Hydrogen Spectrum – Fine Structure
Here is a description of some finer details in the spectrum of hydrogen. It is based on an earlier account of atomic energy levels that are noted by $E$. Let an atom of hydrogen ${\mathbf{H}}$ change from some initial state $i$, to some final state $f$, by emitting a photon $\gamma$. This is usually written as ${\mathbf{H}} _{i} \to {\mathbf{H}} _{f} + \gamma$. And then, by the conservation of energy

$E(\gamma) = E \left({\mathbf{H}}_{i} \right) - E \left({\mathbf{H}}_{f} \right) = - \Delta E({\mathbf{H}})$

This approach works well, to perhaps a few parts in a billion. But experimental science marches on, and now observations of the ${\mathrm{Lyman}} \, \alpha \,$ photons are being reported with a precision of two parts in $10^{15}$. We have to consider ever more subtle possibilities just to stay apace. So for WikiMechanics, we extend analysis to considering what causes a hydrogen atom to change state, and what other effects may be impressed upon the atom in addition to kicking-out a photon. We account for the fine structure of hydrogen using a more detailed thermodynamic process that explicitly includes an interaction with some kind of force field, $\mathscr{F}$. Thus

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

And we also consider that this force may do some work, $W$, on the atom by changing its shape. Then for $\Delta \! W \equiv W ( \mathbf{H}_{f} ) - W ( \mathbf{H}_{i} )$ we write

$E(\gamma) = - \Delta E({\mathbf{H}}) - \Delta \! W ({\mathbf{H}})$

## Atomic Transition Forces

Here is a repertoire of little kicks and torques used to get hydrogen to jump from state to state. These field quanta are generically noted by $\mathscr{F}$. There are no electrochemical quarks in these fields. Energies are typically stated in micro electron-volts. The most important characteristic for classifying these fields is their helicity, $\delta_{z} \,$. The atomic quantum numbers $\, \ell$ and $s$ are also relevant. The letter $Ł$ notes a frequently used collection of quarks that is called the Lamb quantum.

Recall that the internal energy of quark $\sf{z}$ is noted by $U ^{\sf{z}}$. Then a conjugate difference $\; \Delta \hspace{-2px} U^{\sf{Z}}$, and a conjugate mean $\tilde{U}^{\sf{Z}}$, are used to describe the relationship between quarks and anti-quarks

 \begin{align} \Delta \hspace{-2px} U^{\sf{Z}} \equiv \frac{U^{\sf{\overline{z}}} - \, U^{\sf{z}}}{2} \end{align} and \begin{align} \tilde{U}^{\sf{Z}} \equiv \frac{U^{\sf{\overline{z}}} + \, U^{\sf{z}}}{2} \end{align}

Usually we assume that the internal-energies of down-quarks are small enough to be completely negligible. Then we write $\Delta \hspace{-2px} U^{\sf{D}} \! =0$ and $\tilde{U}^{\sf{D}} \! =0$. But actually, the conjugate mean for down quarks is found to be $\; \tilde{U}^{\sf{D}}= -27$ (µeV) which is very small, but not zero. Also, we generally make an assumption of conjugate symmetry so that $\Delta \hspace{-2px} U^{\sf{Z}} \! =0$ and $\tilde{U}^{\sf{Z}} \! = U^{\sf{\overline{z}}} = U^{\sf{z}}$. But again, these assumptions are not good enough for hydrogen where the tiny differences shown in the accompanying table provide a more accurate description of fine structure in the atomic spectrum. The energy of $\mathscr{F}$ is also a function of its spin angular momentum quantum number, $s \,$. The constant of proportionality is $\; k_{\sf{spin}} = -91$ (µeV), so the dependence is slight. But for field quanta, quarks are paired with anti-quarks so phenomena like mass and charge are always absent. Then smaller effects are relevant, and gathering them all together gives an expression for the change in energy due to an interaction. Let $N$ note the quark coefficients of $\mathscr{F}$, then

\begin{align} U ( \mathscr{F} ) \; \equiv \; s k_{\sf{spin}} \; + \; N^{\sf{D}} \hspace{1px} \tilde{U}^{\sf{D}} \; + \; \sum_{\zeta =1}^{16} N^{\zeta} \Delta \hspace{-2px} U^{\zeta} \end{align}

