Pake-doublet

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-... 

This lineshape is often referred to as a Pake doublet or Pake pattern59 and is shown in Fig. 8. 

A special case occurs if the solid contains nuclear spin pairs that are sufficiently far apart from each other such that intrapair coupling dominates interpair coupling. In this situation, instead of a featureless broad lineshape, we observe a well-defined powder pattern known as a Pake doublet (Figure 15.15). Since the dipolar coupling between two nuclei... 

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. 

The wideline spectram, consequently, consists of two overlapping powder spectra, which possess mirror symmetry with respect to cul. This is the Pake doublet. In addition to the two observable single-quantum transitions ( Am = 1), a double-quantum transition ( /iml = 2) exists which can be detected indirectly. Its transition frequency is the sum of both single quantum transitions. 

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]. 

Figure 3.2.3 Pake doublet for a powder spectrum of a single deuterium site.

Temperature dependent lineshape variation of the Pake doublet represents then an experimental evidence of a motion occurring in the solid state. 2H NMR investigations on selectively deuterated compounds, therefore, are important tools for the recognition of molecular motions in solids. This method has been applied to the characterization of molecular motions in molecular solids [8] and inclusion compounds [9]. As an example in Fig. 3.2.4 is reported the effect of the motion about the molecular symmetry axis, either by two-fold or n-fold (n>3) flips, on the deuterium spectrum of a deuterated para-substituted benzene [10]. 

For an isolated pair of nuclei in a polycrystalline system the band shape shows a Pake doublet. When the number of interacting nuclei becomes larger and a polycrystalline material is analyzed, the resulting spectrum is much more complex and much less informative. Nevertheless an expression for the dipolar second moment (or mean square width) of the NMR spectrum in a polycrystalline sample has been suggested by Van Vleck [12] ... 

The disordered nature of the amorphous phase has prevented the direct determination of the structural parameters by the usual structural techniques such as X-ray diffraction. Solid-state NMR can give us those parameters even for amorphous samples. Yannoni and Clark [14] applied nutation NMR spectroscopy, which is specifically designed to measure interatomic distances, to determine the bond lengths in both the cis- and tran -polyacetylenes. They used polyacetylene polymerized from a mixture of 4% doubly C-enriched acetylene and doubly depleted acetylene. In Fig. 7.6, the observed and simulated proton decoupled nutation spectra of the cis sample are shown. The sharp peak in the centre arises from the isolated nuclei in the sample. The remainder of the spectrum is a Pake-doublet arising from the dipolar coupling of adjacent nuclei in the polyacetylene. The best fit to the observed spectrum corresponds to a distribution of bond length with... 

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... 

The dipolar pattern for polystyrene shown in Figure 3 (left) is not a Pake doublet reflecting only the CH static dipolar coupling (as scaled by WAHUHA irradiation of the protons). 

The internuclear distances employed are 1.095 A for the phenyl C-H distance, 1.125 A for the formal C-H distance, 2.4 A foj the 2-3 phenyl proton distance, and 1.75 A for the formal proton - proton distance. The 2-3 phenyl proton distance used here is comparable to the distance of 2.41 A used in the polycarbonate interpretations. The choice of 2.4 A is based on the phenyl proton Tj minimum and the slightly smaller value is confirmed by a larger Pake doublet splitting observed in the solid state spectrum of the phenyl protons in the partially deuterated analogue (10). 

Most studies of water in clays are not necessarily related directly to heterogeneous catalysis, but the state of the water in the interlayer is clearly important in the pillaring process. Woessner at Mobil pioneered this area 20 years ago. It was discovered that interlayer water in clays was undergoing anisotropic motion, the result being a partially motionally averaged Pake doublet spectrum (53-551. Under typical hydration conditions, many clays appear to maintain the equivalent of one or two monolayers of water between the clay sheets, so it is not surprising that the motion is restricted. 

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