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Atomic Models

Here are some models of atoms that bring the notion of physical shape to their description. Aggregations of chemical and thermodynamic quarks are represented as cylinders, spheres and even a morsel of pasta.

Cylindrical Atomic Models

Consider an atom  \mathbf{A} described by a repetitive chain of events written as

\Psi^{\mathbf{A}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3}  \, \ldots \,  \right)

where each repeated cycle \mathsf{\Omega} is a space-time event. Let this particle be characterized by its orbital radius  R and wavelength  \lambda. These properties are related to position and we can use them to make a simple geometric model of P that is shaped like a cylinder. Or to be more exact; like a finite section of a right circular cylinder with its ends closed to form two circular surfaces, oriented along the  z-axis, similar to the one shown in the accompanying diagram. The area  A of the circular cross section is defined by

A \equiv \pi R^{2}

This is just a statement of ancient knowledge about circles going back at least to Archimedes . And to restate another traditional mensuration formula, the volume  V of the cylinder is

V \equiv \lambda  A      = \lambda \pi R^{2}

We use this model to visualize one atomic event. Then it is easy to imagine the chain of events \Psi as a row of cylinders strung-out end-to-end, like a tube or wire.

Spheroidal Atomic Models

The French mathematician RenΓ© Descartes certainly thought that atoms were like little balls spinning around and bumping into each other.1For example he writes that; “The material, as I said, is composed of many small balls which are in mutual contact; and we have sensory awareness of two kinds of motion which these balls have. One is the motion by which they approach our eyes in a straight line, which gives us the sensation of light; and the other is the motion whereby they turn about their own centers as they approach us. If the speed at which they turn is much smaller than that of their rectilinear motion, the body from which they come appears blue to us; while if the turning speed is much greater than that of their rectilinear motion, the body appears red to us. But the only type of body which could possibly make their turning motion faster is one whose tiny parts have such slender strands, and ones which are so close together (as I have shown those of the blood to be), that the only material revolving round them is that of the first element. The little balls of the second element encounter this material of the first element on the surface of the blood; this material of the first element then passes with a continuous, very rapid, oblique motion from one gap between the balls to another, thus moving in an opposite direction to the balls, so that they are forced by it to turn about their centres.” From A Description of the Human Body published in The Philosophical Writings of Descartes, Volume I, page 323. Translated by John Cottingham, Robert Stoothoff and Dugald Murdoch. Cambridge University Press, 1985. So also let atom  \mathbf{A} be described by its wavenumber  \kappa. Then we can model  \mathbf{A} as a spheroid that is mathematically represented using Cartesian coordinates using the equation

\dfrac{x^{2} + y^{2}}{R^{2}} + \left( \dfrac{2\kappa}{3\pi} \rule{0px}{14px} \right)^{2} z^{2} = 1

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

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

\dfrac{x^{2} + y^{2}}{R^{2}} + \left( \dfrac{4}{3\lambda} \rule{0px}{14px} \right)^{2} z^{2} = 1

Spheroidal Shapes
a perfect sphere3 \lambda = 4 R
a prolate spheroid3 \lambda > 4 R
a oblate spheroid3 \lambda < 4 R

Traditional mensuration formulae 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 atomic model. A variety of spheroidal shapes are specified in the accompanying table.

A Rotini Model of an Atom

Principia Philosophiae by Rene Descartes, page 271. Amsterdam 1644.

Let the events of atom  \mathbf{A} be described by their time coordinates  t. Our first spatial conception of such an atom was as a compound quark in quark space. But to implement the hypothesis of spatial isotropy our next view is set in a Cartesian coordinate system where  \mathbf{A} is represented as a rotating atomic clock with a phase angle  \theta given by

\theta \! \left( t \right)  = \theta_{\mathsf{o}} +\omega t

such that  \mathbf{A} is whirling about its polar axis with an angular frequency of  \omega. The rotation supposedly averages-out variations in the electric and magnetic radii leaving an effective orbital radius  R that is then used to represent the atom as a rotating cylinder. This rotating cylinder model smooths out some rough edges, but it is still amiss because the electromagnetic part of the quark metric is larger than the other non-polar components. So one radial direction is predominant and the atom is shaped more like a piece of rotini pasta than a solid cylinder. This corkscrew spiral can be approximated by a geometric curve called a helicoid . It is described mathematically by radii of

\rho_{x} = R \cos{\! 2 \theta}

\rho_{y} = R  \sin{\!  2 \theta }

\rho_{z} = \dfrac{ \lambda \theta}{2\pi}

When moving, the rotini model looks a lot like a machine called the Archimedean screw . Humans have been thinking about screw conveyor mechanisms like this for thousands of years. They were reportedly used to irrigate the Hanging Gardens of Babylon as early as 600 BC. This atomic model is good for understanding the Euclidean metric of the ordinary spaces in our laboratories and classrooms.

References
1For example he writes that; “The material, as I said, is composed of many small balls which are in mutual contact; and we have sensory awareness of two kinds of motion which these balls have. One is the motion by which they approach our eyes in a straight line, which gives us the sensation of light; and the other is the motion whereby they turn about their own centers as they approach us. If the speed at which they turn is much smaller than that of their rectilinear motion, the body from which they come appears blue to us; while if the turning speed is much greater than that of their rectilinear motion, the body appears red to us. But the only type of body which could possibly make their turning motion faster is one whose tiny parts have such slender strands, and ones which are so close together (as I have shown those of the blood to be), that the only material revolving round them is that of the first element. The little balls of the second element encounter this material of the first element on the surface of the blood; this material of the first element then passes with a continuous, very rapid, oblique motion from one gap between the balls to another, thus moving in an opposite direction to the balls, so that they are forced by it to turn about their centres.” From A Description of the Human Body published in The Philosophical Writings of Descartes, Volume I, page 323. Translated by John Cottingham, Robert Stoothoff and Dugald Murdoch. Cambridge University Press, 1985.