The quantum basis of the electron shell structure

We recently had an, I hope, interesting discussion concerning my amateur theory that the electron “shell” or “cloud” structure of the atom is based on laws that do not proceed from and cannot be explained on the basis of more basic physical laws, and that it therefore suggests the working of a higher intelligence in the formation of matter and the universe. The laws I had in mind have to do, first, with the maximum number of electrons in each succeeding concentric electron shell, 2, 8, 18, 32, 50, and so on—why are those the numbers of the maximum electron capacity for each shell and not other numbers?—and second, with the fact that all the elements that have eight electrons in their outermost shell are “inert,” or “noble,” meaning they are stable elements that do not tend to combine with other elements. Whether an element has only two occupied electron shells and its second shell with eight places is “full up” with eight electrons and no “unused” spaces (neon, with atomic number 10), or whether it has three electron shells in which the outer shell has 18 places with eight electrons in that shell and 10 “unused” spaces (argon, with atomic number 18), or whether it has four electron shells and its outermost shell has 32 places with eight electrons in that shell and 24 “unused” spaces (krypton, with atomic number, 36), they are all alike in having exactly eight electrons in their outermost shell and in being inert gases.

This inert or “noble” condition of certain elements happens to be the basis of all chemistry, which can be defined as the study of the ways in which the outer electron shells of various elements combine with the outer electron shells of other elements to form compounds. As explained in the earlier post, the noble elements, with eight electrons in their respective outermost shells, are happy the way they are and don’t want to change. The other, uh, plebian, elements, with a number of electrons in the outer shell that either exceeds or falls short of the magic number eight, are unhappy with themselves the way they are and seek to attain the number eight and the stability it represents by combining with other elements. (In passing I might note that the use of “noble” and “plebian” in this discussion reverses Ortega y Gasset’s use of the same concepts in The Revolt of the Masses, in which he says that what defines mass man is his satisfaction with himself as he is and his lack of any ideals, while the noble man strives for things higher than himself.) Thus the “rule of eight”—by which all elements with eight electrons in the outermost shell are inert—is the basis for all the chemical compounds that exist in the universe, including such common subtances as water and salt. But where does this rule of eight come from? Why does it exist? Why is the number of inertness, nobility, and stability eight, rather than ten or six or four? The rule’s very (so it seems to me) peculiarity and arbitrariness seem to be a sign of a higher level of law at work in the universe that cannot be explained on the basis of more rudimentary levels of matter. (Be sure to see Ben W.’s wonderful commentary below on what I’ve said here.)

In reply to my questions, my friend Paul Nachman, a physicist, has sent the below which explains the atom’s electron structure as resulting from the laws of quantum mechanics. The basic outline of what Mr. Nachman is saying can be more or less followed even by those who, like myself, lack the math to understand it in detail.

He wrote:

I’ll tell you about the quantized energy levels for a hydrogen atom. They are given by three electronic quantum numbers plus another number that’s +1/2 or -1/2, depending upon the direction in which the electron’s own spin axis points. The Pauli principle “works on” the spin. (The fact that the electron has spin magnitude 1/2 makes it a fermion.)

First, there’s the principal quantum number, N = 1, 2, 3, … This tells us something about the size of the orbit, how far out the electron is from the nucleus although ..

- The orbit could be elliptical, so it’s not a unique distance out; and

- The electron position is best thought of as a cloud of probabilities, not a well-defined “orbit.”

Nevertheless, I’ll use the image of orbits henceforth.

An atom in its ground (lowest-energy) state has N = 1.

Next, there’s the angular momentum quantum number L. For a given N, L takes on integer values up to N-1. So for the ground state, only L=0 is possible. For the first excited energy level (i.e. N = 2), L has possible values 0 and 1. Etc. What does L signify? Basically, the extent to which the orbit is circular (maximum L for given N) or elliptical (smaller values of L for given N).

Next there’s the quantum number M, which gives the orientation of the angular momentum or, equivalently, of the electron’s orbital plane. For a given L, M takes on these (2*L+1) possible values: -L, -(L-1), -(L-2), … -1, 0, 1, 2, …(L-2), (L-1), L

What does M signify? Pick your own Z axis. Then think of M as telling you the orientation of the electron’s orbital plane such that when M = L or -L, the plane is perpendicular to your Z axis (the electron revolving in opposite directions about the same orbit for the plus and minus cases). For intermediate values of M, the orbit’s intrinsic shape is the same (still determined by L), but its orientation is between the extremes reached for M = L and -L.

Specification of an electron’s orbit/quantum state thus takes a triplet of integers, (N,L,M). And for single-electron atoms (i.e. intact hydrogen, of helium with one electron missing, or lithium with two electrons missing, …) the energy depends only upon the value of N. For such an atom, the value of N alone basically defines an electron “shell.” (Except there’s only a single electron, so it’s not a matter of filling up shells with multiple electrons at this stage of the description. We’ll get back to this.) So what are the number of quantum states available in the first shell?

