Zero-resolution points

Optimization procedures are used to determine which of these (1)a values will obtain the greatest resolution of the critical pair (36). Figure 5.17b shows a resolution window diagram where the resolution Rs of the critical pair is plotted against the volume fraction of phase A. The zero-resolution points are defined by the (t)A values that result in the crossing of the various plot pairs in Figure 5.17a. Between the zero-resolution points are windows of various amplitude, which... 

The subdiffraction resolution is given by Ar as defined in (19.1) [39,41]. Since it is a far-held approach, the resolution still scales with A, however, it is no longer limited by A, because Imax/Is — oo leads to Ar —> 0 [39,41,69]. The reason for the square-root law is that, in hrst approximation, the intensity /(r) increases quadratically when departing from the zero-intensity point r. ... 

Spherical aberration (coefficient Q) of the objective lens. Calculating the function at Scherzer defocus (optimal CTF, with fewest zeros) the point resolution of a particular TEM can be determined [36, 37]—the results of this procedure, e.g. the resolution limits for the used TEM instmments are stated in Table A.9. 

Another approach to obtain subdifiraction resolution can be classified under RESOLFT. This technique also relies on switching molecules between two states but in this case, the molecules are not switched on or off stochastically. Instead, the sample is illuminated with patterned laser excitation light. For instance, let us assume we have a sample labeled with dye molecules which can be switched between a fluorescent (on) and a nonfluorescent state (off). The sample can now be illuminated with a spatial intensity distribution with zero-intensity points or lines to selectively illuminate certain parts of the sample If this excitation light will convert the molecules to the off state, only the molecules that were not illuminated will still show fluorescence. Despite the theoretical diffraction limit, the size of the zero-intensity points or lines can be made infinitely small, by adjusting the intensity and the duration of the laser pulse. An example of such an intensity distribution is a donuf mode, which is characterized by a zero-intensity point in the center surrounded by high intensity. This means that when exciting with this donut-shaped laser light, all molecules except the one in the center will be switched to the off state. 

The factor limiting the resolution in ultraviolet photoelectron spectra is the inability to measure the kinetic energy of the photoelectrons with sufficient accuracy. The source of the problem points to a possible solution. If the photoelectrons could be produced with zero kinetic energy this cause of the loss of resolution would be largely removed. This is the basis of zero kinetic energy photoelectron (ZEKE-PE) spectroscopy. 

The effective resolution is determined by the number of data points in each domain, which in turn determines the length of t and <2- Thus, though digital resolution can be improved by zero-filling (Bartholdi and Ernst, 1973), the basic resolution, which determines the separation of close-lying multiplets and line widths of individual signals, vdll not be altered by zero-filling. 

In the case of the t domain, since it is only the number N of data points that determines the resolution, and not the time involved in the pulse sequence with various delays, it is advisable to acquire only half the theoretical number of FIDs and to obtain the required digital resolution by zero-filling. Thus the resolution in the Fi domain will be given by R = 2SWi/A i that in the F2 domain is given by / = 1/AQ = 2SW2/A2. 

Because the FFT algorithm requires the number of data points to be a power of 2, it follows that the signal in the time domain has to be extrapolated (e.g. by zero filling) or cut off to meet that requirement. This has consequences for the resolution in the frequency domain as this virtually expands or shortens the measurement time. 

It is still possible to enhance the resolution also when the point-spread function is unknown. For instance, the resolution is improved by subtracting the second-derivative g x) from the measured signal g x). Thus the signal is restored by ag x) - (7 - a)g Xx) with 0 < a < 1. This llgorithm is called pseudo-deconvolution. Because the second-derivative of any bell-shaped peak is negative between the two inflection points (second-derivative is zero) and positive elsewhere, the subtraction makes the top higher and narrows the wings, which results in a better resolution (see Fig. 40.30). Pseudo-deconvolution methods can correct for sym-... 

Second, the resolution achieved in a 2-D experiment, particularly in the carbon domain is nowhere near as good as that in a 1-D spectrum. You might remember that we recommended a typical data matrix size of 2 k (proton) x 256 (carbon). There are two persuasive reasons for limiting the size of the data matrix you acquire - the time taken to acquire it and the shear size of the thing when you have acquired it This data is generally artificially enhanced by linear prediction and zero-filling, but even so, this is at best equivalent to 2 k in the carbon domain. This is in stark contrast to the 32 or even 64 k of data points that a 1-D 13C would typically be acquired into. For this reason, it is quite possible to encounter molecules with carbons that have very close chemical shifts which do not resolve in the 2-D spectra but will resolve in the 1-D spectrum. So the 1-D experiment still has its place. 

Scrutiny of the resolution equation indicates that is controlled by three relatively independent terms retention, selectivity, and efficiency (Figure 10). To maximize R, k should be relatively large. However, a value of k over 10 will approach a point of diminishing returns as the retention term of k /(l + k ) approaches unity. No separation is possible if k = 0, since R must equal zero if k is zero in the resolution equation. 

The first zero transition of the CTF at this defocus is called the point resolution, d, of the TEM and can be shown to be... 

Zero filling the FID more than a factor of two does not contribute to information extraction and any features revealed by this are artefacts. In most instances, zero filling by a factor of two amounts to an interpolation procedure benefiting primarily peak-picking. There are other procedures which can allow peak-picking interpolation between data points and the one used by the author is a simple equation to fit the maximum intensity and one point either side to a parabola and compute the position of its maximum. Bruker peak-pick table positions for instance are not separated by a multiple of the digital resolution and it would seem that they use the same or an analogous procedure. 

As described in Section 10.2, the final output from the NMR spectrometer to the computer is an FID. Typically 2048 096 digital points are accumulated in the FID and the next step is to improve the potential resolution of the FID by zero-filling the FID to 16384 digital points by adding zeroes to the end of the FID. Upon Fourier transformation (FT) the resultant spectrum contains 16384 points describing a spectral width of 2000-4000 Hz depending on the settings in the ADC. 

The question now is what happens when this calculation is performed for more components, i.e. when at each design point the minimum resolution over all solutes is taken. This is shown in Table 6.3. The first consequence is that a large amount of information is lost, since it is not detectable which solutes lack resolution. A second consequence is that sometimes the minimum resolution is zero. This means that this setting results in insufficient separations, but also that the inverse (l/y, ) cannot be calculated. Therefore the signal-to-noise ratios are calculated on the nonzero values. 

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