They point every which way

kw: musings, physics, electrons

This is my 1,000^th post! Recent reading about the behavior of electrons in atoms reminded me of work I did many years ago (as well as triggering my prior post). My first paying job (besides newspaperboy) was in spectroscopy. Though I worked primarily in infrared spectroscopy of molecular species, I did some visible and UV work, and got to know that field a little. Years before I took the relevant physics courses, I became familiar with the S P D F designations for the various series of atomic spectral lines.

It is amazing what early spectroscopy workers were able to discern about electron orbitals from the lines they examined on photographic film. SPDF is the earliest indication of substructure in the electron "orbits". These four letters refer to the terms "sharp", "principle", "diffuse", and "fundamental", as explained by Grant R. Fowles in Introduction to Modern Optics (1989):

The sharp and diffuse series are so named because of the appearance of the spectral lines.

The principal series is the most intense in emission...

...the frequecies [ratios] of the lines of this [fundamental] series are very nearly the same as those of the corresponding series in hydrogen. This is the reason for the name "fundamental."

[The bracketed words are mine.]

This image, from the article "Light and the Modern Atom" at Chemistry Land, shows the original designations of four series of spectral lines as seen in the absorption spectrum of sodium at high resolution. The two dark bands in the warmish yellow portion of the spectrum, one of which is designated as an example of a Diffuse line, are the diagnostic sodium D lines near 589 nm wavelength.

What is really happening here? The four spectroscopic series have a relationship to the quantized energy levels surrounding the nucleus, which allow electrons to have certain electrical and angular energies, but not others. There are shells and subshells in which electrons can reside, both in spectroscopic terms and as more and more electrons are gathered by heavier and heavier nuclei. However, the spectroscopic energy levels do not correspond exactly to electron subshell orbitals, so more modern nomenclature relies on quantum numbers. Shells are numbered n=1,2,3… and subshells as l=0,1,2… and there are other quantum numbers. Detailed quantum designations are a side issue to the point of this post, so please refer to the Wikipedia article Electron Configuration for a thorough discussion.

Now that we can use computers to calculate the probabilities of electron locations quickly, and to visualize them, I find the images we can create show most compellingly what the spectra are telling us. For example, the S series is sharp because the subshell that generates it is the simplest. It has spherical symmetry in all cases, and can be occupied by either one or two electrons, no more. The D series is more diffuse (and indeed, P and F lines are also broader than S lines) because the shell has a substructure, in which pairs of electrons occupy a shape that is distinguished not only by distance from the nucleus, but angle also. A few images will illustrate.

These images of S orbitals (quantum number l=0; that is an "ell") for the L, M, and N shells (n=2,3,4) show the spheres cut in half so you can see the inner shells of alternating phase that nestle inside. The boundary of the colored area is not a hard edge. It represents the 3-D contour of 90% probability that the electron is "located" within it. For any S subshell, there is both an inner sphere and an outer sphere that together define the 90% probability region. However, any electron can actually be anywhere in space with some probability, but that statistic is very close to zero for much larger distances or for locations much closer to the nucleus.

A characteristic of the other subshells that allures me is that a filled subshell has spherical symmetry. A particular orbital within such a subshell is, however, very far from spherically symmetric.

This illustration of the orbital shapes for the P subshell (l=1) shows the shapes of the 90% probability regions of these orbitals for L, M, and N shells. There are no P orbitals for the K shell (n=1). If we call this set of three orbitals a, b and c, then when we work through the periodic table and the P subshell is being filled, one electron goes into each of a, b and c, before a second electron is added to any of the three.

I recall my awe when a physics instructor went through the math to demonstrate that the actual probabilities for the filled P shell of neon actually added up to a purely spherically symmetric entity, with none of the bumps we might expect from seeing the component orbitals. He wryly pointed out that while "it may be shown" that any filled subshell is the same, he personally didn't know anybody who could actually work the math to perform the demonstration.

The orbitals of a D subshell, for M, N and O shells (n = 3,4,5) are more complex, with more lobes. The plethora of ways a D subshell could relate to the ground state of an atom explains the diffuse nature of the spectral lines. The broad D lines of sodium, under very high resolution study, are seen to consist of a veritable forest of lines. The energy splitting that underlies this line broadening is due to quantum effects caused by the other electrons in an atom with D subshells.

Finally, the F orbitals, here just shown for an N shell (n = 4), are dauntingly complex. So, why are the F lines in the spectrum above not even more diffuse than the D lines? It is mainly because the physical size of the F orbital greatly reduces the quantum effects of the other electrons, so that though the F lines have very complex structure, it all fits into a narrower spectral band. This also leads to a simple relationship between the wavelengths (or energies) of a series of F lines, that led to their being called "fundamental", as mentioned in the quote above. By the way, after F, the rather faint spectroscopic series just continue through the alphabet (G, H, etc.) but I've never seen any attempt to illustrate orbital shapes.

I wonder if anyone living can really add up the seven sets of probabilities analytically, to demonstrate that a filled F subshell has spherical symmetry, without any lumps? I know that computer solutions of the appropriate Schrödinger wave equation can be performed numerically, but that is almost cheating.

An atom of platinum (Z=78), with four completely filled shells and the first three subshells of shell 5 filled, is like a Matrushka doll: thirteen concentric spherical regions of high probability for locating any of the 78 electrons. This very high symmetry results in platinum having a less complex spectrum than elements 77 (iridium) or 79 (gold), though all of them have more complex spectra than much less massive elements such as iron.