Spectacular Photons – EthnoPhysics

Outline

For EthnoPhysics, a photon is just a composite quark that satisfies a few specific conditions. These conditions, and a wide variety of photon types, are all defined from the quark content of the photon. So on this page we look at how photonic quarks are described and organized.

Photon Definition

To start, let some generic particle P be characterized by a repetitive chain of events written as $\Psi^{\mathsf{P}} \! = ( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \ldots )$ where each repeated cycle is a finite bundle of quarks

$\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2}, \, \mathsf{q}_{3}, \; \ldots \; \mathsf{q}_{N} \right)$

And let each quark be described by its phase $\delta_{\theta} \hspace{0.5pt} .$ Use this phase to sort quarks into a pair of sets, noted by $\mathcal{A}_{\mdsmwhtcircle}$ and $\mathcal{A}_{\mdsmblkcircle} \hspace{0.5pt} ,$ such that all quarks of the same phase are in the same set. Then $\mathcal{A}_{\mdsmwhtcircle}$ and $\mathcal{A}_{\mdsmblkcircle}$ are called phase components of P, and they are out of phase with each other. We write

$\mathsf{\Omega}^{\mathsf{P}} = \left\{ \mathcal{A}_{\mdsmwhtcircle} \, , \, \mathcal{A}_{\mdsmblkcircle} \rule{0px}{9px} \right\}$

and

$\delta_{\theta} \! \left( \mathcal{A}_{\mdsmwhtcircle} \right) = - \, \delta_{\theta} \! \left( \mathcal{A}_{\mdsmblkcircle} \right) = \pm 1$

Now consider that P is an almost perfectly phase anti-symmetric particle so that $\mathcal{A}_{\mdsmwhtcircle} \! = \overline{\mathcal{A}_{\mdsmblkcircle}}$ for all types of quarks except perhaps down quarks which are noted by $\mathsf{D} .$ Generic quark-types are marked by $\mathsf{Z}$ and up quarks are written using the letter $\mathsf{U} .$ Quark coefficients are noted by $N$ and $\Delta n .$ These coefficients are used to define a photon $\boldsymbol{\gamma}$ as a particle that satisfies the conditions

$N^{\mathsf{D}} = N^{\mathsf{U}} \! \pm 8$

and

${\Delta}n^{\mathsf{Z}} = 0 \ \ \mathsf{\text{if}} \ \ {\mathsf{Z \ne D}}$

Photon Characteristics

The foregoing conditions that define a photon constrain the angular momentum quantum number $\textsl{\textsf{J}}$ such that for all photons

$\textsl{ \textsf{J} } \! (\boldsymbol{\gamma}) \equiv \dfrac{ \, \left| N^{\mathsf{U}} - N^{\mathsf{D}} \right| \, }{8} = 1$

As for other characteristics, recall that the internal energy of a down-quark is so small that it is conventionally taken as zero. Then any imbalance between ordinary down quarks and down anti-quarks can be ignored. Also, the requirement for phase anti-symmetry means that $\Delta n \! = \! 0$ for all other quark types. Then by substitution into definitions for the charge, strangeness, lepton number, baryon number, the work and the enthalpy we obtain

$q(\boldsymbol{\gamma})=0$

$S(\boldsymbol{\gamma})=0$

$L(\boldsymbol{\gamma})=0$

$B(\boldsymbol{\gamma})=0$

$W(\boldsymbol{\gamma}) \simeq 0$

$H(\boldsymbol{\gamma}) \simeq 0$

Finally recall that lepton-numbers, baryon-numbers, charge and the enthalpy are always conserved. So, any particle may interact with countless photons without much change to its own character because for photons, all these conserved quantities are nil or negligibly small.

Photon Types

Next we look at a range of spectacular possibilities. We specify gamma-rays, X-rays and ultraviolet photons. We consider visible sensation, and also some invisible infrared quanta. These phenomena are usually called particles or photons. And the sensory patterns they represent are more-or-less steady and local.

But as we consider larger clusters of quarks, sensory patterns extend over larger spaces, and longer times. Pinpoint locations can get blurry. Snapshots become movies. And motion can be innate. Moreover, patterns must be recurrent to be recognized. So in other words, we encounter particles that show intrinsic repetitive motion. We could say that they vibrate, fluctuate, undulate or oscillate. These sorts of ‘particles’ are historically called waves.

So a full discussion of photon-types must also consider waves. Specifically microwaves and radio waves. These oscillating particles have some special features. So here is a more detailed article about the description and analysis of waves.

Waves

A historical view of waves is advanced for both inertial and dispersive environments. Wavelengths and refraction are defined.

