Spin-Orbit Coupling #
If you’ve come across some XPS spectra in the past, you may have noticed that for certain peaks, we always see twin emissions. This doesn’t mean we have two chemical states, infact both of these peaks are from a single chemical environment. This is due to an effect called spin-orbit coupling, in which the photoemission is split into two peaks. We see this splitting for all orbitals other than S-orbital. So what’s going on?
When we describe electron energy levels, we do so in terms of quantum numbers. These are the basic integers that describe the electronic level and state of an electron.
First, we have the principal quantum number, n, which denotes the electron shell.
Next we have l, the angular momentum quantum number. This denotes the shape of the orbital – whether that is an S orbital, or a P orbital etc. and can take any number from 0 up to n-1. So for example, for our first shell, we can only have a value of 0, in other words an S orbital. When we reach the 4th shell, we have quantum numbers 0, 1, 2, and 3 – in other words, S, P, D, and F orbitals.
Our next quantum number is the magnetic quantum number, ml, which tells us the umber of orbitals in our subshell. This can take the values of –l, l+1, l+2, up to l, through zero.
Finally, we have the electron spin, which can be either spin up or spin down, and takes the values of plus or minus a half, describing the electron momentum.
To understand why spin-orbit coupling occurs, we need to think from the electron’s perspective.
In the electron’s rest frame, it is the nucleus that appears to be in motion — orbiting around it. A moving charge constitutes a changing electric field, and Maxwell’s equations tell us that a changing electric field generates a magnetic field. The electron therefore sits inside a magnetic field generated by the apparent motion of the nuclear charge.
The electron is not passive in this interaction. It carries its own magnetic moment, arising from its intrinsic spin — either spin-up or spin-down. These two magnetic fields interact directly: if the spin magnetic moment is aligned with the orbital magnetic field, the electron sits at a slightly different energy than if it is opposed.
This gives rise to two distinct energy states for the same electron in the same orbital — and in XPS, two distinct peaks.
The reason S orbitals are exempt is straightforward. When l = 0, the electron has no orbital angular momentum — there is no orbital motion, no changing electric field, and therefore no magnetic field is generated. With no field to couple to, spin becomes irrelevant, and only a single peak is observed.
When spin-orbit coupling occurs, the orbital angular momentum and the electron spin angular momentum do not act independently — they combine into a single quantity called the total angular momentum, denoted j.
The total angular momentum is defined as the magnitude of the vector sum of the orbital angular momentum quantum number l and the electron spin quantum number ms (which takes values of +½ or −½). Because we are concerned only with the magnitude of this combined quantity — not its direction — we take the modulus:
j = | l + ms |
For any given orbital, this yields two values of j: one where the spin is aligned with the orbital angular momentum (ms = +½), and one where it is opposed (ms = −½). These two values correspond directly to the two peaks observed in the XPS doublet.
Working through each orbital type:
S orbital (l = 0): j = |0 ± ½| = ½ in both cases — a single value, no splitting
P orbital (l = 1): j = ³⁄₂ or ½
D orbital (l = 2): j = ⁵⁄₂ or ³⁄₂
F orbital (l = 3): j = ⁷⁄₂ or ⁵⁄₂
The notation used to label doublet peaks follows directly from this — Au 4f₇/₂ and Au 4f₅/₂ simply denote the 4f orbital with total angular momentum values of ⁷⁄₂ and ⁵⁄₂ respectively. Once you understand where j comes from, the peak labelling system becomes self-evident rather than something to memorise.
It is also worth noting that as orbital complexity increases — from P through to F — both j values grow larger. This reflects the greater changes in angular momentum experienced by electrons in more complex orbital geometries, and has direct consequences for the energy separation between the doublet components, which is discussed in the following section.
If we have a look at how some of the energy separations track by element, it’s clear there is a trend based on the element identity, and the orbital. As we increase our atomic number, our 2 orbital energy separations increase, but then we come back down again for our 3p orbitals, before rising in the same way. So what’s the relationship here?
Well we have an expression, where doublet separation is proportional to the atomic number, the principle quantum number, and the orbital angular momentum, or azimuthal quantum number. We saw in our previous plot, that for the same element, going from the 2p to 3p orbital decreased the doublet separation, and now if we take a look at 3 elements, iridium, platinum and gold, we can see the effect of atomic mass and orbital angular momentum. So what’s going on here? Well, if we increase our atomic number, we increase our nuclear charge. Stronger nuclear charge means a greater Coulombic force acting on our electron, so this increases the energetic impact of this spin orbit splitting. Principle quantum number descries how close the electron is to the nucleus, where again, these fields will be stronger, and have a bigger impact on our electrons. Orbital angular moment, well a larger azimuthal quantum number means a larger magnetic moment. Recall our orbital shapes, we see greater changes to the angular momentum for higher values of l, and this also impacts our energy separation. Finally, there are some impacts due to potential electron screening processes, this is particularly uncommon, or when we get to very heavy elements, relativistic effects due to high electron velocities, which will greatly increase the separation, but these are well beyond the scope of this course and not something you’d be likely to encounter.
