As I discussed in my post "How I Met the Gaussian Distribution," I am assuming that quantum dot sizes follow a Gaussian bell-curve distribution in terms of size in a single layer of a quantum dot solar cell. The presence of variation in quantum dot sizes, as well as the fact that quantum dot sizes "center" about a face-value size like 3 or 4 nm diameter (which is the "advertised" size) are both confirmed by Santra & Kamat. While the presence of these factors does not altogether confirm a Gaussian nature of quantum dot sizes, it does lend significant evidence that such a phenomenon is occurring.
Thus, to account for this variation in sizes of quantum dots in one layer, it can be assumed that the quantum dot sizes remain centered about the face-value size (i.e. 3 or 4 nm diameter) and follow a normal Gaussian distribution, which can be expressed as
In retrospect, this algorithm looks a little redundant / common-sensical, because it LOOKS like the big expressions in the numerator and denominator cancel each other only to leave A(x) for Ai, but the A(x) is actually changing for every value of 0 to infinity substituted for x. Thus, this is much more complicated than a simple cancellation (and using a cancellation here would be incorrect).
For the purposes of computation, the above algorithm is simply impossible because it involves an integral to infinity, which the computer can NEVER exactly compute! But I solved this problem with Riemann sums, which is essentially the practice of using a bunch of rectangles defined by the function's x increments and y values to approximate the area under the curve.
My next steps are programming my algorithms and determining the number of sigma (confidence level, significance level) needed to define S in the algorithm immediately above. Do you have any recommendations for number of sigma, or feedback on any other parts of these algorithms? My impression is that the scientific community considers 3 sigma significant, 5 sigma "proof", and 6 sigma CERN Higgs boson-level, but I'd be interested in hearing your take.
Thanks so much for your time and feedback,