A key takeaway from the foregoing article about waves is a definition for the wavevector. To be exact, if $\boldsymbol{\gamma}$ is a free photon then its free wavevector in vacuo is given by

$\displaystyle \overline{\kappa}_{\mathsf{o}}(\boldsymbol{\gamma}) = \dfrac{2\pi k_{\mathsf{F}}}{hc} \, \Delta \overline{\rho}$

where $\Delta \overline{\rho}$ is the quark flux vector and the positive constant $k_{\mathsf{F}}$ was discussed earlier. Next we expand $\Delta \overline{\rho}$ in terms of individual quarks to obtain

$\displaystyle \overline{\kappa}_{\mathsf{o}} (\boldsymbol{\gamma}) = \dfrac{2\pi k_{\mathsf{F}}}{hc} \sum_{\mathsf{q} \, \in \, {\boldsymbol{\gamma}} } \delta_{\theta}^{\, \mathsf{q}} \; \overline{\rho}^{\, \mathsf{q}}$

Here $\delta_{\theta}$ notes the phase of each quark $\mathsf{q}$ in $\boldsymbol{\gamma} .$ And radius vectors are marked by $\overline{\rho} .$ This sum can be expressed in terms of the photon’s asymmetric phase-components $\mathcal{A}_{\mdsmwhtcircle}$ and $\mathcal{A}_{\mdsmblkcircle}$ as

$\begin{align*} \displaystyle \overline{\kappa}_{\mathsf{o}} (\boldsymbol{\gamma}) &= \dfrac{2\pi k_{\mathsf{F}}}{hc} \left( \rule{0px}{17px} \delta_{\theta}^{\, \mathcal{A}_{\mdsmwhtcircle} } \hspace{-6px} \sum_{\mathsf{q} \, \in \, \mathcal{A}_{\mdsmwhtcircle} } \hspace{-4px} \overline{\rho}^{\, \mathsf{q}} + \, \delta_{\theta}^{\, \mathcal{A}_{\mdsmblkcircle} } \hspace{-6px} \sum_{\mathsf{q} \, \in \, \mathcal{A}_{\mdsmblkcircle} } \overline{\rho}^{\, \mathsf{q}} \right) \\ &= \dfrac{2\pi k_{\mathsf{F}}}{hc} \left( \rule{0px}{17px} \delta_{\theta}^{\, \mathcal{A}_{\mdsmwhtcircle} } \, \overline{\rho}^{\, \mathcal{A}_{\mdsmwhtcircle} } \, + \; \delta_{\theta}^{\, \mathcal{A}_{\mdsmblkcircle} } \, \overline{\rho}^{\, \mathcal{A}_{\mdsmblkcircle} } \right) \end{align*}$

But $\mathcal{A}_{\mdsmwhtcircle}$ and $\mathcal{A}_{\mdsmblkcircle}$ are out of phase. So $\delta_{\theta} (\mathcal{A}_{\mdsmwhtcircle}) =- \, \delta_{\theta} (\mathcal{A}_{\mdsmblkcircle}) \,$ and therefore

$\overline{\kappa}_{\mathsf{o}} (\boldsymbol{\gamma}) = \delta_{\theta} (\mathcal{A}_{\mdsmwhtcircle}) \; \dfrac{2\pi k_{\mathsf{F}}}{hc} \left( \, \overline{\rho} \! \left( \mathcal{A}_{\mdsmwhtcircle} \right) \, - \; \overline{\rho} \! \left( \mathcal{A}_{\mdsmblkcircle} \right) \rule{0px}{17px} \right)$

Also the radius-vectors of any particle and its conjugate twin are symmetrically opposed. So $\overline{\rho} \! \left( \mathcal{A}_{\mdsmblkcircle} \right) \! =- \, \overline{\rho} \hspace{0.5pt} ( \overline{\mathcal{A}_{\mdsmblkcircle}} )$ and therefore

$\overline{\kappa}_{\mathsf{o}} (\boldsymbol{\gamma}) = \delta_{\theta} (\mathcal{A}_{\mdsmwhtcircle}) \; \dfrac{2\pi k_{\mathsf{F}}}{hc} \left( \, \overline{\rho} \! \left( \mathcal{A}_{\mdsmwhtcircle} \right) \, + \; \overline{\rho} \hspace{0.5pt} ( \overline{\mathcal{A}_{\mdsmblkcircle}} ) \rule{0px}{17px} \right)$

But the phase-components of a photon are anti-symmetric except for some negligibly small down-quarks. So $\mathcal{A}_{\mdsmwhtcircle} \! = \overline{\mathcal{A}_{\mdsmblkcircle}}$ and we can eliminate $\overline{\mathcal{A}_{\mdsmblkcircle}}$ to obtain

$\overline{\kappa}_{\mathsf{o}} (\boldsymbol{\gamma}) = \delta_{\theta} (\mathcal{A}_{\mdsmwhtcircle}) \; \dfrac{4\pi k_{\mathsf{F}}}{hc} \; \overline{\rho} \! \left( \mathcal{A}_{\mdsmwhtcircle} \right)$

A wavenumber is given by the norm of a wavevector. And $k_{\mathsf{F}}$ is positive, so the photon’s wavenumber is

$\kappa_{\mathsf{o}} (\boldsymbol{\gamma}) = \dfrac{4\pi k_{\mathsf{F}} }{hc} \left\| \, \overline{\rho}^{\, \mathcal{A}} \, \rule{0px}{11px} \right\|$

