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Gross Hydrogen Spectrum

a spectrum-like image used as an icon for photons

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  \boldsymbol{\gamma}, we write

{\mathbf{H}}_{i}  \to  {\mathbf{H}}_{f} + \boldsymbol{\gamma}

Particles are described by their principal quantum number  \rm{n}. This quantity is always conserved because quarks are conserved, and  \rm{n} is directly proportional to the number of down quarks. So for any interaction, the conservation of down quarks guarantees that

{\rm{n}} \! \left( {\mathbf{H}}_{i} \right) = {\rm{n}} \! \left( {\mathbf{H}}_{f} \right) + {\rm{n}} \! \left( \boldsymbol{\gamma} \right)

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, quark conservation will automatically be obtained if

{\rm{n}}_{i} = \dfrac{N^{\mathsf{D}} \! \left( \boldsymbol{\gamma} \right) }{8}

and

{\rm{n}}_{f} = \dfrac{\Delta n^{\mathsf{D}} \! \left( \boldsymbol{\gamma} \right) }{8}

Here \Delta n^{\mathsf{D}} and N^{\mathsf{D}} note the photon’s coefficients for down quarks (i.e.  n is different from  \rm{n}). These relationships can be used to understand the hydrogen spectrum. Atomic interactions involving hydrogen emit many ultraviolet, visible and infrared photons. There are also some microwaves. But usually, there are no gamma-rays. We can describe photons that are not gamma-rays by first assessing their momentum as follows.

The gross hydrogen spectrum is shown in this photograph of four lines; red, green, blue and violet.
The first quantized account of hydrogen was achieved by Johann Balmer, a Swiss high-school teacher. Here is a photo of what he described mathematically.
Consider finding the mechanical energy E, of a photon \boldsymbol{\gamma}, that is specified by a pair of phase components \mathcal{A}_{\LARGE{\circ}} and \mathcal{A}_{\LARGE{\bullet}} written as
\mathsf{\Omega} \left( \boldsymbol{\gamma} \right) = \left\{ \mathcal{A}_{\LARGE{\circ}}, \, \mathcal{A}_{\LARGE{\bullet}} \rule{0px}{10px} \right\}
As discussed earlier, the wavenumber of this photon can be written as
\kappa \! \left( \boldsymbol{\gamma} \right) = 2 \left\| \, \overline{\rho}^{\mathcal{A}} \rule{0px}{10px} \right\| \! \left( \dfrac{1}{ \rho_{in}^{2} } - \dfrac{1}{ \rho_{out}^{2} } \rule{0px}{18px} \right)
where  \rho_{in} is the photon’s inner radius,  \rho_{out} is its outer radius and  \overline{\rho} is a radius vector. The subscript on \mathcal{A} is dropped because both phase-components have the same norm. We can use this wavenumber to express the momentum of the photon, in a perfectly inertial reference frame, as
p \! \left( {\boldsymbol{\gamma}} \right) = \dfrac{h}{2\pi} \kappa = \dfrac{h}{\pi} \left\| \, \overline{\rho}^{\mathcal{A}} \rule{0px}{10px} \right\| \! \left( \dfrac{1}{\rho_{in}^{2}} - \dfrac{1}{\rho_{out}^{2}} \rule{0px}{18px} \right)
Writing-out the norm in terms of the radial components of \mathcal{A} gives

    \begin{equation*}   \left\| \,   \overline{\rho}^{\mathcal{A}} \,  \right\|  =  \left(  \begin{split} &    \; k_{mm} \rho_{m}^{2} +  k_{ee} \rho_{e}^{2} +  k_{zz} \rho_{z}^{2}  \\  &  + 2 k_{em}  \rho_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z}    \\ & \hspace{30px} + 2 k_{ez}\rho_{e}  \rho_{z} \;  \end{split} \right)^{\frac{1}{2}} \end{equation*}

This expression can be simplified if the photon is not a gamma-ray. For long-wavelength photons the coefficients of leptonic quarks must all be zero because even one of these high energy quarks is enough to yield a gamma-ray. So the electric and magnetic radii of \mathcal{A} are null.  That is, \rho_{m} = \rho_{e} = 0. And recall that k_{zz} = 1. Then
\left\| \, \overline{\rho}^{\mathcal{A}} \rule{0px}{10px} \right\| = \left| \rho_{z} \right|
and so
p \! \left( {\boldsymbol{\gamma}} \right) = \dfrac{h}{\pi} \left| \rho_{z} \right| \! \left( \dfrac{1}{\rho_{in}^{2}} - \dfrac{1}{\rho_{out}^{2}} \rule{0px}{18px} \right)
The photon’s momentum is proportional to the absolute-value of its polar radius which is defined as
\rho_{z} \equiv \dfrac{ H_{chem} + \Delta n^{\mathsf{U}} U^{\mathsf{U}} - \Delta n^{\mathsf{D}} U^{\mathsf{D}}}{k_{\mathsf{F}}}
where H_{chem} is the enthalpy due to any chemical quarks. This expression can be simplified too because by convention U^{\mathsf{D}} \! =0. Also, for the high-energy up-quarks, \Delta n^{\mathsf{U}} must be zero as well, or else the photon would be a gamma-ray. Thus
\rho_{z} = \dfrac{ H_{chem}^{\mathcal{A}} }{k_{\mathsf{F}}}
and so
p \! \left( {\boldsymbol{\gamma}} \right) = \dfrac{h}{\pi k_{\mathsf{F}} } \left| H_{chem}^{\mathcal{A}} \rule{0px}{13px} \right| \! \left( \dfrac{1}{\rho_{in}^{2}} - \dfrac{1}{\rho_{out}^{2}} \rule{0px}{18px} \right)
The photon is ethereal, its mechanical energy is
E \equiv \sqrt{ c^{2}p^{2} + m^{2}c^{4} \rule{0px}{10px} \; } = cp
So
E \! \left( \boldsymbol{\gamma} \right) = \dfrac{hc}{\pi k_{\mathsf{F}} } \left| \, H_{\sf{chem}}^{\mathcal{A}} \rule{0px}{13px} \right| \left( \dfrac{1}{\rho_{in}^{2}} - \dfrac{1}{\rho_{out}^{2}} \right)
Then substituting-in definitions for the radii gives the photon energy in terms of quark coefficients as

