The surface matching theorem

The surface matching theorem makes it possible to generalize the idea of muffin-tin orbitals to a nonspherical Wigner-Seitz cell r. Each local basis orbital is represented as (p =a x + V on the cell surface a, where y and p are the auxiliary functions defined by the surface matching theorem. An atomic-cell orbital (ACO) is defined as the function — y, regular inside r. By construction, the smooth continuation of this ACO outside r is the function p. The specific functional forms are... 

The notation Wtl defines a Wronskian integral over ct/2. By the surface matching theorem, xm = X> the interior component of 4> Since 4r is a solution of the Lippmann-Schwinger equation, this implies xout = Xm when evaluated in the interior of Tjj. This is a particular statement of the tail-cancellation condition. To show this in detail, after integration by parts... 

This problem can be avoided by expressing the MST equations in terms of the square matrix S C = CVS, Hermitian in consequence of the surface-matching theorem. This matrix has full rank because it is contracted over the larger index lg. From the definitions of the C and S matrices, the matrix product S C is a specific integral involving the Helmholtz Green function [281],... 

This can vanish only if (H — r) b = 0 in both rin and rout. Moreover, this requires that both ( ifrin Wa if/in — fout) and (S j/0Ut Wa i//in — j/out) must vanish when ijrin and fs out are varied independently. By an extension of the surface matching theorem, both these Wronskian integrals must vanish in order to eliminate the value and normal gradient of tf/m — i//""r on o. Practical applications of this formalism use independent truncated orbital basis expansions in adjacent atomic cells, so that the continuity conditions cannot generally be satisfied exactly. 

The variational equations imply [r — t- =a 0 on each cell boundary ct/2. Given independent expansions ijr = E/. within each atomic cell, and (ct) = J2/1.L Ni(To)Pl on the global matching surface, the coefficients are determined by the implied variational equations. The surface matching theorem implies two independent Wronskian integral conditions for each atomic cell,... 

Optimum transmission of secondary ions is achieved when the acceptance of the mass spectrometer is matched with the secondary ion beam. Figure 4a shows schematically the transfer of the secondary ion beam from the sample to the mass spectrometer entrance slit. If d. is the diameter of the area to be analyzed, the entrance slit Xwidth s) has to accept the image d + which is d + 0 1. rf"j is the virtual width of the sample point, given by = V /E where Is the initial kinetic energy and E is the field strength on the surface. This explains the desirability for high extraction fields at the sample surface). Using the Louiville theorem, we obtain... 

The second problem is actually more complex. When considering an infinite array, the terminal impedance will be the same from element to element in accordance with Floquet s Theorem. However, when the array is finite, it is well known that the terminal impedance will differ from element to element in an oscillating way around the infinite array value (sometimes denoted as jitter). We postulated that this phenomenon was related to the presence of surface waves of the same type as encountered in Chapter 4. However, there is a significant difference in amplitude of these surface waves in the passive and active cases. This is due to the fact that the elements in the former case in general are loaded with pure reactances (if any), while the elements in the latter case are (or should be) connected to individual amplifiers or generators containing substantial resistive components (as encountered when conjugate matched). 

As the Ward-Tordai equation contains two independent variable functions (surface excess and subsurface concentration), its application requires a further equation relating the two functions. The first attempt at this was by Sutherland [11], who incorporated a linear adsorption isotherm. This, however, proved to be quite limiting, and so various other isotherms were employed [12, 13]. Even so, these extended theorems accurately matched experimental results only in the case of some nonionic surfactants. 

---