Function complementary error

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

The integral in Eq. (10) is the usual definition of the error function. A closely related function is the complementary error function... 

Fig. 5. The complementary-error-function solution of the simple diffusion equation.

Exposure of bulk GaAs Si wafers to a capacitively coupled rf deuterium plasma at different temperatures generates deuterium diffusion profiles as shown in Fig. 1. These profiles are close to a complementary error function (erfc) profile. At 240°C, the effective diffusion coefficient is 3 x 10 12 cm2/s. The temperature dependence of the hydrogen diffusion coefficient is given by (Jalil et al., 1990) ... 

The quantity erfc(r/) is called the complementary error function, defined as... 

Integrated error functions are repeated integrations of the complementary error function. Define... 

Figure A2-1 Error function and complementary error function 566...

Again we find the complementary error function, erfc(y), with the same argument as in Eq. 18-20. This time the solution applies to both sides of the interface x < 0 for system B, x > 0 for system A. The interface is located at x = 0, where C (0,t) is always equal to (1/2) (Cg -CA), since erfc(0) = 1. Note that the solution is symmetrical around x = 0 in the sense that the losses and gains are equal at equal distances from the boundary (Fig. 18.5e). The transformation back to the original concentrations (see Eq. 18-26) yields ... 

Now you are ready to estimate the relevant diffusion distances. For the deepest depths in which PCNs appear, they are present at 120/18 000 = 0.0067 of the peak concentration at 24—25 cm. As discussed with respect to Eq. 18-23, this implies that you are interested in the argument of the complementary error function where the erfc(y0 0067) = 0.0067. In Appendix A, you find that y0,oo67 is about 1.9. Thus, you can solve ... 

Again, this is a rather unwieldy expression. Since the arguments of the exponential and complementary error functions are generally large at experimentally accessible times, this expression can be simplifed to... 

Equation 2.13 is commonly expressed in terms of the complementary error function such that... 

Thus, for the boundary conditions just described, the concentration of impurity as a function of space and time is given by a complementary error function (erfc) whose argument is x/ Dt. The complementary error function is a tabulated function. 

The FTIR-TPD profiles (A vs. T) for the desorption of benzene, toluene, and ethylbenzene from high-silica H-ZSM-5 is reported in Figure 4.40 [97], These results were fitted with the complementary error function, that is,... 

In test programs, where the numerical solution is compared with the analytical solution, the latter often involves the error function erf or the complementary error function erfc. The latter could be obtained simply by subtracting erf from unity but a better approximation is obtained by the direct algorithm. The two routines, ERF and ERFC, were given to the author by a colleague, who probably obtained them from an IBM collection. They have been adapted to Fortran 90/95 by the author, and coupled to the above module. The comments in capitals are the original comments. 

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