Atoms
 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.

As usual, atoms are defined from combinations of 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 a three-dimensional The locations of larger atoms are even 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

Atoms oscillate and jostle about in ways that are not simple. So many different modes of atomic vibration have been examined in the laboratory, compared with historical standards of timekeeping, and assessed for their practical application for use as clocks. Some are excellent. can be used to make time measurements that are good to one part in 1014 and they are still being improved. High precision laboratory work often uses caesium, but here is a generic way to tell time using any sort of atom, $\mathbf{A}$, as a clock. 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

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

Since $\mathbf{A}$ is being used as a clock, we assume that it is sufficiently isolated and calibrated so that its period $\hat{\tau}$ is known and stable. Then we can measure elapsed time just by counting atomic cycles to determine $k$. Report

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

Without loss of generality, let $t_{\sf{0}} = 0$ so that $t_{k} = k \hat{\tau}$. By definition, the period is $\hat{\tau}\equiv 1/ \nu$ where $\nu$ is the frequency of $\mathbf{A}$. Then $t_{k} = k / \nu$. Also by definition, the frequency is given by $\nu \equiv N \omega / 2 \pi$ 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. However, we often think of time as the independent parameter and write the phase-angle as a function of time. Then we drop the $k$-subscript, and write $\theta \left( t \right)$ instead of $\theta_{ k}$. Substitution and rearrangement gives

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

 Next step: molecules and molecular bonds.
page revision: 304, last edited: 14 Aug 2018 21:52