Phase corrections

4 Phase correction. Phase-sensitive acquisition of the FID with quadrature detection means that the pure absorption lineshape can be formed after FT but it may require some manipulation of the data to achieve this. If the real and the imaginary parts of the signal are Ar and A], the actual absorption spectrum A(w) can be formed as 

A phase difference between the carrier frequency and the pulse leads to a phase shift which is almost the same for all resonance frequencies (u)). This effect is compensated for by the so-called zero-order phase correction, which produces a linear combination of the real and imaginary parts in the above equation with p = po- The finite length of the excitation pulse and the unavoidable delay before the start of the acquisition (dead time delay) leads to a phase error varying linearly with frequency. This effect can be compensated for by the frequency-dependent, first-order phase correction p = Po + Pi((o - (Oo), where the factor p is frequency dependent. Electronic filters may also lead to phase errors which are also almost linearly frequency-dependent. 

For several technical reasons, it is not possible to acquire NMR data with perfect phase. One reason is the inability to detect XY magnetisation correctly another is the fact that we are unable to collect the data as soon as the spins are excited. These limitations mean that we have to phase correct our spectrum so that we end up with a pure absorption spectrum. What we don t want is a dispersion signal (see Spectrum 4.2). 

Spectrum 4.2 An absorption signal (below) and dispersion signal (above). 

The XY problem gives rise to a constant phase error across the spectrum, the delay problem gives a linear phase error. To correct for this, we have two phase adjustment parameters at our disposal zero and first order. 

Modem NMR software comes with very good automatic phase routines so most of the time you should end up with a beautifully phased spectrum. Sometimes, however, the software doesn t quite perform and you may need to tweak the phase manually. It can take a bit of familiarity to get this right but it is just a matter of practise. If you remember that the zero order adjustment works constantly across the spectrum and that the first order doesn t, it is quite easy to see what is going on. Normally the software gives you an option of setting the pivot point of the first order adjustment (i.e., the frequency in the spectrum where there is no effect from the first order adjustment). This pivot point is normally set to the largest peak. 

Spectrum 4.3 shows how the phase can be improved with a manual tweak. Note that in a poorly phased spectrum, the integrals will be distorted such that they are pretty much unusable. 

The real and imaginary spectra obtained by Fourier transformation of FID signals are usually mixtures of the absorption and dispersion modes as shown in Fig. 2.13 (a). These phase errors mainly arise from frequency-independent maladjustments of the phase sensitive detector and from frequency-dependent factors such as the finite length of rf pulses, delays in the start of data acquisition, and phase shifts induced by filtering frequencies outside the spectral width A. 

One method of phase correction assumes a linear dependence of the phase (p on the frequency, as in eq. (2.15) pA is the phase at frequency difference zero, cpB is the phase shift across the total spectral width from zero to zl Hz. 

Typically, both forms of error occur in a spectrum directly after the FT. The procedure for phase correction is essentially the same on all spectrometers. The zero-order correction is used to adjust the phase of one signal in the spectrum to pure absorption mode, as judged by eye , and the first-order correction is then used to adjust the phase of a signal far away from the first in a similar manner. Ideally, the two chosen resonances should be as far apart in the spectrum as possible to maximise the frequency-dependent effect. Experimentally, this process of phase correction involves mixing of the real and imaginary parts of the spectra produced by the FT process such that the final displayed real spectrum is in pure absorption mode whereas the usually unseen imaginary spectrum is pure dispersion. 

Appropriate transformation of interferograms to produce accurate spectra is not restricted to Fourier transform algorithms, and one of the additional procedures, phase correction, is presented in the following section. 

The sources of phase error have been presented in Section 2.5, where it was stated that the phase angle is usually a function of wavenumber. It should also be noted that phase is a complex quantity that is, it has real and imaginary parts. 

When a recorded interferogram S(8) is transformed to produce a spectrum a complex Fourier transform must be used unless the interferogram is symmetric. Hence, from Eq. 4.10, 

The magnitude spectrum exhibits zero phase error but has noise nonlinearities. The magnitude (Eq. 4.32) and complex spectra (Eq. 4.31) are related by the phase angle 0  

The complex spectrum B ( contains all the spectral information, but it is dispersed into two complex planes by the phase. The magnitude spectrum B(v) is a real (in the complex sense) representation of the spectrum, but it is only the absolute value of that representation. The true spectrum B(v), which is also real, lacks the noise nonlinearities of the magnitude spectrum. The object of phase correction is to produce the true spectrum B. Since 0 usually varies slowly with wavenumber, it is possible to factor e from Eq. 4.33. In this case. 

