Distance error function

Rms difference vs. refined structure (A) Distance error function value... 

The following distance error functions have been used/ -52,33 where N is the number of atoms, djj is the distance (A) between two atoms i and j, and and lij are the upper and lower bounds, respectively ... 

The experimental duration and the length of the diffusion couple are designed such that the length of the diffusion profile is short compared to the total length of the diffusion couple. Hence, the diffusion medium may be treated as infinite, meaning that at the ends of the two halves, the compositions are still the initial compositions. If D does not vary with concentration or distance, the concentration profile (C versus x) would be an error function. Hence, the first step to try to understand the profile is to fit an error function to the profile (Equation 3-38) ... 

Note that Equation 4-99 means that the solution is an error function with respect to the lab-fixed reference frame (x = x—2aVSf). In the interface-fixed reference frame, the solution appears like an error function, and its shape is often error function shape, but the diffusion distance is not simply especially... 

That is, if = 0 (normal error function profile), ymM = 0.954(D t), and the midconcentration distance is roughly If is a large positive value, the midconcentration distance is much smaller than If is negative, the... 

Another example for treating concentration profiles during mineral dissolution can be found in Figure 3-32b, which shows a Zr concentration profile during zircon dissolution. In this case, the dissolution distance is very small compared to the diffusion profile length. Hence, the diffusion profile is basically an error function. 

Next we turn to the inference of cooling history. The length of the concentration profile in each phase is a rough indication of (jDdf) = (Dot), where Do is calculated using Tq estimated from the thermometry calculation. If can be estimated, then x, Xc and cooling rate q may be estimated. However, because the interface concentration varies with time (due to the dependence of the equilibrium constants between the two phases, and a, on temperature), the concentration profile in each phase is not a simple error function, and often may not have an analytical solution. Suppose the surface concentration is a linear function of time, the diffusion profile would be an integrated error function i erfc[x/(4/Ddf) ] (Appendix A3.2.3b). Then the mid-concentration distance would occur at... 

This model suggests at once the type of distribution of energy that will result. It must involve the law governing chance events, the Gaussian distribution law. Thus, if one repeatedly shoots abullet at a target, and the causes giving rise to the near-misses of the bullseye are all due to chance (e.g., ticks in the nervous system of the aimer), the number of near-misses is related to distance from the target (x) by a so-called error function ... 

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 ... 

Even though the above method of solution of the diffusion equation (Eq. 7-12) becomes impractical for complicated cases, it illustrates the appearance of the error function in problems where diffusion from an infinite number of sources occurs and the solution is obtained in the form of an infinite series as a result of the overlapping of diffusion streams. The overlapping diffusion streams are due to an infinite number of repeated reflections at the ends of the diffusion path which are spaced finite distances apart. 

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