Interactive displacement parameters

With the aid of the recurrence procedure described in the previous section, the <)) dependence of I2 incorporated in the interactive displacement parameters and Aj as well as in the cross-frequency parameters P21 been investigated. 

Equations, the interactive displacement parameters convert into... 

The explicit separation of vibrations in the multidimensional GF of Theorem 4.4 allows an effective reduction of the number of degrees of freedom and facilitates the calculation of transition rates. In this regard, it cannot be too strongly emphasized that the calculation of transition rates in the parallel-mode approximation has a fundamental inadequacy that is evident from the derivation above. The defect emerges, if we return to the case of N-vibrational modes some of which are not parallel to each other. This situation occurs if the latter have the same symmetry, especially if they are totally symmetric in the molecular group. In this case, the complexity introduced by the reciprocal (4.69a) and interactive displacement parameters (Equations 4.69b and 4.69c) is considerable and this fact cannot be overlooked in any parallel-mode estimate of the transition probability. Even with considerable effort, it is impossible to factor the exact GF into a product of one-dimensional GF. 

Note that the interactive displacement parameters are not quantities with physical significance. They merely represent the error incurred in making a crude approximation, assuming that all modes are parallel. Another remarkable feature of mode mixing is the dependence of the transition rate on the cross-frequency parameters (see Equation 4.71). (Note that in the parallel-mode approximation the transition rate depends solely upon and 3j, = per mode). Both of these findings are... 

Perhaps the most significant complication in the interpretation of nanoscale adhesion and mechanical properties measurements is the fact that the contact sizes are below the optical limit ( 1 t,im). Macroscopic adhesion studies and mechanical property measurements often rely on optical observations of the contact, and many of the contact mechanics models are formulated around direct measurement of the contact area or radius as a function of experimentally controlled parameters, such as load or displacement. In studies of colloids, scanning electron microscopy (SEM) has been used to view particle/surface contact sizes from the side to measure contact radius [3]. However, such a configuration is not easily employed in AFM and nanoindentation studies, and undesirable surface interactions from charging or contamination may arise. For adhesion studies (e.g. Johnson-Kendall-Roberts (JKR) [4] and probe-tack tests [5,6]), the probe/sample contact area is monitored as a function of load or displacement. This allows evaluation of load/area or even stress/strain response [7] as well as comparison to and development of contact mechanics theories. Area measurements are also important in traditional indentation experiments, where hardness is determined by measuring the residual contact area of the deformation optically [8J. For micro- and nanoscale studies, the dimensions of both the contact and residual deformation (if any) are below the optical limit. 

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