Spheroidal Models of an Atom
René DescartesXlink.png certainly thought that atoms were like little balls spinning around and bumping into each other.1 So consider an atom $\mathbf{A}$ described by a repetitive chain of events $\Psi = \left( \sf{\Omega}_{1} , \sf{\Omega}_{2}, \sf{\Omega}_{3} \ldots \right)$ where each repeated cycle $\sf{\Omega}$ is a space-time event in a Euclidean space. Let this atom be characterized by its wavenumber $\kappa$ and an orbital radius $R$. Definition: We can model $\mathbf{A}$ as a spheroid mathematically represented in Cartesian coordinates by

$\begin{align} \frac{x^{2} + y^{2}}{R^{2}} + \left( \frac{2 \kappa}{3 \pi} \right)^{2} z^{2} = 1 \end{align}$

This shape is also known as an ellipsoid of revolution about the atom's polar axis. If $\mathbf{A}$ is in its ground-state then $\kappa = 0$ and the sphere collapses into a circle

Spheroidal Shapes
a perfect sphere $3 \lambda = 4 R$
a prolate spheroid $3 \lambda > 4 R$
a oblate spheroid $3 \lambda < 4 R$

$x^{2} + y^{2} = R^{2}$

However if $\mathbf{A}$ is an excited atom then its wavelength is $\lambda =2 \pi / \kappa$ and its shape can be represented as

$\begin{align} \frac{x^{2} + y^{2}}{R^{2}} + \left( \frac{4 }{3 \lambda} \right)^{2} z^{2} = 1 \end{align}$

Traditional mensuration formulaeXlink.png give the volume enclosed by this curve as

$V = \lambda \pi R^{2}$

So the spheroidal model has been scaled to give $\mathbf{A}$ exactly the same volume as the cylindrical model of an atom. A variety of spheroidal shapes are specified in the accompanying table.

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