Instrumental lineshape

Suppose that a pulse Fourier transform proton NMR experiment is carried out on a sample containing acetone and ethanol. If the instrument is correctly operated and the Bq field perfectly uniform, then the result will he a spectrum in which each of the lines has a Lorentzian shape, with a width given hy the natural limit 1/(7tT2). Unfortunately such a result is an unattainable ideal the most that any experimenter can hope for is to shim the field sufficiently well that the sample experiences only a narrow distribution of Bq fields. The effect of the Bq inhomogeneity is to superimpose an instrumental lineshape on the natural lineshapes of the different resonances the true spectrum is convoluted by the instrumental lineshape. 

In principle we could deconvolute the experimental spectrum with the instrumental lineshape, if that were known, to recover the true spectrum. In our example we have some good experimental evidence as to the form of the instrumental lineshape since the acetone signal is (apart from small carbon-13 satellites) a singlet, its experimental shape is just the instrumental lineshape convoluted by a Lorentzian of width l/(7rr2 ), where is the spin-spin relaxation time of the acetone protons. How can we use this experimental evidence to correct the imperfect experimental spectrum The simplest way to deconvolute one function fi uj) by another f2 ( ) is to Fourier transform the ratio of their inverse Fourier transforms ... 

The corrected free induction decay Sc t) will transform to a spectrum Sc i ) in which not only the acetone signal but also all the ethanol signals have had the instrumental contributions to their lineshapes removed. Provided that the reference region lui to wr gives a complete and accurate representation of the experimental acetone lineshape, our deconvolution process should allow us to obtain a clean corrected spectrum even when the shimming is far from ideal. There are of course limitations on this process. If the experimental lineshape is very broad, it will clearly not be possible to obtain a corrected spectrum in which the lines are very narrow without some sort of penalty. Here the limiting factor is signal-to-noise ratio since S u>) is much sharper than Se u>), the ratio of their inverse Fourier... 

Peak widths. The lineshapes and linewidths of peaks in a powder XRD pattern depend on the crystallinity of the sample, as well as features of the instrumentation and the data collection procedure. In particular, peaks in the powder XRD pattern may be broadened as a consequence of small crystallite size. If the powder XRD patterns of two samples with the same crystal structure have significantly different linewidths, the visual appearance may differ substantially, especially in regions of significant peak overlap. 

The design of EPR spectrometers resembles that of a field-sweep NMR instrument (Section 3.3.2), though pulsed-mode (Fourier transform Section 3.4) EPR spectrometers are now available. Many of the considerations (such as field stability, lineshape, saturation, relaxation, etc.) that were discussed in Chapters 2 and 3 for NMR are also important in EPR,1 but there are some significant differences. 

Fig. 10.17 (a) INS spectrum of /raw-polyacetylene, (b) calculated density-of-states convoluted with a Gaussian lineshape and the instrument resolution function and (c) as (b) including the effects of the Debye-Waller factor and phonon wings. Reproduced from [29] with permission of Elsevier. 

The proton lineshape test uses chloroform in deuteroacetone typically at concentrations of 3% at or below 400 MHz, and 1% at or above 500 MHz. Older instruments and/or probes of lower sensitivity or observations via outer decoupler coils , may require 10% at 200 MHz and 3% at 500 MHz to prevent noise interfering with measurements close to the baseline. A single scan is collected and the data recorded under conditions of high digital resolution (acquisition time of 16 s ensuring the FID has decayed to zero) and processed without window functions. Don t be tempted to make measurements at the height of the satellites themselves unless these are confirmed by measurement to be 0.55%. Since these arise from protons bound to C, which relax faster than those of the parent line, they may be relatively enhanced should full equilibrium not be established after previous pulses. The test results for a 400 MHz instrument is shown in Fig. 3.66. The traditional test for proton resolution which dates back to the CW era (o-dichlorobenzene in deuteroacetone) is becoming less used nowadays, certainly by instrument manufacturers, and seems destined to pass into NMR history. 

It turns out that for instrumental reasons the axis along which the signal appears cannot be predicted, so in any practical situation there is an unknown phase shift. In general, this leads to a situation in which the real part of the spectrum (which is normally the part we display) does not show a pure absorption lineshape. This is undesirable as for the best resolution we require an absorption mode lineshape. 

It is clear from the form of S(t) that this phase introduced by altering the experiment (in this case, by altering the phase of the pulse) takes exactly the same form as the instrumental phase error. It can, therefore, be corrected by applying a phase correction so as to return the real part of the spectrum to the absorption mode lineshape. In this case the phase correction would be k 2. 

Figure 8.8 Relationship between full width at half maximum (FWHM) of individual lines and band contours, (a) Single lines with FWHM consistent with band contour shown below, (b) Band contours are constructed by convolution of individual lines (each with the lineshape shown above) with an instrumental resolution of 0.01 cm-1. The spectrum is calculated for a 2n [case (b)] <— X2] [case (a)] transition of the NO molecule observed in absorption at 78 K. [From Giusti-Suzor and Jungen (1984).]...

A is a proportionality constant to include instrumental factors. The lineshape function f((o) has a maximum at co=coq and it decreases for high power levels (i.e. for large 5j) and when the spin-lattice relaxation is not fast enough to maintain the population difference. This decrease is called saturation. If the saturation factor s is defined by... 

Contrast the above situation with that of wide line FT NMR. First, we need the instrumental capabilities mentioned earlier. Secondly, often we are trying to obtain the shape of a single line or those of a small number of lines which may be complex. Therefore, we cannot presume to know the line shape in order to phase it properly. Similarly, an exponential time window function may alter the lineshape information. Third, the delay time used to avoid the pulse breakthrough in the FID is almost certain to be a significant fraction of the total acquisition time and must be taken into account. Let us deal with each of these difficulties in order. 

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