Well, N=1, so L=0, so M=0. One quantum state. Lowest energy level (ground state)

Second shell? N=2, so L could be 0 ==>M=0 One state

or L could be 1 ==> M=-1,0, or 1 Three states

i.e. Four states altogether in the second shell. First excited energy level.

Third shell? N=3, so L could be 0 … One state

or L could be 1 … Three states

or L could be 2 ==> M=-2, -1, 0, 1, 2 Five states

i.e. Nine states altogether in the third shell. Second excited energy level.

You can do successive ones …

Now we include the electron spin, whose quantum number can be +1/2 or -1/2 (This is really an orientation quantum number directly analogous to the M above but with only these two possibilities for the spin axis’s orientation with respect to your Z axis.)

So each of those numbers of quantum states gets doubled because of the two possibilities for the electron’s spin, so we have 2, 8, 18, 32, 50, .. where you have to work out the last two. Those are your magic numbers.

Now none of the above is arbitrary numerology. The possible values for N, L, and M (and the relations between them) all arise from the differential equation known as the Schrodinger Equation applied to the physical situation of the one-electron atom and the further insistence that the solutions to the equation (many solutions——-one for each N,L,M trio) be “well-behaved” in a sense I won’t go into. But it’s not an arbitrary, after-the-fact rule.

But all the above was for single-electron atoms. Neutral helium has two electrons, neutral lithium has three, neutral uranium has 92 (93?). None of these can be solved exactly, as hydrogen was. But the ideas from hydrogen are the starting point, a first approximation. For more complicated atoms, each electron feels (electrically) the nucleus, but it also feels all the other electrons. It turns out that principal quantum number N is still a fairly good predictor of the electron’s energy, but now the energy depends upon L as well. (But still not upon M.)

Further, the exclusion principle now comes into play. Each of the states specified by N,L,M and +1/2 or -1/2 can be “empty” (or “unoccupied” or “unused”)——-or it can have a single electron in it. That’s the exclusion——-no two electrons can share a full set of quantum numbers.

Enough for now. There are things I would add, but it’s late. And I’d like to hear if you find this to be an understandable description before I attempt anything further.

LA replied:

Ok, the number of possible quantum energy states for each shell does correspond to the 2, 8, 18 32, etc. I don’t understand all this by a long shot. But you’ve provided a plausible basis that the law governing the maximum number of electrons in each shell does proceed from something to do with primary physical law and is not just coming arbitrarily out of the blue.

Why would the possible number of quantum states for each shell equal the highest number of electrons that shell can hold? Is it that each additional electron corresponds with a higher quantum state, so that one electron in the second shell corresponds with the first possible quantum state, two electrons with the second quantum state, … and eight electrons with the eighth quantum state?

The next question is about the rule of eight. Why are elements in which there are eight electrons in the outermost shell inert?

PN replies:

“Ok, the number of possible quantum energy states for each shell does correspond to the 2, 8, 18 32, etc. I don’t understand all this by a long shot.”

This would require being able to follow a development involving partial differential equations. All I can do is describe for you some aspects of the solution to the relevant equation.

“But you’ve provided a plausible basis that the law governing the maximum number of electrons in each shell does proceed from something to do with primary physical law and is not just coming arbitrarily out of the blue.”

When Pauli introduced the Exclusion Principle, one would probably say the idea was “in the air.” I’m not sure if instant identification was made with the idea of electrons spinning on their axes. But still, at that point, one would probably also say that the principle was coming out of the blue, as it was postulated to explain some regularities in atomic spectra plus the chemical facts that especially interest you. So it was an ad hoc idea to provide a way of accounting for some observations.

But recognize that any scientific principle ultimately starts that way. The out-of-the-blue ideas that survive to become “principles” are those that show themselves to be useful far beyond the application they were invented for.

Later, the Exclusion Principle indeed found use far afield, in things like the gross structure of white dwarf stars, electrical conduction in metals, … Now it is considered a bedrock principle.

Nevertheless, Pauli was ultimately able to find a basis for the principle in ideas that are considered even more basic than the principle itself. So now we’d probably be justified in regarding the Exclusion Principle as a consequence of relativistic quantum field theory.

“Why would the possible number of quantum states for each shell equal the highest number of electrons that shell can hold?”

That’s simply the application of the Exclusion Principle. The solutions of the differential equations are the quantum states of the atom. But the solutions themselves don’t tell us that, in a given atom, only zero, one, or two electrons can be in a state. [e.g. Why can’t 17 be in a state?] The Exclusion Principle mandates that a state already occupied by one electron can accommodate a second electron if the second electron’s spin is opposite the first’s. But that’s it. Another way of saying it is that, in any system [not just isolated atoms], no electron can share a full set of quantum numbers with any other electron. In an atom, the complete set of quantum numbers are the N, L, and M that I described previously along with the +1/2 or -1/2 for the electron’s spin direction——-so four quantum numbers per electron.