It has already been mentioned in Section 3.2 that the phase of a spectrum needs correcting following Fourier transformation because the receiver reference phase does not exactly match the initial phase of the magnetisation vectors. This error is constant for all vectors and since it is independent of resonance frequencies it is referred to as the zero-order phase correction (Fig. 3.38). Practical limitations also impose the need for a frequency-dependent or first-order phase correction. Consider events immediately after the 

Typically both forms of error occur in a spectrum directly after the FT. The procedure for phase correction is essentially the same on all spectrometers. The zero-order correction is used to adjust the phase of one signal in the spectrum to pure absorption mode, as judged by eye and the first-order correction is then 

In practice, the interferogram measured is not mirror symmetrical about the point d = 0. Call up the interferogram of the file MIR GLYCIN or ACQUIS in order to verify that. This asymmetry originates from experimental errors, e.g., wavenumber-dependent phase delays of the optics, the detector/ amplifier unit, or the electronic filters. The Fourier transformation of such an asymmetrical interferogram generally yields a complex spectrum C(v) rather than a real spectrum S(r) as known from spectrometers based on the dispersive technique. That is why phase correction is necessary. 

Although it makes no difference where the carrier is located, you should be aware of which edge of the spectrum is defined by the carrier so that you can take the appropriate action for lines which are folded over or aliased in the absence of QD. Be sure to label your spectra, especially for publication, as to where the carrier is and even as to which direction the shielding increases. 

In FT NMR, it is not trivial to set the detector phase so that it will be set to the absorption mode for all lines in the spectrum. In addition to a phase offset error due to improper setting of the detector for the absorption mode, there will be frequency dependent phase changes across the spectrum. Lines at the start of the spectrum might appear absorption-like 

The receiver system may also introduce phase shifts. The most basic cause for frequency dependent phase shifts, though, is that a finite rotates magnetizations at different offsets by different amounts (see II.A.2.). 

All of these phase shifts are approximately linear in frequency for moderate frequency deviation. For example, it was pointed out in II.A.2. that the phase shift due to finite Hj is quite linear for offsets up to vH. Therefore, a linear phase correction of the form A J =(Au)/27i)+B, implemented by software, is quite effective. 

In this connection, we point out that determining the required phase correction parameters at two points may not be sufficient. For example, if there are only two peaks and they seem to be phased correctly, it could be a coincidence with a 360 degree phase shift between them. In fact, this situation has been deliberately created for a two-line spectrum by adjusting the time origin suitably to make the two lines have the same phase and thus allow an accurate determination of their separation (Canet, et al., 1976). Separations measured in this way will be correct as long as the line shapes are symmetric. The shapes and widths themselves will be distorted in such a scheme, however, since there is a dispersion of phase shifts within each peak. 

FID will often show a number of points with an anomalously large amplitude, and may also contain large jumps or discontinuities between adjacent points. 

If we find that the ringdown delay required to produce an artifact-free spectrum yields an unacceptably low signal-to-noise ratio (this is a subjective decision), then we can use backwards linear prediction to correct the first few corrupted points in the FID. Linear prediction is generally accepted in the NMR community as a reasonable method for avoiding choosing between a spectrum with a low signal-to-noise ratio and a spectrum marred by ringdown artifacts. 

A spectrum can only be phased when the data is collected in a phase-sensitive receiver detection mode. For a spectrum to be phase sensitive, we require either two orthogonal receiver channels digitizing 

Phase correction. The balancing of the relative emphasis of two orthogonal data arrays (or matrices for a 2-D spectrum) to generate a frequency spectrum with peaks that have a fully absorptive (or, in some cases, fully dispersive) phase character. 

Anisochronous.Two spins are aniso-chronous when they undergo transitions between their allowed spin states at different frequencies. That is, two spins are anisochronous if their resonances appear at different points in the frequency spectrum. 

In the case of NMR spectra, as the natural abundance of is only 1%, the chances of finding two nuclei bound to one another are 0.01 X 0.01 = 10 i.e., very low. Therefore couplings are not normally observed. Extensive couplings are, however observed. The 

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While vapor-phase corrections may be small for nonpolar molecules at low pressure, such corrections are usually not negligible for mixtures containing polar molecules. Vapor-phase corrections are extremely important for mixtures containing one or more carboxylic acids. 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

This chapter presents quantitative methods for calculation of enthalpies of vapor-phase and liquid-phase mixtures. These methods rely primarily on pure-component data, in particular ideal-vapor heat capacities and vapor-pressure data, both as functions of temperature. Vapor-phase corrections for nonideality are usually relatively small. Liquid-phase excess enthalpies are also usually not important. As indicated in Chapter 4, for mixtures containing noncondensable components, we restrict attention to liquid solutions which are dilute with respect to all noncondensable components. 

At low pressures, it is often permissible to neglect nonidealities of the vapor phase. If these nonidealities are not negligible, they can have the effect of introducing a nonrandom trend into the plotted residuals similar to that introduced by systematic error. Experience here has shown that application of vapor-phase corrections for nonidealities gives a better representation of the data by the model, oven when these corrections... 

These were converted from vapor pressure P to fugacity using the vapor-phase corrections (for pure components), discussed in Chapter 3 then the Poynting correction was applied to adjust to zero pressure ... 

