Hydrogen Spectrum – Gross Structure
 The first quantized mathematical account of hydrogen was achieved by Johann Balmer, a Swiss high-school teacher. Here is a photo of what he described; the visible part of the hydrogen spectrum.

Photons that are absorbed or emitted by atomic hydrogen $\mathbf{H}$, are collectively known as the spectrum of hydrogen. They are mostly involved in the atomic and molecular interactions of everyday experience, not nuclear reactions. Energies are typically measured in (eV) rather than (MeV). All hydrogen-spectrum photons are linked to changes in the excited states of atomic hydrogen. When $\mathbf{H}$ goes from some initial state $i$, to some final state $f$, by emitting a photon $\gamma$ we write

\begin{align} {\mathbf{H}} _{i} \to {\mathbf{H}} _{f} + \gamma \end{align}

Atoms and photons are described by their principal quantum number $\rm{n}$ which is always conserved because $\rm{n}$ is proportional to the number of down quarks, and quarks are indestructible. So for any interaction, the conservation of down quarks guarantees that

\begin{align} {\rm{n}} \left( {\mathbf{H}} _{i} \right) = {\rm{n}} \left( {\mathbf{H}} _{f} \right) + {\rm{n}} \left( \gamma \right) \end{align}

Let us write ${\rm{n}}_{i} \equiv {\rm{n}} \left( {\mathbf{H}}_{i} \right)$ and ${\rm{n}}_{f} \equiv {\rm{n}} \left( {\mathbf{H}}_{f} \right)$. Then, as shown earlier, this conservation law will be automatically satisfied if

 \begin{align} {\rm{n}} _{i} = \frac{N^{\sf{D}}\left( \gamma \right) }{8} \end{align} and \begin{align} {\rm{n}} _{f} = \frac{\Delta n^{\sf{D}}\left( \gamma \right) }{8} \end{align}

where $\Delta n^{\sf{D}}$ and $N^{\sf{D}}$ note the coefficients of down quarks. Next we are going to use these relationships to formulate the photon energy, $E(\gamma)$, in terms of the principal quantum numbers of $\mathbf{H}$.

The hydrogen spectrum has many ultraviolet, visible and infrared photons. There are also some microwaves, but no gamma-rays. So overall, chemical quarks determine the range of photon energies. These chemical quarks are described by their enthalpy which is noted as $H_{\sf{chem}}$. For photons that are not gamma-rays, we can calculate $E(\gamma)$ by first assessing their momenta.

\begin{align} E \left( {\large{\gamma}} \right) = 2 \left| \, H_{\sf{chem}}^{\mathcal{A}} \vphantom{{H_{\sf{chem}}^{\large{\gamma}}}^{9}} \right| \cdot \left[ \frac{64}{\left( \Delta n^{\sf{D}} \right)^{2} } - \frac{64}{ \left( N^{\sf{D}} \right)^{2} } \right] \end{align}

This formula shows the strong influence of any down-quarks on these low-energy photons. Substituting-in the relationships with $\rm{n}$ discussed above gives

\begin{align} E \left( {\large{\gamma}} \right) = 2 \left| \, H_{\sf{chem}}^{\mathcal{A}} \vphantom{{H_{\sf{chem}}^{\large{\gamma}}}^{9}} \right| \cdot \left( \frac{1}{ {\rm{n}} _{f}^{2} } - \frac{1}{ {\rm{n}} _{i}^{2} } \right) \end{align}

This expression is used to make quark-models by adjusting the distribution of electochemical quarks so that

\begin{align} \left| \, H_{\sf{chem}}^{\mathcal{A}} \vphantom{{H_{\sf{chem}}^{\large{\gamma}}}^{9}} \right| = \frac{hc}{2} \mathcal{R} _{\mathbf{H}} \end{align}

where $\mathcal{R} _{\mathbf{H}}$ is the Rydberg constant of hydrogen.1 Using this constraint to eliminate $H_{\sf{chem}}$ then defines the energy of a photon for the so-called gross structure of hydrogen spectroscopy

\begin{align} E_{\sf{gross}} \equiv hc \mathcal{R} _{\mathbf{H}} \left( \frac{1}{{\rm{n}}_{f}^{2}} - \frac{1}{{\rm{n}}_{i}^{2}} \right) \end{align}

Measurements of photons report their wavelength which is related to their energy by $\lambda = hc/E$. Wavelengths may also depend on the surroundings, then the symbol $\lambda _{\sf{o}}$ indicates a wavelength where environmental effects are negligible. We assume this to write

\begin{align} \frac{1}{ \lambda _{\sf{o}} } = \mathcal{R} _{\mathbf{H}} \left( \frac{1}{{\rm{n}}_{f}^{2}} - \frac{1}{{\rm{n}}_{i}^{2}} \right) \end{align}

The description of hydrogen was first put in this form by . We use it to calculate $\lambda _{\sf{o}}$ in the following tables of hydrogen-spectrum photons. Models for these photons are built exclusively from electrochemical and rotating quarks.2 The rotating quarks all provide an angular momentum of $𝘑 \, =1$ in various ways. But the electrochemical quarks are the same for every photon because all these photons are associated with hydrogen. Comparisons are made with experimental observations.3 Some models provide excellent descriptions that are nonetheless outside of experimental uncertainty. This is indicated with an X in the following tables.

## Other Series

Results also depend on how the foregoing quarks are distributed between phase-components. For that level of detail please see the spreadsheets in the wiki-files stored here. In the models above, agreement with experiment is good to a few parts in a million. This is not perfect, partly due to blurring from finer structure. However, agreement is certainly good enough to distinguish between competing quark-models. And so these specific quark combinations are taken to define each photon, and used later to make fine and hyper-fine models of hydrogen.

Related WikiMechanics articles.

page revision: 392, last edited: 15 Nov 2019 07:09