experiment for that matter, consists of a series of ID experiments in which the duration of the evolution time, ti, is systematically incremented in some fashion from one experiment to the next. In the specific case of a 2D J experiment, the incremented parameter is the dwell time, which corresponds to 1/swl, where swl is the desired spectral width of the second frequency domain, Fj in Hz. Typical one-bond heteronuclear couplings range from about 125 to 160 Hz for aliphatic to aromatic compounds, respectively, with some heteroaromatics having one-bond couplings ranging up to about 210 Hz. In most cases, the spectral width in the second frequency domain of a 2D J experiment can be set to a total of 100 Hz, keeping in mind that couplings will be scaled by J/2 since J-modulation occurs for only half of the evolution time. 

Fj = COj = y-axis frequency domain obtained by Fourier transformation with respect to tj F2 = CO2 = axis frquency domain obtained by Fourier transformation with respect to 2 h = evolution time in a 2D pulse sequence = detection period in a 2D pulse sequence Tj = spin-lattice (longitudinal) relaxation time T2 = spin-spin (transverse) relaxation time. 

If our function of time and the corresponding values in the frequency domain are continuous, we will write them as h(t) and H(f). If either is available only at discrete times or frequencies, we will write them as hj (the value at time tf and Hj (the value at frequency) ). In some places it is helpful to emphasize the time and frequency dependence by writing these as hftj) and Hj(fj). 

Therefore, in order to minimize the overlap between G+( ) and Fj+ oo) for general gapped baths, and thereby the transfer infidelity (4.198), we will design a narrow bandpass filter centered on the gap. Since G ( ) has a narrower gap than G+( ), we optimize the filter Fj o)) under the variational E-L method. We seek a narrow bandpass filter, whose form on time domain via Fourier transform decays as slowly as possible, so as to filter out the higher frequencies. This amounts to maximizing Fj- (t) = ( )e (f subject to the variational... 

The constant of proportion between the time dependent phase, 0(fj), and td has been written tuadd madd has the dimensions of rad s 1 i.e. it is a frequency. Following the same approach as before, the time-domain function with the inclusion of this incrementing phase is thus... 

FIGURE 4.3.11 (a) EFM measurement with the tip biased at the surface potential (F = Fj) the cantilever resonant frequency is (b) EFM measurement above the sample surface with Vgp Vg, the cantilever frequency shift A/, j is due to the tip-substrate capacitive force gradients, (c) Additional capacitive frequency shift A/g when the EFM tip passes over a conducting domain of dielectric constant e (A/g = 0 if = F ). (d) Additional frequency shift A/g when an amount of charge Q is located inside the domain. (From Melin, T. et al., Phys. Rev. B 69, 35321, 2004. With kind permission.)... 

Two-dimensional NMR spectroscopy may be defined as a spectral method in which the data are collected in two different time domains acquisition of the FID (<2), and a successively incremented delay (/i). The resulting FID (data matrix) is accordingly subjected to two successive sets of Fourier transformations to furnish a two-dimensional NMR spectrum in the two frequency axes. The time sequence of a typical 2D NMR experiment is given in Fig. 3.1. The major difference between one- and two-dimensional NMR methods is therefore the insertion of an evolution time, fj, that is systematically incremented within a sequence of pulse cycles. Many experiments are generally performed with variable which is incremented by a constant Af]. The resulting signals (FIDs) from this experiment depend... 

SW, which in turn is related to the homonuclear or heteronuclear coupling constants. In homonuclear 2D spectra, the transmitter offset frequency is kept at the center of (i.e., at Fj = 0) and domains. In heteronuclear-shift-correlated spectra, the decoupler offset frequency is kept at the center (F = 0) of the F, domain, with the F domain corresponding to the invisible or decoupled nucleus. 

When we collect a 2-D NMR spectrum, both the second frequency dimension data (fj or Fj) and the first frequency dimension data (f2 or F2) may be phase sensitive. (Note that fj and f2 appear to be reversed but this naming convention derives from the order of their time domain precedents, tj and t2, in the NMR pulse sequence.) Zero-and first-order phasing of the second dimension of a 2-D NMR data set is required in many cases. Some experiments, most notably the gradient-selected heteronuclear multiple bond correlation (gHMBC) experiment, use the absolute value of the signal and hence do not require phasing. 

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