Photon Type	$\lambda_{\mathsf{o}}$ (m)
a gamma-ray	$\lesssim 10^{-12}$
an X-ray	$10^{-11} \sim 10^{-8}$
an ultraviolet photon	$\sim 10^{-8}$
a visible photon	$\sim 10^{-7}$
an infrared photon	$10^{-6} \sim 10^{-3}$
a microwave	$10^{-3} \sim 1$
a radio-wave	$1 \sim 10^{8}$

We can drop the subscript on $\mathcal{A}$ because both phase-components have the same norm. And recall that $W^{\!\! \mathcal{A}} \! = \! k_{\mathsf{F}} \| \, \overline{\rho}^{\mathcal{A}} \|$ is the work required to build one of these phase-components. So in energetic terms, the wavenumber of a free photon can be written as

$\kappa_{\mathsf{o}} (\boldsymbol{\gamma}) = \dfrac{\, 4\pi W^{\! \! \mathcal{A}}}{hc}$

Then the wavelength in vacuo for a free photon is given by

$\lambda_{\mathsf{o}} (\boldsymbol{\gamma}) \equiv \dfrac{2\pi}{\kappa_{\mathsf{o}}} = \dfrac{hc}{ \, 2W^{\! \! \mathcal{A}} }$

Anti-Photons

For EthnoPhysics, all photons have a conjugate twin just like other particles. So $\overline{\boldsymbol{\gamma}}$ is defined from $\boldsymbol{\gamma}$ by exchanging quarks and anti-quarks, while leaving the phase and other relationships unchanged. In a photon, $\Delta n \! = \! 0$ for all types of quarks except down-quarks. So particle characteristics that do not depend on down-quarks, are left unchanged when quarks are swapped. Furthermore, characteristics defined using the absolute-value of $\Delta n^{\mathsf{D}}$ also do not change. So usually, photons and anti-photons have the same characteristics as each other. Specifically

$\textsl{\textsf{J}} \left( \boldsymbol{\gamma} \right) = \textsl{\textsf{J}} \left( \overline{\boldsymbol{\gamma}} \right)$

and

$\mathcal{R}_{in} \! \left( \boldsymbol{\gamma} \right) = \mathcal{R}_{in} \! \left( \overline{\boldsymbol{\gamma}} \right)$

But $\Delta n^{\mathsf{D}} \! \left( \boldsymbol{\gamma} \right) = - \Delta n^{\mathsf{D}} \! \left( \overline{ \boldsymbol{\gamma}} \right) \! .$ And photons also have relative characteristics which may differ between $\overline{\boldsymbol{\gamma}}$ and $\boldsymbol{\gamma}$ depending on their juxtaposition with a frame of reference. For example, the wavevector $\overline{\kappa}_{\mathsf{o}}$ depends on down-quarks such that

$\overline{\kappa}_{\mathsf{o}} \left( \boldsymbol{\gamma} \right) = - \, \overline{\kappa}_{\mathsf{o}} \left( \overline{ \boldsymbol{\gamma}} \right)$

and the two photons have symmetrically opposed wavevectors. So photons and their conjugate twins are mostly the same as each other, but moving in opposite directions.

Photon types are distinguished by lateral and phase relationships, perhaps somewhat like the patterns in this beaded pattern from Indonesia. — Baby Carrier panel, Kayan people. Borneo 20th century, 31 x 27 cm. Photograph by D Dunlop.

Excited States

Some photons are not clearly particles or waves, they are somewhere in between. So here is a generalized way of describing phenomena. Everything from a radio-wave to the stolid lump of a ground-state proton may be included. Quark distributions are analyzed to generically define excited states as follows.

Consider some particle P described by a repetitive chain of events noted by $\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \; \ldots \; \right)$ where each repeated cycle is a bundle of quarks written as

$\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2}, \, \mathsf{q}_{3} \; \ldots \; \mathsf{q}_{i} \; \ldots \; \mathsf{q}_{N} \right)$

And let each quark be described by its phase $\delta_{\theta} \, .$ Use this phase to sort quarks into a pair of sets, $\mathsf{P}_{\! \mdsmwhtcircle}$ and $\mathsf{P}_{\! \mdsmblkcircle} \hspace{0.5pt} ,$ so that all quarks of the same phase are in the same set. Then $\mathsf{P}_{\! \mdsmwhtcircle}$ and $\mathsf{P}_{\! \mdsmblkcircle}$ are called phase components of P, and they are out of phase with each other. We write

$\mathsf{\Omega}^{\mathsf{P}} = \left\{ \mathsf{P}_{\! \mdsmblkcircle} \, , \, \mathsf{P}_{\! \mdsmwhtcircle} \rule{0px}{9px} \right\}$

and

$\delta_{\theta} \! \left( \mathsf{P}_{\! \mdsmblkcircle} \right) = - \, \delta_{\theta} \! \left( \mathsf{P}_{\! \mdsmwhtcircle} \right)$

Some sub-set of the quarks in $\mathsf{P}_{\! \mdsmwhtcircle}$ may be matched with quarks of the same type in $\mathsf{P}_{\! \mdsmblkcircle} .$ These quarks have phase symmetry with each other, so we use $\mathcal{S}_{\mdsmwhtcircle}$ and $\mathcal{S}_{\mdsmblkcircle}$ to symbolize these sub-sets.

