Two-axis jump processes

Contents 1. Introduction 104 2. Theory 106 2.1. Specifically for spin-1 nuclei 112 2.2. Two-axis jump processes 114 2.3. 2-by-2-site jump 114 2.4. 2-by-3-site jump 115 3. Numerical Simulations 117 3.1. Spin-1 nuclei 118 3.2. Half-integer quadrupolar nuclei 118 4. Results and Discussion 119 4.1. Spin-1 nuclei 119 4.2. Dynamic effects in 14N MAS spectra by SQ or DQ coherences 123 4.3. Multi-axis jump processes 124 4.4. Half-integer quadrupolar nuclei 129 5. Conclusions 134 Acknowledgements 135 References 135... 

In this case a two-axis jump process is studied. The two-site jump around the first axis is characterized by a rate constant k, whereas the rate constant for the jump around the second axis is AT During free precession only the + IQ coherence is of interest and therefore the matrix of interest is... 

Likewise the L-matrix for a two-axis jump process performing a three-site jump around one axis and followed by a two-site jump around the other axis having rate constants k2 and k3, respectively, is then defined as... 

Fig. 12. Mossbauer line shape of 57Fe in a constant eqQ field and a random hyperfine field assumed to be a two-state-jump process for different values of the jump rate W. (a) The random hyperfine field along the eqQ axis. (b) Perpendicular to the eqQ axis. (Calculations of Tjon and Blume.)...

The approach described above may be extended to 2-by-2-site jumps or 2-by-3-site jump as observed for 2H in, for example, thiourea-tit or DMS-rfg. From a computational point of view, these two jump processes resembles the 4-site and 6-site jump processes, respectively, regarding matrix dimensions, but the presence of two rate constants k and k2 complicates matters. So far only single axis jump processes have been described. In order to generalize to multi-axis processes the following example may be useful. A more comprehensive description of such processes may be found in the work of Kristensen et al.30/52... 

Dynamical effects of a two-axis 3-by-2-site jump process as can be observed in DMS-d6 were investigated by both QCPMG and MAS simulations. Besides rather interesting line broadening effects when both rate constants were in the intermediate regime, it was observed that the QCPMG experiment is more sensitive towards motional effects than MAS if either of the two rate constants is in the fast regime. 

To check the relevance of the two different dynamic processes we have modeled them under the external constraints of a rigid crystalline environment. According to our results (Fig. 7), the libration motion around the pseudo twofold axis (II) is the softer dynamic mode (so soft that large librations about the equilibrium structure are still possible at 100 K) but in-plane 60° jumps of the Fe3 triangle (I) are also allowed at room temperature (computed E ca. 12 kcalmoC ). 

The dynamic terms in eqn (1) depend upon the assumptions used to describe the motion. For the intermolecular motion a diffusive process is assumed (rotation through a sequence of small angular steps). In that case intermolecular reorientation can be characterized by two rotational correlation times, and Tru. The correlation time for reorientation of the symmetry axis of a molecular diffusion tensor is Tr, while Tru refers to rotation about the axis. For the intramolecular motion a random jump process is assumed. Thus, isomerization occurs through jumps between different conformations with an average lifetime Tj. 

The motional process is anisotropic, but by coincidence, the PAS orientation relative to the C(CH3)-COO bond is such that the resulting fast-limit spectmm corresponds to an axially symmetric tensor. The resulting line shapes are discussed in the early literature and can conveniently be obtained by computer simulation on the web. Note that usually two-site jump motions do not lead to axially symmetric fast-limit tensors (at least three symmetrically arranged sites or frill rotation around a given axis are required to generally obtain an axially symmetric fast-limit spectmm). See Figure 11(a) for an illustration of the more common case of an anisotropic fast-limit tensor for a two-site jump in a molecular crystal. 

The effect of an orientation process on an isolated elliptic flaw is depicted in Figure 7A. Let < i and a2 be the permanent stretch (or draw) ratios (ratio of drawn to undrawn length) to which the master sheet is subjected in two orthogonal directions. If an elliptic flaw is originally at right angles to the ai direction (i.e.y p = ir/2) and a2 = 1, then R will increase with until a particular value of is reached at which R = 1. For larger values of i, R will then decrease but the major axis of the ellipse is now at ft = 0. The critical draw ratio at which the orientation P jumps from tt/2 to 0 and for which the ellipse is circular (R = 1) is described later. 

Fig.4 In the biased reptation model, the biased walk of the chain in its tube, which creates new tube sections, is similar to the motion of a point-like particle between two absorbing walls. A biased Jump ends when the molecule has migrated over a distance a along the tube axis, i.e., when it has reached the next point defining the end of the next pore. This process is similar to the absorbtion of the particle by one of the walls, each at distance a from the starting position. The particle and the chain both have a one-dimensional velocity and diffusion constant D.

In the liquid crystalline phase, the local 180° jumps of the phenyl ring are augmented by diffusive rotation about the C2 axis by arbitrary angles as evidenced by the substantial narrowing of the NMR spectra. It is interesting to note that these two motional processes can be separated on cooling into the glassy state, where the diffusive rotation is frozen, whereas the 180° jumps persist as thermally activated local motions. ... 

---