E \! \left( \boldsymbol{\gamma} \right) = 2 \left| \, H_{chem}^{\mathcal{A}} \rule{0px}{13px} \right| \cdot \left[  \dfrac{64}{\left( \Delta n^{\mathsf{D}} \right)^{2} } - \dfrac{64}{      \left( N^{\mathsf{D}} \right)^{2} } \right]

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

E \! \left( \boldsymbol{\gamma} \right)  =  2 \left| \, H_{chem}^{\mathcal{A}} \rule{0px}{13px} \right| \cdot \left( \dfrac{1}{ {\rm{n}}_{f}^{2} }  -  \dfrac{1}{ {\rm{n}}_{i}^{2} }  \right)

This expression is used to make quark-models by adjusting the distribution1Other electrochemical quark distributions are used to model other atoms and molecular bonds. of electrochemical quarks so that

\left| \, H_{chem}^{\mathcal{A}} \rule{0px}{13px} \right|   =  \dfrac{hc}{2}  \mathcal{R}_{\mathbf{H}}

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

E_{\mathsf{gross}} \equiv hc  \mathcal{R}_{\mathbf{H}} \left( \dfrac{1}{{\rm{n}}_{f}^{2}}  -  \dfrac{1}{{\rm{n}}_{i}^{2}} \right)

Photon Wavelengths

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

\dfrac{1}{ \lambda_{\mathsf{o}} } = \mathcal{R}_{\mathbf{H}} \left( \dfrac{1}{{\rm{n}}_{f}^{2}}  -  \dfrac{1}{{\rm{n}}_{i}^{2}} \right)

The description of hydrogen was first put in this form by Johannes Rydberg . Models for these photons are built exclusively from electrochemical and rotating quarks.2Baryonic and leptonic quarks are not used because we are not considering gamma-rays and nuclear processes. The rotating quarks all provide an angular momentum of  \textsl{\textsf{\rule[-1px]{0.01pt}{1px} 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.3Kramida, 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. For this reference, the lines from transitions between levels designated only by principal quantum numbers correspond to observations in which any finer structure is completely unresolved. Reported energy levels are the centre of groups of all fine-structure levels having the same  \rm{n}. Their values are based on observations of the Sun. Some models provide very accurate descriptions that are nonetheless outside of experimental uncertainty. This is indicated with an X in the following tables.

Lyman Series

The gross spectrum of hydrogen for the Lyman series of lines is listed.

Balmer Series

The gross spectrum of hydrogen for the Balmer series of lines is listed in this spreadsheet screen shot.

Paschen Series

The gross spectrum of hydrogen for the Paschen series of lines is listed in this spreadsheet screen shot.

Brackett Series

The gross spectrum of hydrogen for the Brackett series of lines is listed in this spreadsheet screen shot.

Other Series

The gross spectrum of hydrogen for the Pfund, Humphreys and other series of lines is listed in this spreadsheet screen shot.

Gross Hydrogen Spectrum — Summary

In the models above, agreement with experiment is good to a few parts in a million. This would be outstanding in almost any other scientific discipline. But for atomic spectroscopy, it is mediocre. However, it is good enough to distinguish between competing quark-models. And so these specific quark combinations are taken to define each photon. Then later, after a discussion of atomic hydrogen, we extend these models to make a more accurate description of fine structure in the hydrogen spectrum. Calculated results depend on how quarks are distributed between phase-components. For that level of detail please see the spreadsheets in the files stored here. Next we consider how to make quark models of X-rays.

References
1Other electrochemical quark distributions are used to model other atoms and molecular bonds.
2Baryonic and leptonic quarks are not used because we are not considering gamma-rays and nuclear processes.
3Kramida, 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. For this reference, the lines from transitions between levels designated only by principal quantum numbers correspond to observations in which any finer structure is completely unresolved. Reported energy levels are the centre of groups of all fine-structure levels having the same  \rm{n}. Their values are based on observations of the Sun.