Atoms and Atomic Clocks
 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 conventional 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 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

Atoms oscillate and jostle about in complex ways. So many different modes of atomic vibration have been examined in the laboratory, compared with historical standards of timekeeping, and assessed for their practical use as clocks. Some are excellent. clocks can make time measurements that are good to a part in 1014 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 as $\sf{\Omega} = \left( \sf{q}_{1}, \; \sf{q}_{2}, \; \sf{q}_{3} \; \ldots \; \sf{q}_{\mathit{N}} \right)$. Since $\mathbf{A}$ is being used as a clock, we assume 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 $k$. We write

\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 event $k$ occurs at a time given by

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

Recall that the phase angle $\, \theta$ of $\mathbf{A}$ is given by $\theta_{k} = \theta_{\sf{0}} + 2\pi k/N$. So we can write

 Quantum Numbers for Atoms
 principal \begin{align} {\mathrm{n}} \; \equiv \frac{ n^{\mathsf{d}} }{4} \end{align} azimuthal \begin{align} \ell \; \equiv \frac{ N^{\sf{U}} + N^{\sf{D}} + \left| \, N^{\sf{U}} - N^{\sf{D}} \right| }{8} - \frac{ n^{\sf{d}}}{2} \end{align} spin angular momentum \begin{align} s \; \equiv \frac{ n^{\mathsf{u}} + n^{\mathsf{\overline{u}}} -3 n^{\mathsf{d}}+ n^{\mathsf{\overline{d}}} }{ 8 } \end{align} total atomic angular momentum \begin{align} j \; \equiv \frac{ \, \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| \, }{8} \end{align}

\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 rotating motion of particles when they are framed in a Cartesian view.

## Atomic States

The table above shows some quantum numbers for use in descriptions that gloss over visual sensation. To discuss atoms objectively we adopt $\mathrm{n}$, $\ell$, $s$ and $j$ instead of the four coefficients of rotating quarks that previously accounted for achromatic sensation. These atomic quantum numbers also have historic mechanical interpretations. Taken together, they are said to specify an atomic state. Note that $\mathrm{n}$ and $s$ are defined from simple sums and differences of quark coefficients, so they are always conserved.

 Next step: molecules.
page revision: 120, last edited: 08 Sep 2018 16:20