## Hydrogen Transitions

WikiMechanics understands the fine structure of the hydrogen spectrum by analyzing the thermodynamic processes that lead to the production of photons. For example, the ${\mathrm{Lyman}} \hspace{2px} {\mathrm{\delta}} \,$ quartet of photons supposedly result from the four different interactions

\begin{align} 5{\mathrm{S}} \, + \, {\mathscr{F}} ^{\large{⤸}} \, &\to 1{\mathrm{S}} + \gamma \\ \\ 5{\mathrm{P}}_{1/2} \, + \, {\mathscr{F}}^{\large{⇊}} \, &\to 1{\mathrm{S}} + \gamma \\ \\ 5{\mathrm{P}}_{3/2} \, + \, Ł \, &\to 1{\mathrm{S}} + \gamma \\ \\ 5{\mathrm{D}}_{5/2} \, + \, {\mathsf{2}} {\mathscr{F}}^{\large{↑}} \, &\to 1{\mathrm{S}} + \gamma \end{align}

Each of these processes exhibit a change in the principal quantum number of $\Delta {\mathrm{n}} \equiv {\mathrm{n}}_{f} - {\mathrm{n}}_{i} = -4$. This gross effect characterizes the multiplet, along with a transition mean of ${\mathrm{\tilde{n}}} \equiv \left( {\mathrm{n}}_{f} + {\mathrm{n}}_{i} \right) /2 =3 \;$, and a transition volume of $\tilde{V} \equiv \left( a_{\mathsf{o}} {\mathrm{ñ}} \right) ^{3}$ where $a_{\mathsf{o}}$ is the Bohr radius. Changes in other atomic quantum numbers like $\Delta j$ and $\Delta \ell$ then describe smaller variations in the resulting photon. Different sorts of transitions are associated with different forces. For example ${\mathscr{F}}^{\large{⇊}}$, the 'double down' force mentioned above, also accounts for the following processes which are generically noted by $1{\mathrm{S}} \! - \! X{\mathrm{P}}_{1/2}$

 \begin{align} 3{\mathrm{P}}_{1/2} \! + {\mathscr{F}}^{\large{⇊}} \! \to \! 1{\mathrm{S}} + \gamma \end{align} \begin{align} 4{\mathrm{P}}_{1/2} \! + {\mathscr{F}}^{\large{⇊}} \! \to \! 1{\mathrm{S}} + \gamma \end{align} \begin{align} 6{\mathrm{P}}_{1/2} \! + {\mathscr{F}}^{\large{⇊}} \! \to \! 1{\mathrm{S}} + \gamma \end{align} \begin{align} 7{\mathrm{P}}_{1/2} \! + {\mathscr{F}}^{\large{⇊}} \! \to \! 1{\mathrm{S}} + \gamma \end{align} …

There are two transitions that can occur without any precipitating force, while still conserving momenta. They are $3{\mathrm{S}} \! - \! 3{\mathrm{D}}_{3/2}$ and $4{\mathrm{S}} \! - \! 4{\mathrm{D}}_{3/2}$. These are the simplest interactions, but in general, quark models of photons may be much more complicated. And production processess could possibly involve endlessly more complex loops and wiggles. So to make simple models, we impose a boundary condition: Out of all possible interactions, the only transitions that we actually attend to are described by a few specific values of $\xi$, the transition type where

$\xi \equiv \Delta j + \Delta \ell + \Delta N^{Ⓛ}$

Recall that $N^{Ⓛ}$ notes the quantity of levo quarks in an atomic state. For hydrogen $\xi$ takes on some integer values between -8 and +5. For example, consider the following transitions which are generically noted as $1{\mathrm{S}} \! - \! {\mathit{X}}{\mathrm{S}}$

 \begin{align} 2{\mathrm{S}} + {\mathscr{F}} ^{\large{⤸}} \! \to \! 1{\mathrm{S}} + \gamma \end{align} \begin{align} 7{\mathrm{S}} + {\mathscr{F}} ^{\large{⤸}} \! \to \! 1{\mathrm{S}} + \gamma \end{align} \begin{align} 9{\mathrm{S}} + {\mathscr{F}} ^{\large{⤸}} \! \to \! 1{\mathrm{S}} + \gamma \end{align} …

Mathematically, the entire class is identified by $\xi =+4$. And here is another series, generically written as $1{\mathrm{S}} \! - \! {\mathit{X}}{\mathrm{D}}_{5/2}$ and mathematically described by $\xi =-4$

