The ztot parameter is derived from the parameters of the "small" roughness model. All the small roughness calculations are based on the tanh profile; the parameters for the erf profile are derived from key features of the tanh profile.
In this appendix, I'll be speaking of the profile y(x), and I'll use "vertical" to refer to the y axis and "horizontal" to refer to the x axis (which is a re-scaled depth into the sample: z = RO(i) * x).
In the tanh model, the vertical axis from -1 to 1 is divided into NRough + 2 equally spaced partitions, which determine the weights in the averaging process. More precisely, the model assumes the arithmetic average of QCSQ(i) and QCSQ(i+1) at the interface, and the weights govern how far the value of QCSQ in a microslab is from the arithmetic average. The NRough + 1 unknown weights w_tanh range from -(1-1/(NRough+1)) through 1-1/(NRough+1). We know at the middle of the layers QCSQ(z) = QCSQ(i), implying the weights are +/-1. This occurs at x = +/-infinity. In this sense, all our layers are considered "infinitely thick" in the small roughness model. For the erf model, the weights are given by w_erf = erf(2 * arctanh(w_tanh)).
The nominal x values are given by x = 2 * arctanh(w_tanh). These are rescaled by dividing by CT = 2.292 for the tanh model and CE = 1.665 for the erf model, and are subsequently scaled by RO(i). The value of ztot is given by 2 * ztot = 3 * X_1 - X_3 where the X_i are rescaled versions of the nominal x_i in which tanh(x_i / 2) = 1 - i / (NRough + 1). Well, almost. This is the formula for ztot in the criteria to select models, but not always the one displayed by VE. The displayed ztot is actually the scaled thickness of the vacuum gradation, and agrees with this calculation for odd NRough. However, for even NRough the ztot displayed by VE has underestimated x_3 by 2 * arctanh(2/(NRough + 1)), and hence is too large.
The magic constants CT and CE are derived from the derivatives of the profiles in the "large" roughness model. In these models we have weight given by erf(CE * z / RO) or tanh(CT * z / RO). We start first with CE, used in the erf model.
Let g(x) be the derivative of erf(x), scaled to unity at x = 0. Simply, g = exp(-x*x). Erf(x) is the integral of 2/sqrt(Pi)*exp(-x*x). CE satisfies 1/2 = g(CE/2). CE = 2 * sqrt(ln(2)). So for the erf model, CE represents the full width at one-half maximum.
Let G(x) be the derivative of tanh(x). It is already scaled to unity at x = 0. CT satisfies 1/3 = G(CT/2). CT = 2 * arctanh(sqrt(2/3)). So for the tanh model, CT represents the full width at one-third maximum.
A further problem exists with existing versions of GJ2/MLAYER. For the tanh model with large roughness, the RO parameters correspond to the full-width at one-third maximum (FWTM) as described. For the small roughness case, an design error in calculating the partitions maps the RO parameters to the half-width at one-third maximum (HWTM). Thus the tanh model should be avoided because the meaning of RO can change during the fit.
2003-01-17