Pake spectrum

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. 

Pake spectrum 26, 38, 44 Paraffin crystals 129 Parallel model 130 Partially ordered systems 27... 

In case of a deuterated sample (spin 1 case), the spectra are usually dominated by the quadrupolar interaction, that is, the coupling of the nuclear quadrupole moment with the electric field gradient of the C-2H bond. For deuterons in C-2H bonds this can lead to a splitting of about 250 kHz. As in the case of dipolar interaction, a Pake spectrum is obtained for a powder sample. The z-principal axis of the quadrupolar interaction is oriented along the bond axis which makes deuteron NMR particularly useful for studies of segmental orientations and molecular dynamics (reorientation) [1],... 

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... 

According to the discussion in Section 1.1.1.1, a broad Pake spectrum is observed for low mobile PDMS chain units, whereas a narrow line is recorded if the frequency of chain motions exceeds 10 kHz-1 MHz. The effect of hydrophilic and hydrophobic Aerosil on the chain motion is remarkably different. For PDMS filled with hydrophilic Aerosil, a broad H NMR line is observed over the whole temperature range studied. The motional narrowing is not complete, even at 433 K, nearly 300 K above the fg of PDMS. This means that mobility of PDMS chain units at the surface of hydrophilic Aerosil is hindered by adsorption interactions even at 433 K, although the strength of adsorption interactions decreases with increasing temperature. On the other hand, PDMS chains at the surface of hydrc hobic Aerosil are already desorbed at about 200 K, since the NMR line is completely narrowed at this temperature as shown in Fig. 12. 

Figure 12A illustrates a real situation. The NMR spectrum is very complex, so that a detailed analysis would be a rather imposing problem. The de-Paked spectrum is shown in Figure 12B. The quadrupolar splittings, relative intensities, and approximate linewidths were readily determined (M. Ranee, I. C. P. Smith, and H. Jarrell, unpublished results). With this information the source of the spectral complexity was shown to arise from the presence of more than one lipid species in the membrane. The quadrupolar splitting is sensitive to the effects of the lipid head groups. This observation was rather important in as much as similar observations had previously been attributed to other effects.

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. 

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... 

Note that, just like for the first-order expression in Equation 5.12 also the second-order expression in Equation 5.18 applies to field-swept spectra, and a different expression found in EPR textbooks (Pake and Estle 1973) applies to frequency-swept spectra. The effect of including a second-order contribution to the central hyperfine splitting is illustrated in Figure 5.7 on the spectrum of a not uncommon contaminant of metalloprotein preparations Cu(II) ion coordinated by nitrogens of tris-hydroxy-ethyl aminomethane or Tris buffer. 

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-... 

Fig. 9. Solid-echo 2H NMR spectra of GL-d5 (Tg = 189 K) and TOL-ds (Tg = 117K) for GL a collapse of the solid-state spectrum is observed upon heating for TOL only spectra near and below Tg are displayed, showing the Pake form without significant motional effects. (Adapted from Ref. 77.)...

Figure 2 Top The variation of the doublet splitting as a function of the angle 6 between a given C—D bond in a single crystal and the magnetic field. Bottom The Pake doublet. A powdered crystal will display this characteristic spectrum, which is the superposition of all the component doublets from C—D bonds, which are randomly oriented.

If the single crystal is now pulverized so that all C—D bond orientations are equally probable, the powder will display the well-known Pake doublet lineshape (Fig. 2, bottom), which is the sum of all the individual doublet spectra. The separation of the two sharp peaks, for which 0 = 90°, characterizes the width of the spectrum Av = y4 e qQlh). This quantity is known as the quadrupolar splitting. The functional form of the Pake doublet arises from a combination of two factors. First, the relative number of C—bonds oriented at a given angle with respect to the magnetic field varies as sin 6. Second, the transformation from angular to frequency terms of the function 3(cos 0 - l)/2 yields /i(v) = - 2v)- for < v < v J2 and /jfv) (v, -i- 2v)- /2... 

If there is some molecular motion with characteristic times on the order of ICT sec, the NMR spectrum will no longer have the Pake doublet lineshape discussed earlier. For example, in gel-phase bilayers a perdeuterated lipid acyl chain will have a broad, relatively featureless spectmm, as shown in Fig. 3. These spectra do not lend themselves to easy analysis The molecular motion in the membrane is not rapid enough to be axially symmetric (see the description of the fluid bilayer below) on the NMR time scale but is fast enough to influence the average value of the quadrupolar interaction and thus the splittings of the individual labels. 

In a powder spectrum, where all the orientations are present, the superimposition of two symmetric powder distributions reversed in sign forms the well known Pake doublet (Fig. 3.2.3) [6]. 

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