General surface perturbations

Once again, suppose that the surface S is deflned in a rectangular coordinate system with reference to a flat surface with the interior coordinates X, z. The y—direction is normal to this flat reference surface. The nearly flat material surface can be specified by giving its t/—coordinate as a function of position x, z at any time t, say 

The restriction to small amplitude fluctuations requires that 

The block of material is subjected to an equi-biaxial remote traction so that, when the surface is perfectly flat with h x,z,t) = 0, the state of stress everywhere is Uxx = ( zz = and other stress components are zero. The equi-biaxial strain corresponding to the stress is m = Cra/M. This simple state of stress and deformation is then perturbed by formation of surface waviness. The perturbed stress field determines the surface strain energy density U x,z,t) in (9.27). 

A procedure by which the perturbed stress field can be determined has already been described for both plane strain deformation and general three-dimensional deformation in Chapter 8, all subject to the hmitation of small amplitude surface fiuctuations. To lowest order in the surface fluctuation, the strain energy density disbribution over the surface is 

Based on experience with the simpler case of sinusoidal perturbations, it is again convenient to reduce the governing equation (9.27) to a nondi-mensional form by introducing the natural length scale and the natural time scale r, similar to (9.20). Nondimensional variables are again denoted by a superposed hat. The normalizations that are incorporated are 

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The electrical characteristics of the TSM resonator with a generalized surface perturbation can be described by the equivalent-circuit model of Figure 3.7b [14]... 

The electrical characteristics of the TSM resonator with a generalized surface perturbation can be described by the equivalent-circuit model of Figure 3.7b. Measurements can be made on a dry TSM resonator to determine C o,L, Ci, and R. Fixing these parameters and fitting the equivalent-circuit model to data measured on an immersed device determines R2 and L2. Equations 3.21 can then be used to determine the components of from L2 and R2. 

The above analyses were relatively straightforward the departures from the plane surface were quite simplistic. However, in turns out that any arbitrary surface, provided it is smooth and differs little from the plane, can be treated in essentially the same manner as above since any such surface may be represented either by Fourier series (if periodic) or by Fourier integral (if aperiodic) [91]. In fact, as Goldstein et al. [94] state, for the above approach to be applicable to all orders in the perturbation, a necessary condition is that the general surface shape function (here taken to be a cos(ky)) be infinitely differentiable. The price to pay for allowing more general but smooth surface structure is simply in the amount of tedious labor that must be expended, rather than any demand for new theoretical considerations. 

Since the PE surfaces are anharmonic in this limit, the Franck-Condon contributions can be difficult to treat. The approaches to this limit can be separated into two classes (a) perturbation theory corrections of the weak-coupling limit and (b) quantum-mechanical calculations of the reaction coordinate. The latter tend to be done on a reaction-by-reaction basis this makes it difficult to generalize. The perturbation theory approaches have the advantage that they make use of the parameters used in the weak-coupling limit, and this can provide usefiil insights into general trends and patterns. 

We might also note that the generalization of perturbation theory is applicable to problems in reactor shielding, where the characteristic of interest is the measured dose at the outer surface, provided some care is taken over the form of the surface source at the inside of the shield and the adjoint source at the outside. Perturbation theory should be useful in estimating the effect of departures from idealized shields. 

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

There are no direct, reliable measurements of 0 based upon adsorption of water from the gas phase. Therefore, 4.25 eV applies to a macroscopic water layer as in the electrochemical configuration. The decrease in 0 upon water adsorption is a general occurrence with metals. The value of A0 observed with Hg is the lowest among those available in the literature.35,36 With reference to Eq. (20), this means that the perturbations of the surfaces of the two phases are small for the Hg/water contact. In other words, the interaction between Hg and water is weak (hydrophobic). 

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