Lorentzian-shaped pulses

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. 

As mentioned above, a drawback with the etalon method is that the pulses have Lorentzian-shaped frequency profiles. A better profile would be Gaussian, since Gaussians tail to zero much more quickly than Lorentzians with the same full width at half-maximum. With our shaper, we also collected... 

The shape of NMR lines in a homogeneous magnet is dictated by the decay of the transverse magnetization, which is detected as the free induction decay (FID) following a radiofrequency pulse. If the observed nucleus remains in the same environment (associated to the same characteristic resonance frequency), the NMR lines will have a Lorentzian shape with a width at half height given by the effective transverse relaxation time T2). The faster the transverse relaxation, the broader will be the line (1) ... 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

There is a second relaxation process, called spin-spin (or transverse) relaxation, at a rate controlled by the spin-spin relaxation time T2. It governs the evolution of the xy magnetisation toward its equilibrium value, which is zero. In the fluid state with fast motion and extreme narrowing 7) and T2 are equal in the solid state with slow motion and full line broadening T2 becomes much shorter than 7). The so-called 180° pulse which inverts the spin population present immediately prior to the pulse is important for the accurate determination of T and the true T2 value. The spin-spin relaxation time calculated from the experimental line widths is called T2 the ideal NMR line shape is Lorentzian and its FWHH is controlled by T2. Unlike chemical shifts and spin-spin coupling constants, relaxation times are not directly related to molecular structure, but depend on molecular mobility. 

The free induction decay following 90° pulse has a line shape which generally follows the Weibull functions (Eq. (22)). In the homogeneous sample the FID is described by a single Weibull function, usually exponential (Lorentzian) (p = 1) or Gaussian (p = 2). The FID of heterogeneous systems, such as highly viscous and crosslinked polydimethylsiloxanes (PDMS) 84), hardened unsaturated polyesters 8S), and compatible crosslinked epoxy-rubber systems 52) are actually a sum of three... 

In an attempt to investigate the phase structure of this sample, the line shape analysis of the CH2 resonance line in the DD/MAS spectrum at 87 °C that is shown in Fig. 25 was examined. The result is shown in Fig. 26-(a). The elementary line shape of the crystalline phase was obtained as the line shape of the longest Tic component by Torchia s pulse sequence [53]. It was a doublet and was represented approximately by two down- and upheld Lorentzians with an intensity ration of 2 1 (Spectrum A shown by dotted line in Fig. 26). Since all methylene carbons in the a-crystalline form of this polymer are equivalent in the intramolecular helical conformation, the origin of the doublet could be attrib-... 

Spin-Lattice Relaxation. In order to determine whether each resonance line comprises a single component, we first measured the spin-lattice relaxation time Tic by the pulse sequence developed by Torchia [53] or by the standard saturation-recovery pulse sequence. The Tic values thus obtained were 2560,263 and 1.7 s for resonance line I and 0.37 s for line II. As reported by several investigators, the line at 33 ppm is associated with three different Tic values [ 17,54,55]. This means that this line is contributed to by three components with different molecular mobilities. However, since each component was represented by a single Lorentzian line shape at 33 ppm, they are all assignable to methylene groups in the orthorhombic crystalline form or in the trans-trans conformation. The component with a Tic of s can be assigned to methylene groups with a some-... 

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. 

The conducting phase of TMQ /16/. Microwave conductivity experiments, performed at low temperature on the samples used for the pressure experiment, have succeeded in showing an increase by a factor 10 from 300 K down to loo K, /41/. The possibility of susceptibility measurement via low field method is limited by the EPR line broadening occurring at low temperature and under pressure. The spin susceptibility was derived from a fit of the EPR absorption line shape with a Lorentzian curve. The proton relaxation time was measured under pressure at low field lOe- with a pulse spectrometer. Figures... 

The middle trace in Fig. 8 is the line shape observed using the spin echo technique at a pulse separation of t = 0.25 msec (180°-t-90° pulse sequence). Reimer and Duncan (1983) have determined that the FID and echo line shapes of Fig. 8 are different, and the difference spectrum is plotted at the bottom of Fig. 8. Furthermore the T2 as measured by the echo technique is very different for these two components. The spectrum with the longer T2 (—1.4 msec) in the middle of Fig. 8 is best fit with a Lorentzian line shape centered at 175 ppm with respect to a standard reference ( P in 85%H3P03). The broad line at the bottom of Fig. 8 is best fit with a Gaussian line shape centered at 70 ppm. 

To set up the lifetime experiment, fluorescence excitation spectra were recorded using pulses of 90 ns duration corresponding to a spectral bandwidth of 15 MHz at a repetition rate of 1 MHz. The bandwidth was limited by the pulses rise and fall times, the pulse shapes and the frequency jitter of the laser. Fig. 8(a) shows a typical spectrum with four individual molecular resonances A, B, C and D. On average, the lines are about 25 MHz wide as compared to a homogeneous linewidth of about 8 MHz measured by earlier experiments using cw radiation and lower excitation energies. The best fit of the absorption profile of molecule C was obtained using a Lorentzian profile with a FWHM of 27 MHz. 

Spin-lattice relaxation times (Tx) were obtained by the saturation-recovery method. Transverse relaxation times (T2) for the narrow central component of the line shapes were measured directly from the Lorentzian line shapes. The line shapes are Fourier transforms (FT) of quadrupole echo (QE) or free induction decay (FID) transients. Complete line shapes and long Tx measurements were obtained with a composite pulse QE sequence with a 3.5 psec tt/2 pulse length. Long Tx and weak signal strength made data accumulation tedious A single Tx measurement or spectral line shape often required over 12 hours of data acquisition. 

This function is termed a Lorentzian line shape and is the NMR spectral result for a single nuclear type. For actual chemical systems, a number of different nuclear types will be present, and the FID are quite complicated [6]. The Fourier transform process separates the different contributions and produces the proper spectrum in the frequency domain [7]. This is the pulsed NMR Fourier transform experiment. 

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