 $5{\mathrm{D}}_{5/2} \! + {\mathsf{2}} {\mathscr{F}}^{\large{↑}} \! \to \! 1{\mathrm{S}} + \gamma$ $7{\mathrm{D}}_{5/2} \! + {\mathsf{2}} {\mathscr{F}}^{\large{↑}} \! \to \! 1{\mathrm{S}} + \gamma$ $9{\mathrm{D}}_{5/2} \! + {\mathsf{2}} {\mathscr{F}}^{\large{↑}} \! \to \! 1{\mathrm{S}} + \gamma$ …

There is one transition that is different from the others considered so far; changes between the ground-states $1{\mathrm{S}}^{\large{↑}}$ and $1{\mathrm{S}}$. In addition to having $\xi =0$, this change is mathematically described using

\begin{align} \Delta _{\mathsf{Z}} \equiv \frac{ \delta_{z} ( {\mathbf{H}} _{f} ) + \delta_{z} ( {\mathbf{H}} _{i} ) }{-2} \end{align}

For most transitions $\Delta _{\mathsf{Z}} =1$. A signed unit volume is specified by $\hat{V} \equiv \left( a_{\mathsf{o}} \Delta _{\mathsf{Z}} \right)^{3}$. All the foregoing interactions yield a photon and nothing else, no debris. They may also do some work $W$, on the atom by changing its shape. We use $\xi$ to describe $\Delta \! W$ by assessing changes to the norm of the radius vector; $\Delta \rho \equiv \left\| \, \overline{\rho}_{f} \right\| - \left\| \, \overline{\rho}_{i} \right\|$. Field quanta are usually $\mathsf{q \bar{q} }$ pairs that are presumed to have perfect conjugate symmetry. For this ideal case $\Delta \rho =0$. But in the finely-balanced mechanical system of a hydrogen atom, we notice a small change described by

\begin{align} \Delta \rho = {\mathrm{h}} \hspace{1px} {\mathrm{ñ}}^{\hspace{1px} 3\epsilon} \! + {\mathrm{b}} \end{align}

where $\epsilon \equiv (-1)^{\xi}$. The distances given by ${\mathrm{h}}(\xi)$ and ${\mathrm{b}}(\xi)$ characterize each transition-type. They are tiny, effects are measured in micro electronvolts as shown in the adjoining table. The change of shape is combined with the change of internal-energy caused by absorbing $\mathscr{F}$ to define the transition density as

\begin{align} \tilde{\varrho} \equiv \frac{ U({\mathscr{F}}) + k_{\mathsf{F}} \Delta \rho }{\tilde{V}} \end{align}

where the constant $k_{\sf{F}}$ was introduced earlier. Then the work done by $\mathscr{F}$ on $\mathbf{H}$ is given by

$\Delta \! W = \tilde{\varrho} \hspace{1px} \hat{V}$

## Quark Models

Quarks are indestructible. So models of the fine-structure photons can be obtained by combining the quark coefficients of their components, as prescribed by their thermodynamic production processes. Coefficients of the excited $\mathbf{H}$-states are known, and coefficients for the transition forces $\mathscr{F}$ are given above. Adding and subtracting them together automatically conserves charge, momentum, etc. Note that some photons contain the same quarks, but nonetheless have different wavelengths. This is because photon energies also depend on how quarks are brought together and internally arranged, as described by $W$. Click to unfold more detail below:

## Wavelengths

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

\begin{align} \lambda_{\sf{o}} = \frac{hc}{\Delta E + \Delta \! W } \end{align}

Values of $E({\mathbf{H}}_{i})$ and $E({\mathbf{H}}_{f})$ can be obtained from the description of atomic hydrogen. And, as shown above, shape changes are described by $\Delta \! W = \tilde{\varrho} \hspace{1px} \hat{V}$. The wavelengths calculated from these formulae are compared with experimental observations1,2,3 and shown below. Results that are outside of experimental uncertainty are marked by X. There are two of them, $3{\mathrm{P}}_{1/2} \! - \! 3{\mathrm{S}}$ and $4{\mathrm{P}}_{1/2} \! - \! 4{\mathrm{P}}_{3/2}$. For more detail about these calculations, please see the spreadsheets in the wiki-files stored here.

Related WikiMechanics articles.

page revision: 677, last edited: 25 Mar 2020 19:12