A different sub-set of quarks might be matched with anti-quarks of the same type. These quarks have phase anti-symmetry with each other, so we note them as $\mathcal{A}_{\mdsmwhtcircle}$ and $\mathcal{A}_{\mdsmblkcircle} \, .$

Some quarks in $\mathsf{P}_{\! \mdsmwhtcircle}$ might not correspond with any quarks in $\mathsf{P}_{\! \mdsmblkcircle}$ and vice versa. But such lopsided possibilities seem to be superfluous so we do not consider them further.

Thus P is represented by the union of two entirely symmetric, and two purely anti-symmetric components. Quarks in the symmetric sets may vary independently of the quarks in the anti-symmetric sets. This is expressed mathematically as

$\mathsf{\Omega}^{\mathsf{P}} = \left\{ \left\{ \mathcal{S}_{\mdsmblkcircle} \, , \, \mathcal{A}_{\mdsmblkcircle} \right\}, \ \left\{ \mathcal{S}_{\mdsmwhtcircle} \, , \, \mathcal{A}_{\mdsmwhtcircle} \right\} \rule{0px}{11px} \right\}$

where

$\mathcal{S}_{\mdsmblkcircle} = \mathcal{S}_{\mdsmwhtcircle}$

$\mathcal{A}_{\mdsmblkcircle} = \overline{\mathcal{A} _{\mdsmwhtcircle} }$

$\delta_{\theta} \! \left( \mathcal{S}_{\mdsmblkcircle} \right) = - \, \delta_{\theta} \! \left( \mathcal{S}_{\mdsmwhtcircle} \right)$

and

$\delta_{\theta} \! \left( \mathcal{A}_{\mdsmblkcircle} \right) = - \, \delta_{\theta} \! \left( \mathcal{A}_{\mdsmwhtcircle} \right)$

This arrangement provides a general way of describing particles that are moved or excited by the absorption of additional quarks. P is defined as an excited particle, or said to be in an excited state if it contains at least one anti-symmetric pair of quarks

$\mathcal{A}_{\mdsmblkcircle} = \overline{\mathcal{A}_{\mdsmwhtcircle}} \ne \left\{ \varnothing \right\}$

These anti-symmetric quark-pairs may be due to the absorption of a photon. Or more generally, to interactions with any field quanta.

Ground States

We say that any particle P is in its ground state if it has perfect phase symmetry. Then P has no phase anti-symmetric quark-pairs

$\mathcal{A}_{\mdsmblkcircle} = \overline{ \mathcal{A}_{\mdsmwhtcircle} } = \left\{ \varnothing \right\}$

A photon cannot satisfy this condition because photons are defined from pairs of phase anti-symmetric quarks. So photons do not have ground-states, and the term is reserved for other kinds of particles.

When in its ground-state, all of a particle’s quark coefficients must be integer multiples of two because quarks are matched between two phase-components. That is why so many nuclear particles display quark-coefficients in patterns like 2-4-6 instead of 1-2-3. A ground-state also limits the free wavevector in vacuo of P. Recall that for any free particle

$\displaystyle \overline{\kappa}_{\mathsf{o}} = \dfrac{2\pi k_{\mathsf{F}}}{hc} \, \Delta \overline{\rho}$

where $\Delta \overline{\rho}$ is the quark flux vector and the positive constant $k_{\mathsf{F}}$ was discussed earlier. We can expand $\Delta \overline{\rho}$ into sums over individual quarks. And since the asymmetric sets are empty, we can evaluate this wavevector by considering the sum over just symmetric sets

$\begin{align*} \overline{\kappa }_{\mathsf{o}} &= \dfrac{2\pi k_{\mathsf{F}}}{hc} \left( \delta_{\theta}^{\, \mathcal{S}_{\mdsmwhtcircle} } \hspace{-5px} \sum_{\mathsf{q} \, \in \, \mathcal{S}_{\mdsmwhtcircle} } \! \overline{\rho}^{\, \mathsf{q}} \; + \; \delta_{\theta}^{\, \mathcal{S}_{\mdsmblkcircle} } \hspace{-5px} \sum_{\mathsf{q} \, \in \, \mathcal{S}_{\mdsmblkcircle} } \! \overline{\rho}^{\, \mathsf{q}} \right) \\ &= \dfrac{2\pi k_{\mathsf{F}}}{hc} \left( \delta_{\theta}^{ \mathcal{S}_{\mdsmwhtcircle} } \, \overline{\rho}^{ \mathcal{S}_{\mdsmwhtcircle} } \; + \; \delta_{\theta}^{ \mathcal{S}_{\mdsmblkcircle} } \, \overline{\rho}^{ \mathcal{S}_{\mdsmblkcircle} } \right) \\ &= \dfrac{2\pi k_{\mathsf{F}}}{hc} \left( \delta_{\theta}^{ \mathcal{S}_{\mdsmwhtcircle} } + \delta_{\theta}^{ \mathcal{S}_{\mdsmblkcircle} } \right) \, \overline{\rho}^{ \mathcal{S}_{\mdsmwhtcircle} } \ \ \text{\sf{because}} \ \ \overline{\rho}^{ \mathcal{S}_{\mdsmwhtcircle} } \! = \overline{\rho}^{ \mathcal{S}_{\mdsmblkcircle} } \\ &= (0, 0, 0) \ \ \ \ \text{\sf{because}} \ \ \ \ \delta_{\theta}^{\mathcal{S}_{\mdsmwhtcircle}} = - \, \delta_{\theta}^{\mathcal{S}_{\mdsmblkcircle}} \rule{0px}{16px} \end{align*}$

