The final episode of our article, in which we try to develop an understanding of atomic structure and the periodic table, from the beginning of secondary school/junior-high level, to first/second year undergraduate physics/chemistry, via one fictional conversation. Find Parts 1 – 5 on our blog page. Sigrid has just asked Sally whether ‘l=0’ corresponds to the s-orbitals…
SALLY: Yes! n tells you the main energy level, and l tells you the type of orbital.
SIGRID: So l=1 represents p, l=2 represents d…
SALLY: … Exactly.
SIGRID: But isn’t this just relabelling stuff? I mean is there any deeper understanding to be gained from this?
SALLY: Maybe!
SIGRID: Oh hang on, for n=2, we said that l could be 0 or 1. And l=0 is an s-orbital. But there are 3 p-orbitals. So it would look like this.
But how does l=1 give you three p-orbitals.
SALLY: Good question, and one that will get us somewhere… Would you believe me if I said there is a third quantum number?! If l represents orbital angular momentum, then ml is the component of the angular momentum orientated in a particular direction.
SIGRID: Which direction?
SALLY: Sort of any direction we choose, but it becomes important in energy terms if there is something like a magnetic field in a particular direction.
SIGRID: So what’s the rule this time? So far, we’ve got that n can be any whole number starting at 1, and l can be any whole number starting at 0 and going up to n-1. What’s the rule for ml?
SALLY: Maybe if you look at this diagram, and if I tell you that ml is also quantised as a whole number, then you will be able to work it out.
SIGRID: Well, if that picture is to be believed, in the magnetic field the angular momentum in a particular direction can be parallel to the axis of interest, anti-parallel or any whole number in between.
SALLY: So what’s the rule?
SIGRID: ml can range from –l through 0 to +l.
SALLY: And now can you see why there are 3 p-orbitals for a given value of n?
SIGRID: Yes! I know that p orbitals correspond to l=1. And if l=1, then ml can have values -1, 0 and 1. Three different values! So the three different p-orbitals have the same value of angular momentum (l=1) but different values of ml.
SALLY: -1, 0 and 1. Exactly – but only for n=2 and above, because for n=1, l can only be 0.
SIGRID: And that explains the five d-orbitals, too. If l=2, then ml can take values -2, -1, 0, 1 and 2. Five values, five d-orbitals!
SALLY: Which is only possible for n=3 and above.
SIGRID: l=3 gives seven f-orbitals! But only for n=4 and above – I get it!
SALLY: And remember you can fit two electrons into each orbital.
SIGRID: I suppose there’s a quantum number for that too?
SALLY: Yes, you’ve half met it before. It’s the spin quantum number, ms.
SIGRID: You mean the ‘intrinsic angular momentum number’, since a particle with no size can’t really ‘spin’.
SALLY: OK! It’s good to see you realising that analogies from everyday life sort of break down on the atomic scale. I’m still going to call it ‘spin’ though!
SIGRID: We called the possible values of spin ‘up’ and ‘down’. So they are sort of opposite each other. Does that mean their quantum numbers are +1 and -1?
SALLY: Good thinking, but no! We aren’t free to give spin any value we like – it has to compare correctly to the other angular momenta, l and ml. So actually the possible values of ms for an electron in an atom are +½ and -½.
SIGRID: Weird.
SALLY: Yeah – that’s another conversation again. We’d need to talk about the standard model, fermions and bosons to say more about that. Let’s not for now. After all, we’ve been talking a long time, and it will be a miracle if anyone is still reading.
SIGRID: OK, but we haven’t finished yet. We need to talk about how we build atoms in this new language.
SALLY: You’re right. OK, here goes… Each electron in an atom has its set of four quantum numbers, n, l, ml, ms. And in a given atom, no two electrons can have the same set of four quantum numbers – that’s a rule. So ‘building atoms’ is about allocating quantum numbers to electrons, and minimising their energies.
SIGRID: And the lower the quantum numbers, the lower the energies?
SALLY: At first, yes. But we will find that there are overlaps between n=3 and n=4 and so on, just like when we filled the 4s orbitals before the 3d orbitals. Anyway let’s start easy – what would be the ground state of hydrogen in this notation?
SIGRID: Well, we are using the first main energy level (shell). So n=1. Then that forces l to be 0, which in turn forces ml to be 0, and shall I choose ms to be +½?
SALLY: Exactly! {1,0,0,+½}. And what’s the orbital representation of that?
SIGRID: 1s1.
SALLY: Good, and why is it important to specify that it is the ground state?
SIGRID: Because an excited state could have any four quantum numbers, but the atom would quickly return to {1,0,0,+½}. Then for the ground state of helium, we can fit the other electron in the 1,0,0 state, so it would have one electron in {1,0,0,+½} and one in the state {1,0,0,-½}. And we could write that in terms of orbitals as 1s2.
SALLY: Can you do the same for elements up to number 10? Remember Hund’s rule!
SIGRID: Ah, so I use up all the orbitals before adding the second electron to an orbital. Right, here we go…
I like the way both representations show the same thing. But the quantum number version is for each electron, and you need a full list of them, whereas the orbital version summarises the electrons within an atom.
SALLY: And there’s a whole other nomenclature which gives the information about spins and angular momenta for the atom as a whole, rather than the individual electrons, but I don’t think that would give you any fresh insights. I’m just telling you so that when you see hydrogen written 2S1/2 and fluorine 2P3/2, you’ll know that’s what’s going on, and you’ll have to look it up.
