Angular ordering

F. Hare, Simplified derivation of angular order and dynamics of rodlike fluorophores in models and membranes, Biophys. J. 42, 205-218 (1983). 

Figure 4.4 Dependence of NOE, nTi and nT2 upon rotational correlation time Tr, for a proton-decoupled resonance in a rigid rotor, in spectrometers with proton resonance frequencies of 250 and 500MHz. The dotted curve represents nT for a non-rigid rotor with an internal motion which rapidly halves the angular order parameter, S. The curves for other values of may be estimated by interpolation.

In order to be able to respond fully to the study of the spatial and angular relations between the back-ground of the image and the defects, we opted for the Co-Occurrence Matrix using spatial gray level dependence method. 

This angular dependence is different from the first-order perturbations so that the conventional teclmique of removing linebroadening in solids, MAS (see below), caimot completely remove this interaction at the same time as removing the first-order broadening. Flence, the resolution of MAS spectra from quadnipolar nuclei is usually worse than for spin-2 nuclei and often characteristic lineshapes are observed. If this is the case, it is... 

The interpretation of MAS experiments on nuclei with spin / > Fin non-cubic enviromnents is more complex than for / = Fiuiclei since the effect of the quadnipolar interaction is to spread the i <-> (i - 1) transition over a frequency range (2m. - 1)Vq. This usually means that for non-integer nuclei only the - transition is observed since, to first order in tire quadnipolar interaction, it is unaffected. Flowever, usually second-order effects are important and the angular dependence of the - ytransition has both P2(cos 0) andP Ccos 9) terms, only the first of which is cancelled by MAS. As a result, the line is narrowed by only a factor of 3.6, and it is necessary to spin faster than the residual linewidth Avq where... 

The zeroth-order Gaussian function has s-orbital angular symmetry the three first-order iTiiissian functions have p-orbital symmetry. In normalised form these are ... 

I liese second-order functions do not all have the same angular symmetry as the 3d atomic othitals, but a set comprising g y, ai d gy, together with two linear combinations of the gxx< gyy 3nd gzz, does give the desired result ... 

The Tersoff potential was designed specifically for the group 14 elements and extends the basic empirical bond-order model by including an angular term. The interaction energy between two atoms i and j using this potential is ... 

The function/c is a smoothing function with the value 1 up to some distance Yy (typically chosen to include just the first neighbour shell) and then smoothly tapers to zero at the cutoff distance, by is the bond-order term, which incorporates an angular term dependent upon the bond angle 6yk- The Tersoff pofenfial is more broadly applicable than the Stillinger-Weber potential, but does contain more parameters. 

---