And so the wavenumber of a free particle in its ground-state is always zero

$\kappa_{\mathsf{o}} \equiv \left\| \; \overline{\kappa}_{\mathsf{o}} \, \right\| = \left\| \, (0, 0, 0) \, \vphantom{<meta charset="utf-8">\overline{\kappa}^{\mathsf{P}} } \right\| = 0$

Collections of Photons

Photons have been defined from quarks that are paired with their matching anti-quarks. For example quark models of X-rays employ hundreds of these $\mathsf{q \overline{q}}$ pairs. They are all ethereal and neutral. They are elusive, and difficult to count. So if there are a lot of these slippery quanta in a description, then it may be easier to treat them collectively, and call them all a field. Different field-types are defined from different quark distributions. For example, collections of photons are called electromagnetic fields.

Next we are going to talk specifically about hydrogen. But there are some far-reaching general ideas about fields and forces that branch out from here. So more detail about fields can be found on this page.

Fields

Fields are specified from quark pairs. Field quanta are specified for electric, magnetic, gravitational and other forces.

The Spectrum of Hydrogen

Photons that are absorbed or emitted by atomic hydrogen are collectively known as the spectrum of hydrogen. The collection is large and assorted. It includes many ultraviolet, visible and infrared photons. There are also plenty of microwaves and radio waves. These photons are mostly involved in the atomic and molecular interactions of everyday activity, not nuclear reactions. Energies are typically measured in (eV) rather than (MeV).

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 photograph of what he described mathematically. Can you see four lines?

The hydrogen spectrum is relevant because hydrogen atoms are theoretically accessible and experimental observations are remarkably exact. For example, in 2018 wavelength measurements of the $\mathrm{Lyman} \hspace{0.5pt} \alpha$ photon were reported¹Kramida, 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. with an uncertainty of less than two parts in $10^{15}.$

To mathematically describe the spectrum, we represent a photon $\boldsymbol{\gamma}$ with a repetitive chain of events noted by $\Psi \! \left( \boldsymbol{\gamma} \right) \! = \! ( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \ldots )$ where each repeated cycle is a finite bundle of quarks written as

$\mathsf{\Omega} \! \left( \boldsymbol{\gamma} \right) = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2}, \, \mathsf{q}_{3} \; \ldots \; \mathsf{q}_{i} \; \ldots \; \mathsf{q}_{N} \right)$

The hydrogen spectrum is described using a simple particle model for atomic hydrogen. This model involves two concentric spheres focused on the atomic proton. There is an inside sphere and an outside sphere. They are characterized by their radii $\mathcal{R}_{in}$ and $\mathcal{R}_{out}$ which constrain the atomic electron. We assume that all hydrogen spectrum photons interact with this electron in $\mathbf{H} .$ So the range of photon locations is also constrained. They are not free. So to describe their range we use the constrained wavenumber-in-vacuo

$\kappa_{\mathsf{o}} \! \left( \boldsymbol{\gamma} \right) = \dfrac{1}{2\pi} \left( \dfrac{1}{\mathcal{R}_{in}^{2}} - \dfrac{1}{\mathcal{R}_{out}^{2}} \right) \left\| \, \overline{\rho}^{\, \mathcal{A}} \right\|$

Writing-out the norm in this formula using terms from the terrestrial metric and the radial components of $\mathcal{A}$ gives

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

This expression can be simplified if $\boldsymbol{\gamma}$ is not a gamma-ray. For these lower energy photons, the coefficients of leptonic quarks must all be zero because the energy in even one leptonic quark is enough to yield a gamma-ray. So $\rho_{m} \! = \! \rho_{e} \! = 0 \hspace{0.5pt} .$ Also recall that $k_{zz} \! = \! 1 .$ Then we can write $\| \, \overline{\rho}^{\mathcal{A}} \| = \left| \rho_{z}^{\, \mathcal{A}} \right|$ and

So if $\boldsymbol{\gamma}$ is not a gamma-ray, then its wavenumber is proportional to the absolute-value of the central radius of a phase-component. This central-radius has been defined as

$\rho_{z} \equiv \, \mathcal{R}_{core} + \dfrac{ \Delta n^{\mathsf{U}} U^{\mathsf{U}} - \Delta n^{\mathsf{D}} U^{\mathsf{D}} }{ k_{\mathsf{F}} }$

We can simplify this expression because by convention $U^{\mathsf{D}} \! = 0 .$ Moreover, for the up-quarks $\Delta n^{\mathsf{U}}$ must be zero or else the photon would be a gamma-ray. So for hydrogen photons $\rho_{z} \! = \! \mathcal{R}_{core} \hspace{0.5pt} .$