T were eliminated beyond that point, the vapor-phase correction, as calculated here, is inadequate and the liquid molar volume is no longer constant with pressure. 

I l = Stator DC resistance per phase corrected to operating temperature. r = Rotor resistance per phase at rated speed and operating temperature referred to stator. 

It gives the crossover between diffraction-limited and turbulence-limited resolution. For aperture diameters smaller than ro, close to diffraction limited imaging is possible without phase correction, for aperture diameters larger than ro, the resolution is limited by the turbulence. For a circular aperture of diameter D, the phase variance over the aperture is... 

In order to compensate for the distortions in the wavefront due to the atmosphere we must introduce a phase correction device into the optical beam. These phase correction devices operate by producing an optical path difference in the beam by varying either the refractive index of the phase corrector (refractive devices) or by introducing a variable geometrical path difference (reflective devices, i.e. deformable mirrors). Almost all AO systems use deformable mirrors, although there has been considerable research about liquid crystal devices in which the refractive index is electrically controlled. 

The Bohr quantization condition, with quantum number v, corresponds to choosing the lowest Riemann sheet, which requires an additional phase correction of —2ti on crossing the branch cut, which is taken along the positive -axis. Thus the phase term cj) rises to ti at = 0+ and > 0, drops abruptly to —71 on crossing the positive fe = 0 axis, and returns to zero on the negative... 

Figure 6. The Bohr-Sommerfeld phase corrections t)(8, ) for k = 0, 1, and 2. The ratio r z,k)lK estimates of the error of primitive Bohr-Sommerfeld eigenvalues as a fraction of their local vibrational spacing.

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

Fourier transformation in (Fti), spectra are obtained with real (R) and imaginary (/) data points. For detection in the quadrature mode with simultaneous sampling, a complex Fourier transformation is performed, with a phase correction being applied in F. (c) A normal phase-sensitive transform P— RR and I- RI. (d) Complex FT is applied to pairs of columns, which produces four quadrants, of which only the RR quadrant is plotted. 

The next step after apodization of the t time-domain data is Fourier transformation and phase correction. As a result of the Fourier transformations of the t2 time domain, a number of different spectra are generated. Each spectrum corresponds to the behavior of the nuclear spins during the corresponding evolution period, with one spectrum resulting from each t value. A set of spectra is thus obtained, with the rows of the matrix now containing Areal and A imaginary data points. These real and imagi-... 

If phase-sensitive spectra are not required, then magnitude-mode Pico) (or absolute-mode ) spectra may be recorded by combining the real and imaginary data points. These produce only positive signals and do not require phase correction. Since this procedure gives the best signal-to-noise ratio, it has found wide use. In heteronuclear experiments, in which the dynamic range tends to be low, the power-mode spectrum maybe preferred, since the S/N ratio is squared and a better line shape is obtained so that wider window functions can be applied. 

Phasing A process of phase correction that is carried out by a linear combination of the real and imaginary sections of a 1D spectrum to produce signals with pure absorption-mode peak shapes. 

Fig. 5.5.5 1 D CSI datasets showing the extent of conversion during a batch reaction. The form of the feature identified as peak B is associated with a single chemical shift i.e., it is of constant form at all positions across the bed, and therefore shows that the extent of conversion is uniform throughout the bed. The low intensity horizontal streaking" effect observed in these datasets and that shown in Figure 5.5.6 are artifacts arising from the automatic phase correction applied to the data ...

The two-phase correction factor fc is obtained from Figure 12.56 in which the term l/Xtt is the Lockhart-Martinelli two-phase flow parameter with turbulent flow in both phases (See Volume 1, Chapter 5). This parameter is given by ... 

Take a moment to survey the spectrum and ask yourself if it is fit for purpose Of course, if you have run it yourself, then it should be fine but this may not always be so with walk-up systems. Is the line shape and resolution up to standard Has the spectrum been phased correctly Is the vertical scale well adjusted so that you can see the tops of all the peaks (except perhaps, obvious... 

By taking into account the scaling factor in the pulse sequence, the linear BSPS can be effectively removed. Additionally, this linear BSPS can also be corrected by a first-order phase correction. 

Reference Deconvolution, Phase Correction and Line Listing of NMR Spectra by the ID Filter Diagonalization Method. 

Two coordination shells were found for Co in SAN, one at 2.33 and the other at 3.50 A (phase corrected). These two peaks correspond to Co-Si direct bonds (first coordination shell) and Co-Co scattering (second coordination shell). Only one major... 

Z neighbors. Phase correction of the Fourier transform by the backscattering phase shift of one of the absorber—neighbor pairs is also extensively used. This has the effect of correcting the distances observed in the radial structure function as well as emphasizing the contributions from the chosen ab-... 

Figure 6. (a) weighted EXAFS of Pt foil collected at the Pt Ls edge and (b) the corresponding weighted Pt phase corrected Fourier transform of the EXAFS data. 

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