Matrix effects relaxation

The observation of large Stokes shifts for the 2P-2S transition of entrapped Ag atoms indicates that the guest-host interactions are markedly different for the 2S and 2P states of this system and can be explained in terms of matrix cage relaxation effects. 

Matrix Effects. pH. Numerous factors such as sample pH, ionic strength, humic substances, and relaxation agents can modify the NMR spectrum. For example, monoester phosphate chemical shifts are pH-de-pendent (44-46), and we showed (44) for several monoester phosphates that as the sample pH is increased, their chemical shifts also increase (Figure 3). This behavior is caused by the ability of monoester phosphates to undergo protonation-deprotonation. Because the monoester phosphate chemical shift is pH-dependent, the curves resulting from plotting pH versus chemical shift are analogous to a titration curve. Thus, monoester phosphate pKa values can be measured from these pH-chemical shift curves (44-46). 

When the applied stress a is less than Su, creep of the matrix will commence after application of the load. During this creep, the matrix will relax and the stress on the fibers will increase. Therefore, further fiber failure will occur. In addition, the process of matrix creep will depend on the extent of prior fiber failure and, as mentioned previously, on the amount of matrix cracking. The details will be rather complicated. However, the question of whether steady-state creep or, perhaps, rupture will occur, or whether sufficient fibers will survive to provide an intact elastic specimen, can be answered by consideration of the stress in the fibers after the matrix has been assumed to relax completely. Clearly, when the matrix carries no stress, the fibers will at least fail to the extent that they do in a dry bundle. It is possible that a greater degree of fiber failure will be caused by the transient stresses during creep relaxation, but this effect has not yet been modeled. Instead, the dry bundle behavior will be used to provide an initial estimate of fiber failure in these circumstances. 

It is our opinion that these results provide a powerful impetus towards the further development of the egg model (already developed in its main features by Stepanov and Shliomis [80]) for the purpose of gaining a simple analytic formula for the effective relaxation time (analogous to Eqs. (8.1) and (8.5)). This approach, as it is essentially based on a two particle model, allows one to take account of the Interdependence of the Neel and Debye relaxation mechanisms and simultaneously eliminates the undesirable features of the present theory which is based on the two extremes of a frozen liquid matrix [17] and a frozen magnetic moment [16]. 

Sdnchez-Almazdn, R, Napolitani, E., Camera, A., Drigo, A.V., Isella, G, von Kfinel, H., Berti, M. (2004) Matrix effects in SIMS depth profiles of SiGe relaxed buffer layers. Applied Surface Science, 231-232,704-707. 

Aveston eta/. [50] suggested that this is not necessarily always the case, and that the overall orientation efficiency would also depend on the response of the matrix to the local flexural stresses. If the matrix is sufficiently weak, it will crumble, and the flexural stresses will be effectively relaxed. They thought this to be the case in the carbon fibre reinforced cement tested in their work. Stucke and M umdar [52] applied this mechanism to account for the embrittlement of glass fibre reinforced cement and suggested that the densification of the ageing matrix around the fibres leads to a build-up of flexural stresses in the fibres in the cracked zone, which in turn results in premature failure. In the younger composite, the matrix interface is more porous and weaker, and crumbles before any significant flexural stress can develop in the fibres. 

Thus in an isotropic environment the number of different relaxation rates is reduced from (21 +1) to (2Jg+l), giving one relaxation rate for each multipole moment of the excited state. Experimentally the number of effective relaxation rates might still be embarrassingly large were it not for the fact that in resonance fluorescence we prepare and monitor the excited atoms through the absorption and emission of electric dipole radiation. The electric-dipole matrix elements have the properties associated with rank one tensors and consequently the observable multipole moments in these experiments are limited to those corresponding to tensors of rank 0, 1, and 2 respectively. 

Erom a given structure, the NOE effect can be calculated more realistically by complete relaxation matrix analysis. Instead of considering only the distance between two protons, the complete network of interactions is considered (Eig. 8). Approximately, the... 

Figure 8 Effects of spin diffusion. The NOE between two protons (indicated by the solid line) may be altered by the presence of alternative pathways for the magnetization (dashed lines). The size of the NOE can be calculated for a structure from the experimental mixing time, and the complete relaxation matrix, (Ry), which is a function of all mterproton distances d j and functions describing the motion of the protons, y is the gyromagnetic ratio of the proton, ti is the Planck constant, t is the rotational correlation time, and O) is the Larmor frequency of the proton m the magnetic field. The expression for (Rjj) is an approximation assuming an internally rigid molecule.

Another principal difficulty is that the precise effect of local dynamics on the NOE intensity cannot be determined from the data. The dynamic correction factor [85] describes the ratio of the effects of distance and angular fluctuations. Theoretical studies based on NOE intensities extracted from molecular dynamics trajectories [86,87] are helpful to understand the detailed relationship between NMR parameters and local dynamics and may lead to structure-dependent corrections. In an implicit way, an estimate of the dynamic correction factor has been used in an ensemble relaxation matrix refinement by including order parameters for proton-proton vectors derived from molecular dynamics calculations [72]. One remaining challenge is to incorporate data describing the local dynamics of the molecule directly into the refinement, in such a way that an order parameter calculated from the calculated ensemble is similar to the measured order parameter. 

In order to study lattice relaxation effect by ASR we assume a simple model. As a first step we consider the terminal point approximation. Here the distortion of the lattice taken into account is the stretching or the contraction and angular distortion of the bond connecting two sites in a lattice and the effect of neighbouring site is neglected. As a result of such distortion the structure matrix takes the form ... 

The Eik/TDDM approximation can be computationally implemented with a procedure based on a local interaction picture for the density matrix, and on its propagation in a relax-and-drive perturbation treatment with a relaxing density matrix as the zeroth-order contribution and a correction due to the driving effect of nuclear motions. This allows for an efficient computational procedure for differential equations coupling functions with short and long time scales, and is of general applicability. 

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