Pake line-shape

Fig. 2.18 The Pake line-shape for quadrupolar broadening. The infinite values at V = 3Cq/4 and the discontinuities at V = 3Cq/2 are removed in practice by other broadening effects.

A similar Pake line-shape would arise for a pair of like spins interacting by magnetic dipole coupling, as shown by equation (2.9), but in this case observed lines are usually broadened even in the absenee of motion because each spin in a solid does not interact with only one neighbouring spin but with a large number of surrounding spins. 

Solid state materials have been studied by nuclear magnetic resonance methods over 30 years. In 1953 Wilson and Pake ) carried out a line shape analysis of a partially crystalline polymer. They noted a spectrum consisting of superimposed broad and narrow lines which they ascribed to rigid crystalline and amorphous material respectively. More recently several books and large articles have reviewed the tremendous developments in this field, particularly including those of McBrierty and Douglas 2) and the Faraday Symposium (1978)3) —on which this introduction is largely based. 

In an isotropic sample, where the C—H bond directions are uniformely destributed in space, the resulting powder line shape is the famous Pake spectrum of the I = 1 spin system depicted for convenience in Fig. 1. 

Fig. la-c. Theoretical 2H NMR line shapes for axially symmetric FGT (r = 0) in rigid solids, cf. Equ. (1). a Line shapes for the two NMR transitions b 2H spectrum (Pake diagram) in absorption mode as obtained by Fourier transform methods c 2H spectrum in derivative mode as obtained by wide line methods... 

If the molecular tumbling rate is slow enough that larger electron-electron dipolar couplings are not motionally averaged, Fourier deconvolution can be used to analyze dipolar interactions in fluid solution.18 Distances in doubly spin-labelled rhodopsin were measured by Fourier deconvolution of CW line-shape changes in room temperature solution.78 The broadening function was modelled as the sum of Pake patterns from a distribution of distances. As a reference point for the distance measurements one label was attached at the cytoplasmic termination of transmembrane helix 1. The second label was attached near the cytoplasmic termination of transmembrane helix 7 or in the short helix 8. The distances and conformational flexibility in the dark state are... 

In the temperature range below room temperature, the fully relaxed and partially relaxed 2H NMR spectra [36] are shown in Fig. 41. Whereas at - 113 °C the fully relaxed spectrum only shows a rigid Pake doublet, at higher temperatures the line shapes can be considered as a superposition of spectra in the rigid and rapid exchange limit with a weighting dependent on temperature. The simulated spectra (Fig. 41) have been calculated by consid-... 

The basic principles are well established isolated pairs of nuclei in a rigid solid experience a mutual dipolar interaction directly proportional to the average of the inverse cube of the internuclear distance. For a powder sample of an H2 complex, a normal Pake doublet line shape results (Fig. 20.3C), which can be used to calculate internuclear H-H distances within 1%. The patterns are quite sensitive to anisotropic motion, that is hindered rotation or torsion of the side-bound H2 about the M-H2 axis. This will not affect the H2 dipolar splitting when this axis is parallel to the applied magnetic field. One pair of temperature-independent... 

Characteristic line shape functions can be observed in special cases as is shown in Fig. 1 by two examples. The so-called Pake doublet [11] is foimd for isolated spin pairs (see Fig. la). The spectrum shown in Fig. lb is observed if an NMR signal is mainly broadened by the chemical shift anisotropy. The analysis of the line shape function, /(v), allows the direct measurement of parameters like the internuclear distance, the chemical shift anisotropy, Aa, or the asymmetry parameter, i]csa> in these cases. 

Table 1. The total hydrogen content, n H), and the line shape parameters for the broad and the narrow components of the NMR line for eight plasma-deposited a-Si H samples prepared under different conditions as listed in column 1 (RF power density, substrate temperature, gas composition and deposition electrode). For the first four deposited on room-temperature substrates, the broad line is a Pake doublet due to SIH2. For the second four, deposited on heated substrates, the broad line is a Gaussian.

NMR spectra of deuterated immobile molecules or their fragments (eg, methyl CD -groups in hydrocarbons) represent Pake-type powder patterns [97,98]. The dominant features of these line shapes are two strong peaks separated by the splitting (3/4)(2(, and two shoulders separated... 

While cooling, when the limit xa = 1 /, (cf. Eq. 15) is reached, the central Lorentzian NMR line, which is characteristic of a liquid (t-,8, solid-state spectrum, in the case of 2H NMR the Pake spectrum. The breadth of the solid-state spectra makes it difficult to measure the corresponding (short) free induction decay (FID), so that it is necessary to use echo-techniques (cf. Section II.D.2). Figure 38 (left) shows solid-echo 2H NMR spectra of glycerol-. The crossover from a Lorentzian line to the Pake spectrum is observed some 20% above Tg. Below Tg the spectrum is independent of temperature. In Fig. 37 (right), the corresponding 31P NMR spectra of m-tricresyl phosphate (m-TCP) are displayed. The characteristic spectral shape is now determined by the anisotropic chemical... 

The line spfitting (shape) observed on the Pake-fike pattern, in the presence of molecular motions, depends on the geometry of motion (Fig. 2.20). This is due to the averaging of the quadrupolar parameters [126]. Molecular motion are classified with respect to the time scale of the quadrupolar cou-phng constant (Cq) [127] a number of models was proposed for describing the molecular motions one of the simplest is a cone model [128]. It must be stressed that variations in details of the geometry underlying the motional... 

Now we shall consider a semiquantitative theory of the shape of the broad component of the melt spectrum. Since over 70% of the protons are on para-substituted aromatic rings in PET/60PHB and PET/30PHB/30THQ, these protons will contribute most of the spectrum. As a first approximation, the neighboring protons on each ring can be considered to be an isolated proton pair. Pake has discussed the spectrum for an isolated pair of protons. Each pair contributes two lines centered at the same position and separated by... 

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