Two by Four Tampan, Paminggir people. Lampung region of Sumatra, 19th century, 55 x 59 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop. |

In accordance with everyday chemistry, various **atoms** are defined from protons, electrons and neutrons. These subatomic components have all been specified from quarks and sensations. So for WikiMechanics, atoms are ultimately defined from sensations too.

Atoms are generically represented using bold, serified upper-case letters like $\mathbf{A}$ or $\mathbf{B}$. The smallest atom is hydrogen, noted by $\mathbf{H}$. Hydrogen atoms are also the smallest stable particles defined from a fully three-dimensional octet of space-time events. And so, they are also the smallest particles to be precisely represented in three-dimensional space. The locations of larger atoms are easier to establish. So we assume that a position $\overline{r}$, and a time of occurrence $t$, can be associated with anything that happens to an atom. Assigning a position to an atom is discussed later in an article about molecular models. But we can readily assign a time-of-occurrence to atomic events by thinking of atoms as little clocks.

## Atomic Clocks

^{14}and they are still being improved. High precision laboratory work often uses atoms of caesium. But here is a way to tell time using a generic atom, $\mathbf{A}$. Let $\mathbf{A}$ be described by a repetitive chain of events

$\Psi ^{ \mathbf{A} } = \left( \, \sf{\Omega}_{0}, \ \sf{\Omega}_{1}, \ \sf{\Omega}_{2} \ \ldots \ \sf{\Omega}_{\mathit{k}} \ \ldots \ \right)$

where orbital cycles are composed of $N$ quarks

$\sf{\Omega} ^{ \mathbf{A} } = \left( \, \sf{q}_{1}, \ \sf{q}_{2}, \ \sf{q}_{3} \ \ldots \ \sf{q}_{\mathit{N}} \, \right)$

Since $\mathbf{A}$ is being used as a clock, we assume it has been sufficiently stabilized and isolated so that its vibrations are steady and regular. We assume their period $\hat{\tau}$ has been measured, so that $\mathbf{A}$ is calibrated. Then we can determine an elapsed time just by counting atomic cycles to determine $k$. Report

$\begin{align} \Delta t \equiv t_{k} - t_{\sf{0}} = k \hat{\tau} \end{align}$

Without loss of generality, let $t_{\sf{0}} = 0$ so that $t_{k} = k \hat{\tau}$. And recall that the period is given by $\hat{\tau}= 1/ \nu$ where $\nu$ is the frequency of $\mathbf{A}$. Then $t_{k} = k / \nu$. Also, by definition the frequency is $\nu \equiv N \omega / 2 \pi$ where $\omega$ is the angular frequency of $\mathbf{A}$. And so

$\begin{align} t_{k} = \frac{ 2 \pi k }{ N \omega } \end{align}$

Recall that the phase angle $\, \theta$ of $\mathbf{A}$ is defined by $\theta_{k} \equiv \theta_{\sf{0}} + 2\pi k \delta_{z} \, / \, N$. And by convention, the helicity $\delta_{z}$ of any ordinary particle is one. So we can write

$\begin{align} t_{k} = \frac{ \theta_{ k} - \theta_{\sf{0}} }{ \omega } \end{align}$

This relationship expresses the time-of-occurrence of the $k$^{th} event as a function of the phase-angle. Time is thus told. However, we often think of time as the independent parameter and write the phase-angle as a function of time; the $k$-subscript is dropped, and $\theta \left( t \right)$ is substituted for $\theta_{ k}$. Then rearranging gives

$\begin{align} \theta \left( t \right) = \theta_{\sf{0}} + \omega t \end{align}$

This form is good for describing the spinning motion of particles when they are framed in a Cartesian view.