Wilson-Sommerfeld Application: Atomic Structure
![[url=https://pixabay.com/en/atom-physics-atomic-nucleus-neutron-618331/]"Atom"[/url] by Geralt is in the [url=http://creativecommons.org/publicdomain/zero/1.0/]Public Domain, CC0[/url]](https://www.geogebra.org/resource/BvTRvZdw/oW4lCwFcbsJUELcW/material-BvTRvZdw.png)
A Brief History
In 1862 Anders Jonas Ångström identified three of the four visible lines in the emission spectrum of Hydrogen. He cataloged the sun's (solar) spectrum in 1868, and then measured all four lines in hydrogen in 1871. Ångström measured the wavelengths of these emissions in terms of units named after himself where 1Å=10-10m. Both the discrete nature of the emission lines as well as their values remained a mystery into the early quantum age.
Niels Bohr provided an explanation for the four emission lines in 1913 in a model that incorporated the quantum hypothesis of Max Planck. Bohr's model, however, treated electrons like little planets orbiting the nucleus. In 1915, Bohr's model was improved by Sommerfeld by using the WS quantization condition. The electrons were viewed as standing waves. You will certainly see the standard Bohr model in chemistry classes, so we will discuss the improved WS approach instead. If you wish to read further on the historical development of early atomic models, there is a nice article below.
Extended History of the Sommerfeld Model
Bohr-Sommerfeld Atomic Structure
The application of the Wilson-Sommerfeld quantization criterion to atomic structure is generally known as the Bohr-Sommerfeld atomic model since it was an extension of Bohr's model proposed by Sommerfeld.
As mentioned in the linked paper above, Sommerfeld's model extended the Bohr model such that it allowed electron orbits that were not only spherical (as in the Bohr model), and among those non-spherical levels showed variations in energy - something that every chemistry student knows today. Further, while the Bohr model only included a single quantum number which we call the principal quantum number, Sommerfeld's model included three quantum numbers, and the fourth number, which we attribute to the intrinsic electron spin, was suggested later by Sommerfeld himself.
The whole model relies on application of the WS quantization condition, but the resulting integrals are very difficult to solve. For this reason we will only derive the spherical orbitals which become identical to the results of the more commonly discussed Bohr model.
Spherically Symmetric Orbitals and the BS Model
There are exactly two things we need to use to solve for the radii and energy levels associated with electron orbitals in hydrogen: The WS quantization condition and centripetal force from first semester which will be equal to the electrostatic attraction between the nucleus and the electron.
The WS condition is, as before . Here we should understand the momentum to be the orbiting electron's momentum and the path to be the circular path that describes the location of the electron standing wave. If you recall the standing waves from the last section, these will look the same except that they will circulate around the nucleus.
If we assume a constant radius (the only case we'll do the math for, while others exist), then the potential energy will be constant, which leads to a constant momentum such that it may be removed from the integral. The math leads us to , or . While this is not the conclusion of our calculation, it does already allow us to relate the momentum of the electron to its distance from the nucleus.
The only other calculation we need to do is to relate the centripetal force to the electrostatic attraction between the proton in the nucleus to the electron, . The electrostatic force of attraction is given by . The centripetal force is . Equating the two forces give us . The constant Combining this equation with the expression for momentum above gives the expression for the allowable radii of the electron wave:
.
The ground state radius of the electron in hydrogen is . This value is referred to as the Bohr radius, and is often given the symbol .Energy States
To find the energies associated with these quantum states requires us to use a fact that relates to orbital motion in stable systems like this one and like planetary orbits. It is always true of such systems that the total system energy E is equal to the negative of the kinetic energy K. Or E=-K. In case you are wondering how this comes about, you might consider taking the time to calculate for yourself the fact that in an orbit, U=-2K also. This means that E=K+U=K+(-2K)=-K.
In our present system, since , we can use , and with slight modification (negate the expression, divide by 2 and multiply by r), we have the total energy written as . Plugging in the allowable radii from above yields the quantized energy levels:
The integer represented by the natural number sequence serves as a quantum number of the atom. In this context it is the principal quantum number. The other quantum numbers that you expect from chemistry arise when we consider other shapes of paths and when we consider the intrinsic spin of the electron which does not arise in any straightforward way from WS quantization.
Take note that while the energy levels in the 1D electron box led to quadratically increasing energy states as the quantum number grew, here we see a very different behavior. The first thing to note is that all states are negative energy states! Negative energies are associated with bound states - or ones in which the electron is not really free to roam. While that is true both in the atomic system and in the 1D box, the difference is that the box is artificial in the description of its potential energy and so we can't attribute a finite, negative energy to the ground state. The atomic system here is described by the real potential energy function, which by definition from electrostatics was negative. Thus the calculation automatically gives rise to negative bound state energies.
Notice that freedom for the electron comes with a quantum state of . The energy to go from to is the ionization energy for hydrogen. Calculating this is trivial since and so the result is just the negative of the ground state energy since If you plug in values for the constants, you will see this leads to the well-known value of 13.6eV for the ionization energy of hydrogen.
Since the constants together equal 13.6eV, it is also common to see the energy equation written as
The other big difference between these results and the 1D electron in a box is that the energy spacing between quantum states in the atomic system gets smaller and smaller in higher quantum states rather than larger and larger. So the biggest gap in energy is between N=1 and N=2. The gap from N=2 to N=3 is smaller, and the trend continues all the way up.
Electron Standing Waves in the Atom
The standing waves associated with these electron states can be seen in the interactive diagram below. The dotted path is a circle of constant radius. If you straighten out the standing waves, as seen in the diagram, they look just like solutions to waves on strings. The only difference is that the phase at the beginning and the end of the wave just needs to match.
Electron Standing Waves in Atom
Absorption and Emission Lines
Once these energy levels are known, we can calculate the expected absorption and emission lines for hydrogen using the same procedure as with the 1D electron in a box in the last section. The absorption of light serves to promote the system into a higher energy state and the emission of an electron occurs when the system's energy state is demoted. Therefore the allowable energies of photons always correspond to differences in atomic energy levels.
Because of the history involved in the investigation of the emission lines of hydrogen, there are names associated with sequences of emissions ending on certain quantum numbers. For instance, when a hydrogen atom starts in an excited state and ends in the first excited state (N=2), the series of lines is known as the Balmer series. These are the most widely discussed of the hydrogen emission lines because they fall in wavelengths visible to the human eye.
Here are the names given to the series leading to their respective final quantum states:
Finding the emitted wavelengths making up the Paschen series means calculating, for instance , where N>3. The product of can be used to facilitate quick calculation where energy levels are known in electron volts and wavelengths in nanometers are desired.
Nfinal | Name of Series |
| 1 | Lyman |
| 2 | Balmer |
| 3 | Paschen |
| 4 | Brackett |
| 5 | Pfund |