Finally recall that $\mathcal{R}_{core}$ is defined by the ratio $H_{\! chem} / k_{\mathsf{F}}$ where $H_{chem}$ is the enthalpy due to any chemical quarks in the photon. Thus the constrained wavenumber-in-vacuo for photons in the hydrogen spectrum is given by

$\kappa_{\mathsf{o}} \! \left( \boldsymbol{\gamma} \right) = \dfrac{1}{ 2\pi k_{\mathsf{F}} } \left( \dfrac{1}{\mathcal{R}_{in}^{2}} - \dfrac{1}{\mathcal{R}_{out}^{2}} \right) \left| H_{chem}^{\mathcal{A}} \right|$

The Principal Quantum Number

Whenever a particle absorbs or emits a photon, its level of excitation changes too because photons contain anti-symmetric quark-pairs. We account for this relationship by defining the principal quantum number as

$\mathrm{n} \equiv \dfrac{ \; n^{\mathsf{d}} \; }{4}$

Here $n^{\mathsf{d}}$ marks the quantity of ordinary down quarks in a particle. Note that the italic letter $n$ is employed for quark coefficients whereas the upright font $\mathrm{n}$ is reserved for the principal quantum number. And remember that $n^{\mathsf{d}} \ge 0$ so the principal quantum number is never negative. Thus down-quarks retain an important descriptive role despite having almost no internal energy.

Next we use this principal quantum number to describe hydrogen spectrum photons. If some hydrogen atom $\mathbf{H}$ goes from an initial state $i ,$ to some final state $f ,$ by emitting a photon $\boldsymbol{\gamma}$ , we write

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

Then consider that the principal quantum number is conserved because down quarks are conserved so

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

But the photon’s principal quantum number can also be written as

$\begin{align*} {\rm{n}} \! \left( \boldsymbol{\gamma} \right) \; &\equiv \frac{ n^{\sf{d}}\left( \boldsymbol{\gamma} \right) }{4} \\ &= \left[ \frac{n^{\sf{d}}\left( \boldsymbol{\gamma} \right) }{8} + \frac{n^{\sf{d}}\left( \boldsymbol{\gamma} \right) }{8} \right] + \; 0 \\ &= \left[ \frac{n^{\sf{d}}\left( \boldsymbol{\gamma} \right) }{8} + \frac{n^{\sf{d}}\left( \boldsymbol{\gamma} \right) }{8} \right] + \left[ \frac{n^{\sf{\overline{d}}}\left( \boldsymbol{\gamma} \right) }{8} - \frac{n^{\sf{\overline{d}}}\left( \boldsymbol{\gamma} \right) }{8} \right] \\ &= \left( \frac{n^{\sf{d}}\left( \boldsymbol{\gamma} \right) }{8} + \frac{n^{\sf{\overline{d}}}\left( \boldsymbol{\gamma} \right) }{8} \right) - \left( \frac{n^{\sf{\overline{d}}}\left( \boldsymbol{\gamma} \right) }{8} - \frac{n^{\sf{d}}\left( \boldsymbol{\gamma} \right) }{8} \right) \\ &= \frac{N^{\sf{D}}\left( \boldsymbol{\gamma} \right) }{8} - \frac{\Delta n^{\sf{D}}\left( \boldsymbol{\gamma} \right) }{8} \end{align*}$

Then comparing the last two expressions for ${\rm{n}} \! \left( \boldsymbol{\gamma} \right)$ shows that the conservation law is always²The conservation law is also satisfied by the indicated quantum-numbers, plus a common constant. We reserve this more general format for the description of large atoms and their complex nuclei. satisfied when principle quantum numbers are related as

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

and

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

Excited states are a mix of symmetric and asymmetric patterns, like this bead panel from Kalimantan. — Baby Carrier panel, Kayan or Kenyah people. Borneo 20th century, 27 x 27 cm. Photograph by D Dunlop.

Photon Energy

Photons may be described by their size. This quantity is called the gross photonic energy. It is defined by

$\mathtt{E} \! \left( {\boldsymbol{\gamma}} \right) \, \equiv \, \dfrac{hc}{2 \pi} \kappa_{\mathsf{o}}$

where $\kappa_{\mathsf{o}}$ is the photon’s wavenumber. For hydrogen spectrum photons, $\kappa_{\mathsf{o}}$ has already been determined such that

$\mathtt{E} \! \left( {\boldsymbol{\gamma}} \right) = \dfrac{hc}{ 4\pi^{2} k_{\mathsf{F}} } \left( \dfrac{1}{\mathcal{R}_{in}^{2}} - \dfrac{1}{\mathcal{R}_{out}^{2}} \right) \left| H_{chem}^{\mathcal{A}} \right|$

Recall that the spherical radii have been defined in terms of quark coefficients as