SIGRID: OK!
SALLY: I like the way you have shown in the diagram that an electron in a p-orbital has angular momentum l=1. I wonder whether you could work out a similar thing for scandium?
SIGRID: Oh, because that’s the first transition metal, so we are using the d-block. And that must then be the first element with an electron with l=2.
SALLY: Exactly! Go on – work it out!
SIGRID: OK!
SIGRID: I know all this seems to work, but you could still be making it all up. I mean, is there any actual experimental evidence for all this?
SALLY: There is, actually! From measuring the energy needed to rip an electron of an atom for successive elements, you can tell how tightly bound the electrons are from one element to the next. And the pattern gives evidence for the shells (n values) and subshells (l values, or if you prefer, s, p, d and f orbitals).
SIGRID: But here’s something that’s troubling me. Angular momentum is not the same thing as energy. So why would different orbitals within the same shell (n value) have different energies?
SALLY: Ah, you are right to say that angular momentum is not the same thing as energy. But depending on the circumstance, it can affect the energy. Remember we spoke about ‘degeneracy’? If l had no effect on energy, then all the orbitals within a shell would be degenerate, and have the same energy.
SIGRID: So what?
SALLY: When we solve the Schrodinger equation, we need to add information about the ‘electrical potential’. That’s a measure of the electrical potential energy of an electron at any point due to the attraction of the nucleus and repulsion from other electrons, and it is function of the distance, r, from the nucleus.
SIGRID: Again, so what?
SALLY: It turns out that if the electric potential is spherical and inversely proportional to r, then solutions to the Schrodinger equation for the energy, E, do not depend on l. That is, there is l degeneracy, and all orbitals within a shell have the same energy.
SIGRID: When is the potential inversely proportional to r?
SALLY: Pretty much only in the hydrogen atom! And we have already discussed the l-degeneracy in the hydrogen atom. If the potential is still spherically symmetric but not inversely proportional to r, and that is true in multi-electron atoms, then different values of l have different energies.
SIGRID: Which is why it matters in what order you fill the orbitals within a shell for multi-electron atoms! So that s-orbitals for a given n fill before the p-orbitals.
SALLY: Precisely! Even then, there is still ml-degeneracy, so that, say the three p-orbitals within a shell are at the same energy.
SIGRID: Then the energy depends on l, but does not depend on the value of ml within that sub-shell (orbital type). And is it possible to get rid of the ml-degeneracy so that ml does affect the energy?
SALLY: Yes, and then the different p orbitals within a shell will also have different energies. We talk of the degeneracy being ‘lifted’. And that happens if the potential is not spherically symmetric. For example, when the system is in an external magnetic field.
SIGRID: And what’s so special about a magnetic field?
SALLY: In one sense, nothing, except that it is an easily achievable way to lift the degeneracy! But in another sense, everything! Because electrons and nuclei themselves act like tiny magnets, since they are spinning charges, and moving charges create magnetic fields. Electrons and protons have a ‘magnetic moment’ which is a measure of how they are affected by a magnetic field. Those ml arrows that we drew before are a representation of that magnetic moment.
SIGRID: Right, but why is there any limit at all on the number of electrons in a given orbital. Ages ago I thought that the 79 electrons of gold would all live in the first shell. Now I see that they don’t but I don’t really know why. I guess my question is: why can’t all 79 electrons have the quantum numbers {1,0,0,+1/2}?
SALLY: This is another one of those times where I’ll give you an answer, and you’ll just question my answer. But here it is, anyway – electrons are fermions. That means they have spin ½. But having half-integer spin, rather than integer spin, means that they follow a certain type of behaviour, called Fermi-Dirac statistics (rather than Bose-Einstein statistics for integer-spin particles called bosons). Fermions take up space, rather than being able to pile on top of each other. And in an atomic context that leads to the Pauli Exclusion Principle, which says that no two electrons in an atom can have the same set of four quantum numbers.
SIGRID: I feel like have learned a lot here. But I still get the impression you haven’t told me everything!
SALLY: I suppose. The spectra of atoms other than hydrogen are all more complicated than we would predict from our discussion. Remember we said that electrons act like tiny magnets? Well, the magnetic behaviour due to the spin of an electron can interact with its magnetic behaviour due to orbital angular momentum, and that complicates things – it’s called spin-orbit coupling. Then in multi-electron atoms, the magnetic moments of a single electron interact with those of the other electrons. Then there’s the effect of the nucleus. The electrons and nucleus orbit a common centre of mass, much like the planets and the Sun (the Sun wobbles too, due to the influence of the planets), and that complicates things. And the nucleus acts like a tiny magnet too, influencing the electrons.
And that’s just for single atoms, where we can concentrate on the interactions between a nucleus and its electrons. There are extra interactions in molecules. Or in crystal lattices of millions of atoms lined up in rows.
But it’s good that there’s always more to look at another day, right? 🙂
Finally, we come to the end! If you are still with us, congratulations, and we hope you have learned something or been entertained, or both. As a reward, here is a pdf of the full article:
AtomicStructureAndThePeriodicTable
Remember, there is eternal honour and glory (and absolutely no material reward whatsoever) if you can tell us why the characters are so named…