Arjen Toni Dijksman | Searcharxiv

arXiv2026

The Angular Seed Power Map: A Constructive Approach to Recursive Scaling Spirals

We present the ''Power Spiral Map'', a continuous angular evolution of the linear coordinate grid established in our previous work. While that previous Power Map utilized a seed value translating along a horizontal axis, this work builds upon a seed angle ($θ$) projected onto a unit diameter circle. This operation controls two coupled geometric behaviors: an internal area-preserving partition of unity within a reference square (cos 2 $θ$ + sin 2 $θ$ = 1) and an external recursive scaling mechanism (sec $θ$ and cos $θ$) that dictates the expansion or contraction of successive generations of squares unfolding as a spiral in the 2D plane. We demonstrate that continuous variation of this angular parameter generates discrete geometric alignments that yield polynomial identities, with examples of the Golden Ratio ($Φ$) and the Plastic Ratio ($ψ$) defined through purely planar intersections.

math.MG↗

arXiv2013

Accurate energy-size dependence of excitonic transitions in semiconductor nanocrystals and nanoplatelets using a phase jump approach

Accurate energy-size dependence of excitonic transitions in semiconductor nanocrystals in the strong confinement regime using classical theoretical approaches such as effective mass approximation, tight binding, or empirical pseudo-potential is difficult. We propose a simple empirical expression with three fitting parameters that accurately relates the size dependence of most known excitonic transitions in CdSe and in InAs nanocrystals. We show that this empirical expression can be deduced from a phase jump approach if the charge carriers are considered to travel on the atomic lattice of the nanocrystal and gain energy upon bouncing at the nanoparticle boundaries. This empirical expression is also tested on the atomically flat CdSe nanoplatelets without any adjustment of the parameters obtained with the CdSe spherical nanocrystals, and provides an estimation of the CdSe nanoplatelets thickness that matches exactly the experimental observations. These results suggest that a phase shift approach could be useful to describe the electronic transitions in semiconductor nanocrystals.

cond-mat.mtrl-sci↗