$\mathcal{R}_{in} \equiv \sqrt{\dfrac{hc}{8 k_{\mathsf{F}}} } \; \dfrac{ \, \left| \Delta n^{\mathsf{D}} \right| \, }{8\pi}$

and

$\mathcal{R}_{out} \equiv \sqrt{ \dfrac{hc}{8 k_{\mathsf{F}}} } \dfrac{ \, N^{\mathsf{D}} \, }{8\pi}$

Then by susbstitution

$\mathtt{E} \! \left( {\boldsymbol{\gamma}} \right) = 2\left( \dfrac{64}{\left( \Delta n^{\mathsf{D}} \right)^{2} } - \dfrac{64}{ \left( N^{\mathsf{D}} \right)^{2} } \right) \left| H_{chem}^{\mathcal{A}} \right|$

And recall that a discussion about the principle quantum number determined that

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

and

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

So by substitution, the gross-energy of a hydrogen-photon depends on the excitation-levels of a hydrogen-atom such that

$\mathtt{E} \! \left( {\boldsymbol{\gamma}} \right) = 2\left( \dfrac{1}{{\rm{n}} ( {\mathbf{H}}_{f} )^{\, 2} } - \dfrac{1}{{\rm{n}} ( {\mathbf{H}}_{i} )^{\, 2} } \right) \left| H_{chem}^{\mathcal{A}} \right|$

This expression is used to make quark-models of photons in the hydrogen spectrum by specifying the distribution³Other quark distributions are used to model other atoms. That is, the Rydberg number of some atom $\mathbf{B}$ is represented by the chemical quarks in any photon $\boldsymbol{\gamma} (\mathbf{B})$ that is in the spectrum of $\mathbf{B} .$ These chemical quarks are distributed over $\mathcal{A}_{\mdsmwhtcircle}$ and $\mathcal{A}_{\mdsmblkcircle}$ the phase-components of $\boldsymbol{\gamma} (\mathbf{B}) .$ Let $\Delta n \! \left( \mathcal{A}_{\mdsmwhtcircle}^{\mathbf{B}} \right)$ note the quark coefficients for one of these phase-components. Then to be more exact

$\displaystyle \mathfrak{R}_{\mathbf{B}} = \dfrac{2}{hc} \left| \, \sum_{\zeta=11}^{16} \Delta n \! \left( \mathcal{A}_{\mdsmwhtcircle}^{\mathbf{B}} \right)^{\zeta} U^{\zeta} \, \right|$

of chemical quarks quarks in the photon such that

$\left| \, H_{chem}^{\mathcal{A}} \right| = \dfrac{hc}{2} \hspace{0.5pt} \mathfrak{R}_{\mathbf{H}}$

where $\mathfrak{R}_{\mathbf{H}}$ is the Rydberg constant of hydrogen. Using this condition to eliminate $H_{chem}$ then gives the gross energy of a photon in the hydrogen spectrum as

$\mathtt{E} \! \left( {\boldsymbol{\gamma}} \right) = hc \hspace{0.5pt} \mathfrak{R}_{\mathbf{H}} \hspace{-0.5pt} \left( \dfrac{1}{{\rm{n}} ( {\mathbf{H}}_{f} )^{\, 2} } - \dfrac{1}{{\rm{n}} ( {\mathbf{H}}_{i} )^{\, 2} } \right)$

Experimental Comparison

Measurements of photons report on their wavelength in vacuo which is noted by $\lambda _{\mathsf{o}} \hspace{0.5pt} .$ And photons cannot move slowly so $\kappa_{\mathsf{o}} \! \ne \! 0$ and $\lambda_{\mathsf{o}} \! = 2\pi \! / \! \kappa_{\mathsf{o}} .$ Also the gross energy of a photon has been defined above as $\mathtt{E} \! \left( {\boldsymbol{\gamma}} \right) \! \equiv h c \kappa_{\mathsf{o}} / 2 \pi .$ Then eliminating the wavenumber yields

$\mathtt{E} \! \left( {\boldsymbol{\gamma}} \right) = \dfrac{hc}{\lambda_{\mathsf{o}}}$

Finally we can eliminate the gross-energy from the last two equations to give the wavelength in terms of hydrogen’s principal quantum-numbers as

$\dfrac{1}{\lambda_{\mathsf{o}}} = \mathfrak{R}_{\mathbf{H}} \hspace{-0.5pt} \left( \dfrac{1}{{\rm{n}} ( {\mathbf{H}}_{f} )^{\, 2} } - \dfrac{1}{{\rm{n}} ( {\mathbf{H}}_{i} )^{\, 2} } \right)$

The description of hydrogen was first expressed this way by Johannes Rydberg in 1888. Next we present some quark models of photons and determine their wavelengths using his formula. Comparisons are made with experimental observations.⁴Kramida, 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

Balmer Series

Paschen Series

Brackett Series

Other Series

In the models above, agreement with experiment is good to a few parts in a million. This would be great in most scientific endeavours, but it is mediocre for atomic spectroscopy. Nonetheless it is more than good enough to distinguish between competing quark-models. So these specific quark combinations are confirmed to define each photon. Then later, after a discussion of atomic hydrogen, we use these photon models to make a more accurate description. For more detail, please see the page about fine structure in the hydrogen spectrum.

