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Atomic Hydrogen

The letter H is used as an icon for atomic hydrogen.

Atomic hydrogen is formed by the union of a proton  \mathsf{p}^{+} with an electron  \mathsf{e^{-}}, bound together by a force-carrying collection of field quanta \mathscr{F}. An atom of hydrogen is noted by \mathbf{H}, so we write

\mathbf{H} \equiv \left \{ \mathsf{p^{+}}, \, \mathsf{e^{-}}, \, \mathscr{F} \right\}

The proton is represented by the quarks \mathsf{p^{+}}  \leftrightarrow 4\mathsf{d} + 4\mathsf{b} + 4\overline{ \mathsf{t} }. The electron is modeled from this selection \mathsf{e^{-}} \leftrightarrow 4\overline{\mathsf{u}} + 2\overline{\mathsf{b}} + 2\mathsf{t} + 2\overline{\mathsf{s}} + 2\mathsf{c} + 4\overline{\mathsf{g}}+ 4{\mathsf{e}}. And the field for hydrogen in its spin-down ground-state is given by

\mathscr{F} \! \left( 1\mathbf{S} \right) \leftrightarrow 4\overline{\mathsf{d}} + 2\mathsf{m \overline{m}} + 2\mathsf{a \overline{a}} + 2\mathrm{ l \overline{l}}

Atoms of hydrogen are objectified from space-time events as shown in the accompanying table and movie. Stereochemical quarks are about a million times smaller than thermodynamic quarks, so they are not shown in the movie.

Atomic Hydrogen is modeled by the array of quarks shown here.
Click on this image for a look around a quark-model of hydrogen in its ground-state.

Excited States of Atomic Hydrogen

The excited states of atomic hydrogen are built-up from the components shown in this table. The set of quarks noted by \text{\L} is a particle of orbital angular momentum. It is called the Lamb quantum. Absorbing or emitting \text{\L} changes the azimuthal atomic quantum number by \Delta \ell = \pm 1 without altering  \rm{n} or  j. It explains the Lamb shift because, as shown below, excited states can be accurately constructed as

2\mathrm{S} \equiv \mathrm{1S \! \! \uparrow} + \mathrm{Lyman} \hspace{1px} \alpha - \text{\L}

2\mathrm{P_{1/2}}  \equiv \mathrm{1S \! \! \uparrow} + \mathrm{Lyman} \hspace{1px} \alpha

Quark coefficients for the Lyman photons are obtained from the gross structure of the hydrogen spectrum, they bring wet and dry quarks into the description. Quarks are conserved so the following models are obtained by adding together the quark-coefficients of all components. This automatically conserves charge, momentum, etc.

Energy Levels of Atomic Hydrogen

Let E^{\prime} \mathsf{(} \mathrm{H}^{\ast} \mathsf{)} note the mechanical energy of some excited state \mathrm{H}^{\ast}, on a scale where the null-value of E^{\prime} =0 is obtained when the electron is very far from the proton. For this case1Hydrogen Atom – Chemistry WebBook , NIST Standard Reference Database Number 69. National Institute of Standards and Technology, Gaithersburg MD, USA. the ground-state has a value of E^{\prime} ( {\mathrm{1S}} ) = -13.6 (eV). There are some challenges to making accurate measurements in this context. So we also specify another quantity E ( {\mathrm{H} ^{\ast}} ) as the energy on a scale where the ground-state always has a value of exactly zero by definition

E ( \mathrm{H}^{\ast}   ) \equiv  E^{\prime} ( \mathrm{H}^{\ast} ) - E^{\prime}  ( \mathrm{1S})

Then the excited states of hydrogen are described by the sum

E^{\prime} \equiv \,  E_{\mathsf{\, fine}} + E_{\mathsf{\, hyperfine}} + E_{\mathsf{\, Lamb}}

The terms in this expression depend strongly on the principal quantum number  \mathrm{n}. But there is also a weaker dependence on  j,  \ell and  s. These atomic quantum numbers are defined from quark coefficients. So the energy of an atomic state is a function of its quark content, and can be formulated as follows. The fine structure of the hydrogen spectrum is given by

E_\mathsf{\, fine} \equiv - \dfrac{hc \, \mathcal{R}_{\mathrm{H}}}{\mathrm{n}^{2}} \left(1+ \,  f_{\mathsf{fine}} \right)

where

f_{\mathsf{fine}} \equiv \dfrac{\alpha^{2}}{\mathrm{n}} \left[ \dfrac{1}{\, j + \text{½} } -  \dfrac{3}{4\mathrm{n}} \right]

These expressions were developed by Arnold Sommerfeld and Paul Dirac . The number  \alpha is a constant and \mathcal{R}_{\mathrm{H}} is another constant called the Rydberg number. The hyperfine energy term is given by

E_{\mathsf{hyperfine}} \equiv \dfrac{1 + \delta_{z}}{\, {\mathrm{n}}^{3}} \, k_{\mathsf{hyperfine}}

where  \delta_{z} is the helicity. Describing the hyperfine splitting is easy for EthnoPhysics because we explicitly retain accounts of both up-quarks and down-quarks in our atomic models.2The constant  k_{\mathsf{hyperfine}} = h \Delta \nu^{\mathbf{H}} /2 and the  {\mathrm{n}}^{3} factor is optional because  {\mathrm{n}}=1. The assessment of  E_{\mathsf{\, Lamb}} has been strongly influenced by Hans Bethe . This formula is based on his work.3Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag OHG, Berlin 1957. Bethe’s approach depends on assumptions that EthnoPhysics cannot accept uncritically, including; continuous spaces and fields; a structureless conception of the proton (page 100); and Dirac’s theory about electrons which features the vacuum as an infinite sea of electrons (pages 2 and 77). Bethe and his followers explain the Lamb shift using complicated devices including perturbations, renormalization, vacuum fluctuations and Feynman diagrams. EthnoPhysics does not logically incorporate any of these ideas. But nonetheless, we gratefully adapt Bethe’s formulation just because it provides a concise, accurate recap of many laboratory observations.

