As folks here know by now, Im always looking for simple number-theoretical descriptions of mathematical phenomena found in quantum systems. Thus far the Pascal Triangle has been the best bet for combinatorics whose results are magic numbers in atomic systems. I was reviewing some of the basics in nuclear models and noticed something that may become of use: For very light nuclei nucleon distribution takes Gaussian form, so related to the harmonic oscillator potential. Ive had best luck with the simple harmonic oscillator, where magics (even when deformed) can be easily understood in terms of Pascal combinations. But for larger nuclei Fermi distribution makes more sense, and is the basis of the Woods-Saxon potential, which includes a term in its denominator based on e, the base of the natural logarithm. A couple of months ago we read about how e can be derived via the Pascal Triangle also, by taking a) the product of terms in row n multiplied by b) the product of terms in row n+2 and then divided by c) the product of terms in row n+1. As n goes to infinity the result becomes e. This has got me thinking about whether one could vary the actual value of the e term in the Woods-Saxon potential depending on the value of row n that relates to the magic numbers of the nuclei in question. Havent tried it yet, but it might be worth exploring. Wouldnt it be something if this tracked actual experimental data?
Posted on: Wed, 12 Feb 2014 00:14:00 +0000
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