X-Rays

The following quark-models of X-rays use only rotating quarks and electrochemical quarks. These sorts of arrangements are called atomic X-ray models. It is also possible to model X-rays using leptonic quarks, but we reserve that design for the X-rays coming from nuclear-decay.

Atomic X-rays are typically produced by bombarding some metallic atom $\mathbf{A}$ , with electrons that have been accelerated to high speeds by absorbing photons with vast quantities of electrochemical quarks. So the total number of quarks used for these models is not constrained, and $N_{ \! \mathsf{q}}$ soars into the thousands. Pauli’s principle is not violated because all photons presumably have different phase angles.

As shown earlier the X-ray’s wavelength is given by $\lambda_{\mathsf{o}} \! = hc / 2 W^{\! \! \mathcal{A}}$ where $W^{\! \! \mathcal{A}}$ notes the work required to assemble one phase component. This wavelength is related to the gross energy by

$\mathtt{E} \! \left( \boldsymbol{\gamma} \right) = \dfrac{hc}{\lambda_{\mathsf{o}}} = 2W^{\! \! \mathcal{A}}$

Atomic X-rays have distinct peaks in their observed energies. This phenomenon is linked to a few specific combinations of rotating quarks. Like other photons, the distribution of rotating quarks is described by the principal quantum number $\mathrm{n}$ , and the total angular momentum quantum number $\textsl{\textsf{J}} .$ These quantities are expressed using the spectroscopic notation for X-rays. They are shown in the adjoining table of X-rays obtained by bombarding zinc, copper, iron and calcium.

These models are demonstrative, rather than definitive. The very close agreement with observed values⁵R.D. Deslattes, E.G. Kessler Jr., P. Indelicato, L. de Billy, E. Lindroth, J. Anton, J.S. Coursey, D.J. Schwab, C. Chang, R. Sukumar, K. Olsen, and R.A. Dragoset (2005), X-ray Transition Energies Database (version 1.2). National Institute of Standards and Technology, Gaithersburg, MD, USA. is easy to obtain because the total number of quarks is huge. Many similar models with slightly different numbers of electrochemical quarks also fit within the limits of observation. So these models⁶Quark models of X-rays could be improved by making some additional requirement for equilibrium. Then, electrochemical quark distributions could be further constrained and explained. show how descriptions that employ a large number of quarks can begin to look like continuous phenomena.

Gamma Rays

Here are some quark models of gamma-rays produced during the decay of radioactive atomic nuclei. Their gross energy has been calculated and compared with experimental observation.⁷S.Y.F. Chu, L.P. Ekström and R.B. Firestone, Table of Radioactive Isotopes (version 1999-02-28). Lund University, Sweden and Lawrence Berkeley National Laboratory, USA. Results are shown in the table below, and there is close agreement. But this consistency is easy to obtain because the total number of quarks is so large. Similar models with slightly different numbers of rotating quarks can also fit inside experimental uncertainty. So like the X-rays, these gamma-ray models show how using large numbers of quarks can approach traditional presumptions about continuity.

Pointy Nuclear Particles

Nuclear particles are described. Their lifetimes and widths are defined. Various classification methods are linked.

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References
1	Kramida, 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.
2	The conservation law is also satisfied by the indicated quantum-numbers, plus a common constant. We reserve this more general format for the description of large atoms and their complex nuclei.
3	Other quark distributions are used to model other atoms. That is, the Rydberg number of some atom $\mathbf{B}$ is represented by the chemical quarks in any photon $\boldsymbol{\gamma} (\mathbf{B})$ that is in the spectrum of $\mathbf{B} .$ These chemical quarks are distributed over $\mathcal{A}_{\mdsmwhtcircle}$ and $\mathcal{A}_{\mdsmblkcircle}$ the phase-components of $\boldsymbol{\gamma} (\mathbf{B}) .$ Let $\Delta n \! \left( \mathcal{A}_{\mdsmwhtcircle}^{\mathbf{B}} \right)$ note the quark coefficients for one of these phase-components. Then to be more exact $\displaystyle \mathfrak{R}_{\mathbf{B}} = \dfrac{2}{hc} \left\| \, \sum_{\zeta=11}^{16} \Delta n \! \left( \mathcal{A}_{\mdsmwhtcircle}^{\mathbf{B}} \right)^{\zeta} U^{\zeta} \, \right\|$
4	Kramida, 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.
5	R.D. Deslattes, E.G. Kessler Jr., P. Indelicato, L. de Billy, E. Lindroth, J. Anton, J.S. Coursey, D.J. Schwab, C. Chang, R. Sukumar, K. Olsen, and R.A. Dragoset (2005), X-ray Transition Energies Database (version 1.2). National Institute of Standards and Technology, Gaithersburg, MD, USA.
6	Quark models of X-rays could be improved by making some additional requirement for equilibrium. Then, electrochemical quark distributions could be further constrained and explained.
7	S.Y.F. Chu, L.P. Ekström and R.B. Firestone, Table of Radioactive Isotopes (version 1999-02-28). Lund University, Sweden and Lawrence Berkeley National Laboratory, USA.