Powder line shape

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. 

Here, ak is the isotropic chemical shift referenced in ppm from the carrier frequency co0, SkSA is the anisotropy and tfk SA the asymmetry of the chemical-shielding tensor, here also expressed in ppm. Note that for heteronuclear cases different reference frequencies co0 are chosen for different nuclei (doubly rotating frame of reference). The two Euler angles ak and pk describe the orientation of the chemical-shielding tensor with respect to the laboratory-fixed frame of reference. The anisotropy dkSA defines the width and the asymmetry t]kSA the shape of the powder line shape (see Fig. 11.1a). 

Fig. 11.1 (a) Powder line shapes of chemical-shielding tensors, dCSA/(2 n) = 5 kHz, for three different values... 

Fig. 1. Powder line shapes for an anisotropic chemical shift (a) arbitrary chemical shift tensor (tr, a22 <733).

Three membrane-bound adenosine triphosphatase enzymes have been characterized using Mn(II) and Gd(III) electron paramagnetic resonance (EPR) and a variety of NMR techniques. Mn(II) EPR studies of both native and partially delipidated (Na+ + K+)-ATPase from sheep kidney indicate that the enzyme binds Mn2+ at a single, catalytic site with Kq = 0.21 x 10- M. The X-band EPR spectrum of the binary Mn(II)-ATPase complex exhibits a powder line shape consisting of a broad transition with partial resolution of the 55 n nuclear hyperfine structure, as well as a broad component to the low field side of the spectrum. ATP, ADP, AMP-PNP and Pj all broaden the spectrum, whereas AMP induces a substantial narrowing of the hyperfine lines of the spectrum. 

In the case of MAS, the spectra typically display a number of spinning sidebands spaced by the spinning frequency. The intensities of the spinning sidebands approximately represent the intensity of the static powder spectrum and hence the overall envelope of the sideband intensities represents the powder line shape. While analytical solutions have also been derived for the intensity of the sidebands as a function of the spinning frequency and anisotropic shielding parameters, it is typically much easier to determine these interaction parameters from numerical simulations - again considering that this method directly takes the experimental errors and spectral noise into account. 

Figure 3.2.10 Schematic representation of theoretical powder line shapes for the chemical-shift tensor, (a) - asymmetric shift anisotropy, (b) axially symmetric shift anisotropy.

The powder ENDOR spectra in this book were calculated with software developed by Erickson [53]. The main equations employed are reproduced, in slightly different notation from that used previously [12, 53]. The powder line shape at the ENDOR frequency v and static magnetic field B is thus given as ... 

The quadrupolar-echo experiment represents the most widely used experiment for the observation of quadrupolar nuclei.For half-integer nuelei, it may be tuned to observe only the central transition (1/2-1/2), which is perturbed by the quadrupolar interaction only to second order, thus allowing the observation of less dominant anisotropic interactions. A significant improvement in sensitivity can be obtained by ncorporating a spin-echo method such as the Carr-Purcell-Meiboom-Gill sequence into the detection period. " The powder line-shape splits into a manifold of sidebands, from which information on the homogeneous and inhomogeneous interactions can be extracted from the line-shape of the sidebands and their envelope, respectively. 

Fig. 4. Powder line shapes in continuous wave (CW) ESR (derivative absorption spectra) and echo-detected ESR (absorption spectra), (a) Rhombic g-tensor. (b) Axial g-tensor. (c) Axial hyperfine coupling tensor with dominating isotropic contribution.

In powder samples, for instance, polydomain liquid crystals, the individual signals from all orientations of the coupling tensors in the sample are superimposed to yield the powder line shape, as shown for the case of... 

Aasa, R. J. Chem. Phys. 52 (1970) 3919. Powder Line Shapes in the Electron Paramagnetic... 

Fig. 8.6. The powder line shapes of an axially symmetric chemical-shift tensor. (Reproduced with permission from Ref. [26]. 1984 Pergamon Press, Inc.)...

These powder line shapes have been convoluted with Lorentzian broadening functions of different widths of A = [cri i — (T22] >o, where cuq is the NMR frequency. 

When the molecule contains several chemically different carbons, the powder pattern is a superposition of powder line shapes of the individual carbons. This overlap causes additional problems. 

Thus the chemical shift will have the same value for the field anywhere in the 2-3 plane but a different value when the field is perpendicular to the plane the chemical-shift tensor is axially symmetric. The chemical shift expected when the field is parallel to the unique axis is labeled a, whereas that expected for the field perpendicular to this axis is Figure 2B demonstrates this averaging and the resultant powder spectrum. Note the characteristic shape vwth the buildup of intensity at <7x- Both a, and individual components of the powder line shape (Seelig, 1978). 

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