Motional averaging

The averaging of nuclear spin Hamiltonians under rotations may be easily studied when it is expressed in terms of irreducible spherical tensor operators, TL,m and In liquid crystals, the main interest is in time averag- 

Ri m will be non-zero only when L = 0 and 2. If i L,m is considered in its principal axis system, only components with m = 0, L are non-zero. The components of the irreducible spherical tensor operators will be denoted in their respective principal axis systems by which are given in Table 2.1 together with the constants C. Table 2.2 lists Ti m for each spin interaction. 

If the interaction Hamiltonian H is expressed in the laboratory frame, the spin parts are constant and the spatial parts i L,m are time- 

The dipole-dipole interaction involves multiple spins. Therefore, it may be complicated to handle in comparison with chemical shift and electric quadrupolar interactions. However, it is simple to examine two interacting proton spins in the high-field approximation as 

FIGURE 2.1. Energy level diagram for a pair of proton spins (/ = 1 triplet note that the singlet / = 0 is not shown) and a deuteron spin (tj = 0) in a large magnetic field. a o/27r is the Larmor frequency. 

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This result, when substituted into the expressions for C(t), yields expressions identieal to those given for the three eases treated above with one modifieation. The translational motion average need no longer be eonsidered in eaeh C(t) instead, the earlier expressions for C(t) must eaeh be multiplied by a faetor exp(- co2t2kT/(2me2)) that embodies the translationally averaged Doppler shift. The speetral line shape funetion I(co) ean then be obtained for eaeh C(t) by simply Fourier transforming ... 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Figure 4 ESR of nitroxide radicals (a) Molecular axes of the nitroxide (b) ESR spectra of oriented nitroxide radicals, with the molecular axes x, y, and z along the external magnetic field, respectively and (c) ESR spectra of randomly oriented nitroxide radicals motionally averaged (upper) and rigid limit (lower) regimes.

The estimated correlation times for the loop domains of the order of 10 4 s are obtained for the suppressed peaks in the [l-13C]amino-acid-labelled bR, including Gly, Ala, and Leu residues as shown in Figure 24C. The loop dynamics can be also examined by measurements of the 13C-1H dipolar couplings by DIPSHIFT experiment in which fluctuations of the Co,-Cp vector result in additional motional averaging as order parameters, in addition to the rotation of Ala methyl groups which scales the dipolar... 

MgO, respectively. Photolysis by an incandescent lamp is necessary to produce the radical on MgO, and the light significantly alters, both in amplitude and shape, the spectrum of the species on silica-alumina. Both spectra are interpreted in terms of very anisotropic hyperfine interactions. This is consistent with work on radical anions such as nitrobenzene on MgO (90) however, it is a bit surprising in light of the motional averaging found for most cations on silica-alumina. 

When a sodium cation jumps between two sites with the same EFG tensor components and the same residence times on both positions, but different orientations, then the two-site exchange process can be described by tensor averaging of the two positions. Depending on the jump angle of the tensor, the motionally averaged principal components, Vxx, Vyy, and Vzz, may eventually have to be redefined in ordering to fulfill the requirement in parenthesis in (4). 

Figure 8 a shows the motionally averaged quadrupole coupling constant, (Cq)/Cq, and asymmetry parameter, ( ), for a two-site jump between axially symmetric equivalent sites. At jump angles of 70° and 109° the principal components (V, Vyy, Vzz) have to be rearranged in order, which leads to the discontinuities in the curve shapes of Fig. 8a. 

Using the approach described in the previous section, Van de Walle (1990) also calculated the isotropic hyperfine constant for muonium at T in Si. The calculated value for iK0)]x/ experimental result (0.45) for normal muonium in Si. Motional averaging slightly lowers the theoretical value, bringing it in even closer agreement with experiment. 

The effect of zero-point motion on the hyperfine constant has been the subject of some controversy. Manninen and Meier (1982) argued that motional averaging (over all positions sampled in the vibrational ground... 

Another method to determine the magnitude and rhombicity of the alignment tensor is based on the determination of the Saupe order matrix. The anisotropic parameter of motional averaging is represented by this order matrix, which is diagonalized by a transformation matrix that relates the principal frame, in which the order matrix is diagonal,... 

This does not mean that minor conformers and/or motional averaging are unimportant. The presence of even small amounts of minor conformers or limited conformational averaging as is suggested by the lowest energy two state structural solution could significantly affect the observed cross relaxation rate. The inverse sixth power dependence of the cross relaxation rate on interproton distance serves to strongly weight contributions from conformers with short interproton distances. Therefore the presence of even small amounts of conformers with short interproton distances can exert a disproportionate amount of influence on the observed cross relaxation rate. 

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