Fine Structure in the Hydrogen Spectrum

Here is a detailed description of some fine structure in the hydrogen spectrum. It is based on an account of atomic energy levels that are noted by . Let an atom of hydrogen change from some initial state , to some final state , by emitting a photon This is usually written as And then, for we write This approach works to perhaps a few parts in a billion. But experimental science marches on, and now observations of the photons are being reported with a precision of two parts in We have to consider ever more subtle possibilities just to stay apace. So for EthnoPhysics, 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 emitting 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 field . Thus we write

And we also consider that the force due to absorbing these field quanta may do some work on the atom by changing its shape. Then for we have

Subatomic Quanta

Here is a repertoire of little kicks and torques used to get hydrogen to jump from state to state. These subatomic quanta are sometimes called fields, and are often noted using the letter There are no electrochemical quarks in these quanta. Energies are typically stated in micro electron-volts. The most important characteristic for classifying these particles is their helicity . The atomic quantum numbers and are also relevant. The letter notes a frequently used collection of quarks that is called the Lamb quantum.

Recall that the internal energy of quark is noted by . Then a conjugate difference and a conjugate mean describe the relationship between quarks and anti-quarks

Usually we assume that the internal-energies of down-quarks are small enough to be completely negligible. Then we write . But the conjugate mean for down quarks, as found from hydrogen observations, is small but not zero. Currently, our best estimate is (µeV).

Also, we generally make an assumption of conjugate symmetry so that and But again, these assumptions are not good enough for hydrogen where the differences shown in the accompanying table provide a more accurate description of fine structure in the spectrum.

The energy of is also a function of its spin angular momentum number, . The constant of proportionality is (µeV), so the dependence is slight. But for field quanta, quarks and anti-quarks are paired, mass and charge are always absent, and smaller effects are relevant. Let note the quark coefficients of . Then the internal energy is obtained from a sum of these small terms

Atomic Transitions

EthnoPhysics understands fine structure in the hydrogen spectrum by analyzing the atomic transitions that lead to the production of photons. For example, the quartet of photons supposedly result from the four different atomic transitions

Each of these processes exhibit a change in the principal quantum number of This gross effect characterizes the multiplet, along with a transition mean of and a transition volume of where is the Bohr radius. Any spin-flips are described by

And a directed unit volume is specified as . Changes in other atomic quantum numbers like and then describe smaller variations in the resulting photon.

Different atomic transitions are associated with different forces. For example consider , the double-down force shown in the table above. It accounts for the following processes which are generically noted by

There are two atomic transitions that occur without any precipitating force. They are and These are the simplest processes, but in general, quark models of photons may be much more complicated. And interactions 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 , the transition type where

Recall that notes the quantity of levo quarks in an atomic state. For hydrogen takes on some integer values between -8 and +5. For example, consider the following transitions which are generically noted as

Mathematically, the entire class is identified by . And here is another series, generically written as and mathematically described by

All the foregoing interactions yield a photon and nothing else, no debris. They may also do some work on the atom by changing its shape. We use to describe by assessing changes to the norm of the radius vector which is written as

Field quanta are pairs that are usually presumed to have perfect conjugate symmetry. For this ideal case But in the finely-balanced mechanical system of a hydrogen atom, we notice a small change described by

where . The transition characteristic is named the shell thickness, and is called the shear deformation. These distances are tiny, effects are measured in micro electronvolts as shown in the adjoining table.¹These adjustable parameters are supplementary fine-structure constants. Since there are 14 of them, data compression is limited. But they enable a systematic description that is succinct. The change of shape is combined with the change of internal-energy caused by absorbing to define the transition energy density as

where the constant was introduced earlier. Then the work done by on is given by

Hydrogen Spectrum Photons

Quark coefficients for the excited states of hydrogen are known. And coefficients for the transition forces that cause hydrogen to jump from state to state are specified in the table above. Then since quarks are conserved we can make photon models just by adding and subtracting quark coefficients as prescribed by the following processes which are generically written as This method automatically conserves charge, momentum, etc.

Lyman Series

Balmer Series

Other Multiplets

Fine Structure Wavelengths

Measurements of photons report on their wavelength, which is related to their energy by . Wavelengths may also depend on a photon’s surroundings. Then the symbol is used to indicate a wavelength where environmental effects are negligible. We assume this to write

Values of and are obtained from the description of atomic hydrogen. And, as shown above, shape changes are described by

The wavelengths calculated from the foregoing formulae are compared with experimental observations²Helmut 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 . Institute of Electrical and Electronics Engineers – Transactions on Instrumentation and Measurement, Volume IM-19, Number 4, November 1970.^{, 3}Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2015). NIST Atomic Spectra Database (version. 5.3). National Institute of Standards and Technology, Gaithersburg, MD, USA. and shown below. Results marked by X are outside of experimental uncertainty. For more detail about these calculations, please see the spreadsheets in the files stored here.

Lyman Wavelengths

Balmer Wavelengths

Other Transitions

The transition has been assigned by Kramida et al. And all adjustable parameters have been tested to represent that datum. But still, the observation does not completely agree with the theory presented above. However, it surprisingly fits perfectly with calculations for the transition, even within an experimental uncertainty that is 4 or 5 times tighter. For additional detail about these calculations, please see the Atoms and Photons spreadsheet.