The Shell You Say

Everyone figures Eddie started his pizza place because he likes to eavesdrop.  No surprise, he wanders over to our table.  “I heard you guys talking about atoms and stuff and how Sy here don’t like Bohr’s model of electrons in atoms even though Bohr’s model and the shell model both account for hydrogen’s spectrum.  Why’s the shell model better?”

Vinnie comes back quick.  “Because it’s not physically impossible, for one thing.”

I’m on it.  “Because the shell model extends smoothly to atoms and ions in an electric or magnetic field.  Better yet, shell methods can be applied to molecules.”

“What do fields have to do with it?”

“It’ll help to know that some of those electron patterns come in sets.  The 2-node shell has three dumbbell shapes, for instance — one each along the x, y and z axes. Think about an atom all alone in space with no fields around.  How does it know which way z goes?””

“It don’t.  Everything’s gotta be in all directions, like spherical.”

Vinnie’s back in.  “I’m seeing an atom in an electric field, say up-to-down, it’s going to pull charge in one direction, say down.  So now the atom don’t look like no ball no more, right?”

orbital in a field

Vertical field on the right

“Right.  Once the atom’s got a special direction, those three dumbbells stop being equivalent.  We say that the field mixes together the spherical pattern (in atoms we’d call it an s-orbital) with that direction’s dumbbell (we’d call it a p-orbital) to make two combination orbitals.  One combination has a lump of charge stretched downwards and the other combination has a bowl of diminished charge stretched upwards.  The stronger the field, the wider the energy split between those two.”

“What about the other two dumbbells?”

“They’re still equivalent, Eddie.  If there’s charge in them it’s spread evenly around the equator like a doughnut.  Energy-wise they’re in between the two s±p combinations.”

IF there’s charge, like maybe there ain’t?”

“Ever suspicious, eh, Vinnie?  You’re right, and that’s a good point.  Orbitals are only a way to describe the chaos inside the atom, like notes are a way to describe music.  There are 3-node orbitals and 47-node orbitals, all the way up, but most of the time they’re not charge-activated just like a piano’s top note hardly ever gets played.”

“How do we know whether an orbital’s activated?”

“We’ve got rules for that, Eddie.  Maximum of two units of charge per orbital, lowest energy first.  Unless some light wave has deactivated a deeper orbital and activated a higher one.”

“You’re being careful again, not saying an electron’s here or an electron’s there.”

“Darn right, Vinnie.  It’s that chaos thing — charge is smeared all over the atom like air molecules jiggle all over the place to carry a sound wave.  Chemists and physicists may talk about ‘the electron in the 2s-orbital’ but that’s shorthand.  They know it’s really not like that.”

“I’m doing arithmetic over here.  So there’s two electrons, OK, call it two units of charge for that 1-node ball orbital, plus two units for the 2-node ball, plus two units each for the three dumbbells, uses up five orbitals.  That’s the same 2+8 stable mix that Bohr came up with.”

“Yeah, Eddie, but that field Sy talked about could be any strength.  Run the energy  equations backwards and the astronomers get a way to check a star’s fields.”

“Exactly, Vinnie.  Transitions involving combination orbitals have slightly different energy jumps than the ones we see in isolated atoms.  Electric and magnetic fields split each line in an element’s spectrum into multiplets.  Measure their splittings and you can work back to the field strengths that caused them.  The shell theory offers more predictions and more scientific insights than Bohr’s model ever dreamed of.”

“You said shell theory can handle molecules, too.  How’s that work?”

“Same as that electric field, but a lot messier.  Every nucleus exerts a field, mostly electric, on the rest of the molecule.  So does all the electron charge, but it’s more diffuse and includes more magnetism.  Molecular orbitals span the whole thing.  Works like atoms but much harder to calculate.”

“Figuring tips is easier,” hints Eddie.

~~ Rich Olcott


Shells A-poppin’

We step into Eddie’s.  Vinnie spots Jeremy behind the gelato stand.  “Hey, kid, you studying something Science-y?”

“Yessir, my geology text.”

“Lemme see it a sec, OK?”

“Sure.  Want a gelato?”

“Yeah, gimme a pistachio, double-dip.  I’ll hold your book while you’re doing that.  Ah-hah, Sy, lookie here, page 37 — new textbook but this atom diagram coulda come right out of that 1912 Bohr paper you don’t like.  See, eight dots in a ring around the nucleus.  Can’t be wrong or it wouldn’t have survived this long, right?”

<sigh>  “What it is isn’t what it was.  Bohr proposed his model as a way to explain atomic spectra.  We’ve got a much better model now — but the two agree on three points.  Atoms organize their electronic charge in concentric shells, innermost shells deepest in the nuclear energy well.  Second, each shell has a limited capacity.  Third, when charge moves from one shell to another, light energy is absorbed or emitted to match the energy difference between shells.  Beyond those, not much.  Here, this diagram hints at the differences.”Better Bohr

“The scrambled-looking half is the new picture?”

“Pure chaos, where the only thing you can be sure of is the averages.  These days the Bohr model survives as just an accounting device to keep track of how much charge is in each shell.  That diagram — what kind of atom is it describing?”

“I dunno, two electrons inside, eight outside, ten total.”

“Could be neon, or a fluoride, oxide, sodium or magnesium ion.  From a quantum perspective they all look the same.”

“Here’s your gelato, sir.”

“Thanks, kid, here’s your book back.  But those are different elements, Sy.”

“The important thing, Vinnie, is they all have an outer shell with eight units of charge.  That’s the most stable configuration.”

“What’s so special about eight, Mr Moire?  If it’s pure chaos shouldn’t any number be OK?”

“Like I said, Jeremy, it’s the averages that count.  Actually, this is one of my favorite examples of what Wigner called ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences.’  Back in 1782, a century and a quarter before anyone took atoms seriously, Laplace did some interesting math.  Have you ever waited for a pot of water to boil and spent the time tapping the pot to see the ripples?”

“Who hasn’t?  Doesn’t boil any faster, though.”

“True.  Looking at those waves, you saw patterns you don’t see with flat reflectors, right?”

“Oh, yeah — some like dumbbells, a lot of circles.”

“Mm-hm.  In a completely random situation all possible patterns could appear, but the pot’s circular boundary suppresses everything except wave patterns that match its symmetry.  You don’t see hexagons, for instance.”

“That’s right, I didn’t.”

“So there’s Laplace in the 1790s, thinking about Newton’s Law of Gravity, and he realizes that even in the boundaryless Solar System there’s still a boundary condition — any well-behaved standing wave has to have the same value at the central point no matter what direction you come from.  He worked out all the possible stable patterns that could exist in a central field like that.  Some of them look like what you saw in the water.  We now classify them by symmetry and node count.”

“Node?”Disk orbitals

“A region where the pattern hits zero, Vinnie.  Density waves range from zero to some positive value; other kinds range from positive to negative values.  A spherical wave could peak at the center and then go to zero infinitely far away.  One node.  Or it could be zero at the center, peak in a spherical shell some distance out and then fade away.  That’d be two nodes.  Or it could be zero at the center, zero far away, and have two peaks at different distances with a spherical third node in between.  Here’s another two-node pattern — that dumbbell shape with nodes at the center and infinity.  You can add radial nodes partway out.”

“I’m getting the picture.”

“Sure.  You might think Laplace’s patterns are just pretty pictures, but electron charge in atoms and ions just happens to collect in exactly those patterns.  Combine Laplace’s one-node and two-node patterns, you get the two lowest-energy stable shells.  They hold exactly ten charge units.  The energies are right, too.  Effective?”


~~ Rich Olcott