“Is it that each additional electron corresponds with a higher quantum state, so that one electron in the second shell corresponds with the first possible quantum state, two electrons with the second quantum state…”

The first two electrons in the second shell will have N=2, L=0, M=0 and one will be +1/2, the other will be -1/2, and (speaking sloppily) the two electrons will have the same energy as each other, …

“… and eight electrons with the eighth quantum state?”

The next six electrons will have N=2, L=1, the three possible values of M and the + and—possibilities, one of each, making six total. Speaking sloppily again, these six electrons will have the same energy as each other [but generally not the same energy as the first two, with N=2, L=0].

“The next question is about the rule of eight. Why are elements in which there are eight electrons in the outermost shell inert?”

I anticipated your question, but I don’t have an answer now. And it might be a lot of work for me to come up with one. I just don’t know. I never actually heard of the “rule of eight” before, though I knew instantly what you meant. Where did you come across the expression? It’s a good term: I looked up the electronic configurations (distribution of electrons over shells) for the noble gases (neon, argon, krypton, xenon, and radon) and they all fit the rule of eight.

LA replies:

I got the idea of it in the same book that gave rise to my argument, Introducing Chemistry, by Hazel Rossotti, 1973, a Penguin paperback I picked up and read parts of many years ago. She describes in a very accessible way the electron clouds and the inert gases and the way the non-inert elements look to combine with others in order to become “inert-like.” “The rule of eight” was just a convenient expression I coined for this in the recent discussion. It was the existence of that rule, when I first came upon it in her book, that struck me as indicating that there is this whole other level of physical law beyond electrons and protons, structural laws that determine the shape and characteristics of the atom and of matter, and that these laws do not exist at the lowest level and cannot be predicted at the lowest level, they only come into play at higher levels.

- end of initial entry -

Ben W. writes:

Some speculation as to why the number eight may have the meaning you attribute to it in the formation of things.

The eighth day in God’s chronology is the day of rest.

You write that: “The fact that all the elements that have eight electrons in their outermost shell are ‘inert,’ or ‘noble,’ meaning they are stable elements that do not tend to combine with other elements …”

Those who will experience the eighth day of rest will have come into God’s rest and cease from all their labours (Book of Hebrews). Thus they are “noble” (saints) and “stable” (have come into their eternal form). They do not combine with other elements because their essence has become pure and singular (not mixed with evil).

Always a maximum of two electrons in the first shell. That is how God proceeded to create man—one male and one female—Adam and Eve.

“They are all alike in having exactly eight electrons in their outermost shell and in being inert gases.”

Gases are cloudlike, and the heavenly host is described as a cloud of witnesses—spirit.

“The noble elements, with eight electrons in their respective outermost shells, are happy the way they are and don’t want to change.”

Thus those in heaven will be happy with their form and essence and will not want to change their spirit to be a compound of good and evil—ie. they will not challenge God and undo or redo their forms (e.g. transsexuality on this earth).

“The other, uh, plebian, elements, with a number of electrons in the outer shell that either exceeds or falls short of the magic number eight, are unhappy with themselves the way they are and seek to attain the number eight and the stability it represents by combining with other elements.”

Thus those “unhappy” with God’s form (not having entered His rest), essence, rules of organization, etc., attempt to reformulate the human condition and try to reconstruct “progressively” what a human being is (and society) but fall short of the eighth day. They “combine” with other elements (liberal universalism) to try and achieve a sort of universality.

“The rule’s very (so it seems to me) peculiarity and arbitrariness seem to be a sign of a higher level of law at work in the universe that cannot be explained on the basis of more rudimentary levels of matter.”

Yes, and its symbolic or analogical significance is to be found in the way God orders time and being.

These are, of course, just my speculations…

LA replies:

This is marvelous, inspired. I have a quibble on only one point, where Ben says that the “dissatisfaction” of the “non-noble” elements and their desire to form new, stable combinations symbolizes secular liberal man’s will to remake his own nature in violation of our God-given nature. In fact these unstable elements are seeking to become stable, they are seeking to emulate the noble gases by becoming stable and thus noble themselves. However, Ben very cleverly turns that argument back on me (before I even made it), by equating the combinations that produce stable compound substances with the striving of liberalism to create “diverse” societies, and thus with liberalism’s false universality and false order.

Paul Nachman writes:

I’m confident that no solutions of any partial differential equation have ever before received such an ecclesiastical exegesis!

LA writes:

A reader pointed out that Ben’s use of the eighth day seemed to be incorrect. Ben will be posting something in the next day or two to explain his meaning.

Posted by Lawrence Auster at March 26, 2007 07:57 PM | Send