E_{\mathsf{\, Bethe}} \equiv \dfrac{\alpha^{5} m^{\mathsf{e}} c^{2}}{4\mathrm{n}^{3}} \begin{cases} \hspace{35px} k_{\mathsf{\, Lamb}} &\mathsf{\text{if}} \; \ell=0 \\ \\ \; \dfrac{-2s}{\pi \left(\, j + \text{½} \right) \left( \ell + \text{½} \right)} + k_{\mathsf{Bethe}} &\mathsf{\text{if}} \; \ell \ne 0 \end{cases}

This expression can accurately represent energy levels, but transition energies are mostly outside of experimental uncertainty. So to improve accuracy we include another summand

E_{\mathsf{\, Lamb}}  \equiv E_{\mathsf{\, Bethe}}  +   E_{\mathsf{\, chiral}}

where

E_{\mathsf{\, chiral}}  \equiv \dfrac{\; s N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{L}}}}}  }{\, \mathrm{n}^{6}} \, k_{\mathsf{chiral}}

and N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{L}}}}} notes the total number of levo quarks in the atom. All the foregoing equations, together with formulae for the atomic quantum numbers, are combined with the quark-models to describe the energy levels of atomic hydrogen. Results are compared with experimental observations4Peter J. Mohr, Barry N. Taylor, and David B. Newell, CODATA Physical Constants: 2010 , Rev. Mod. Phys. 84, 2012., 5Helmut Hellwig, Robert F. C. Vessot, Martin W. Levine, Paul W. Zitzewitz, David W. Allan, and David J. Glaze. Measurement of the Unperturbed Hydrogen Hyperfine Transition Frequency , IEEE Transactions on Instrumentation and Measurement, Volume IM-19, Number 4, November 1970., 6Kramida, A., Ralchenko, Yu., Reader, J., and the NIST ASD Team (2018). NIST Atomic Spectra Database (version. 5.6.1). National Institute of Standards and Technology, Gaithersburg, MD, USA. in the table below.

The EthnoPhysics quark-models of hydrogen thus reproduce the quantum-numbers of excited states correctly. And calculated energy levels for all 28 measured states are within experimental uncertainty. So the description of an inert hydrogen atom is complete. But transition energies are not fully in agreement with observation, so this analysis is continued later for some finer details in the hydrogen spectrum.

Stability of Atomic Hydrogen

The stability of a particle is described by its mean life which is a function of its thermodynamic temperature,  T. And the temperature of a hydrogen atom in its ground state is supposed to be very close to zero. Quark models have been carefully adjusted to obtain this. But, despite much effort, the closest to be had for  \mathbf{H} ( 1\mathrm{S} ) is T=-8.9 \times 10^{-6} (K). We doubt that this number has physical significance. Rather, it shows the limit of our computing techniques. Temperature calculations depend on small differences between large numbers. Some rounding errors are inevitable. And if the temperature is near zero, then these errors can be significant. The non-zero result for hydrogen suggests that our temperature calculations are questionable for any result more exact than a few parts in a million. This seems to be near the limit of what we can obtain from our present computing arrangements

For more detail about any of the foregoing calculations please see the Atoms & Photons spreadsheet. As of January 2019 calculations are being done using; Microsoft Excel for Mac, Version 15.16, running on an iMac Model 16.2 with a Intel Core i5, 3.1 GHz processor and 8 GB of memory.

References
1Hydrogen Atom – Chemistry WebBook , NIST Standard Reference Database Number 69. National Institute of Standards and Technology, Gaithersburg MD, USA.
2The constant  k_{\mathsf{hyperfine}} = h \Delta \nu^{\mathbf{H}} /2 and the  {\mathrm{n}}^{3} factor is optional because  {\mathrm{n}}=1.
3Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag OHG, Berlin 1957. Bethe’s approach depends on assumptions that EthnoPhysics cannot accept uncritically, including; continuous spaces and fields; a structureless conception of the proton (page 100); and Dirac’s theory about electrons which features the vacuum as an infinite sea of electrons (pages 2 and 77). Bethe and his followers explain the Lamb shift using complicated devices including perturbations, renormalization, vacuum fluctuations and Feynman diagrams. EthnoPhysics does not logically incorporate any of these ideas. But nonetheless, we gratefully adapt Bethe’s formulation just because it provides a concise, accurate recap of many laboratory observations.
4Peter J. Mohr, Barry N. Taylor, and David B. Newell, CODATA Physical Constants: 2010 , Rev. Mod. Phys. 84, 2012.
5Helmut Hellwig, Robert F. C. Vessot, Martin W. Levine, Paul W. Zitzewitz, David W. Allan, and David J. Glaze. Measurement of the Unperturbed Hydrogen Hyperfine Transition Frequency , IEEE Transactions on Instrumentation and Measurement, Volume IM-19, Number 4, November 1970.
6Kramida, A., Ralchenko, Yu., Reader, J., and the NIST ASD Team (2018). NIST Atomic Spectra Database (version. 5.6.1). National Institute of Standards and Technology, Gaithersburg, MD, USA.