So for the most part, spin-orbit doublets have an equivalent peak width, and when you’re modelling them you should keep this in mind and keep this peak property locked in your fitting.
Remember this width is related to the core-hole lifetime. There is no reason for this to be different for most systems, since we’re just talking about electrons in the same chemical environment. There are some cases where this is not true, Ti 2p is probably the most common example, which is due to Coster-Kronig type transitions.
Having established that spin-orbit coupling produces a doublet of peaks, the next question is why the two components have different intensities. Taking Au 4f as an example, quantifying the two peaks gives approximately 57% for the 7/2 component and 43% for the 5/2 — but where does this ratio come from?
The answer lies in the number of degenerate states available to each total angular momentum value. Recall that the total angular momentum quantum number j can take values between −j and +j in integer steps, giving a set of mj states that differ only in their orientation around the nucleus. Because these states differ only in orientation — not in energy — they are degenerate, and each contributes equally to the intensity of that peak.
The number of mj states for a given j is simply 2j + 1. For the Au 4f doublet, this gives 2(7/2) + 1 = 8 states for the 7/2 component, and 2(5/2) + 1 = 6 states for the 5/2 component. Taking the ratio of these — 8:6, or 4:3 — and comparing against the measured spectral area ratio yields 1.3336. Precisely as predicted.
This degeneracy argument gives us fixed, orbital-specific area ratios that must be respected in any peak fit:
P orbital — 2:1
D orbital — 3:2
F orbital — 4:3
These ratios are not adjustable parameters. They are a direct consequence of quantum mechanics, and departing from them in a peak model is not justified without strong physical reasoning. Violations do appear in the published literature and can lead to incorrect chemical state assignments — particularly in complex, overlapping regions such as Pt 4f or Al 2p where correct doublet constraints are essential for reliable quantification.
Alright, so what do we need to remember when we get to this process? Well the first thing we discussed was the multiplicity of the two states, and how this impacts the peak area ratio. Remember this is determined by the number of total angular momentum states, and is completely orbital specific. So for our orbitals we have S-orbitals, which show no spin-orbit splitting, so we can forget about these for now. P orbitals, which have an area ratio of 2:1, D-orbitals which have an area ratio of 3:2 and F-orbitals, with an area ratio of 4:3. There are very very few occasions where this relationship breaks down, probably the most common examples are titanium and vanadium, which are actually mostly due to the complex background and secondary spectral features in these regions which are often not included in the fitting models, causing different area ratios. We will go into more detail about these effects later in the course, and to be clear what we are talking about here is a practical issue, not a theoretical issue, and can be accounted for with robust analysis techniques – although for simple chemical analysis, it is typically easier to just use a slightly adjusted peak area ratio. Best advice is, if you start to model something and think your peak area ratios are deviating form the expected, just check out one of the element pages on our guru site, or on the Thermo XPS simplified website, or Mark Biesingers XPS fitting website, and just check if there is some unusual behaviour you should be aware of. OK, so the second rule, FWHM, they should be the same, so always lock these together when starting your fits. Again, there are some deviations, pretty much confined to first row transition metals for the standard orbitals you’d want to analyse, where the 2p ½ slightly broadens, and specifically, once again titanium and vanadium are the biggest culprits. This is caused by a combination of special type of secondary electron emission, and we have a page and publication that goes into this in more detail for those interested, I’ll link below – and a number of final state effects that only impact the ½ peak, as was found by Paul Bagus and coworkers in 2018, but again this is beyond the scope of this course. And finally, our doublet separation. This should be consistent for a particular element and orbital, regardless of chemistry, so should be locked. BUT, there are 2 notable exceptions. Can you guess what they are? Our old friends titanium and vanadium do exhibit a bit of deviation in the doublet separation! This is effectively due to empty d-orbitals for higher oxidation states, which cause big differences in screening, so valence electrons protecting core electrons from chemical changes, and more covalent bonding character. In comparison, Scandium never really has much d-electron density to begin with, and when we get to chromium we’re adding enough electrons and nuclear charge that these effects are diminished, so it’s just a bit of a perfect storm. Don’t worry, if you plan on analysing titanium or vanadium by XPS, they really aren’t difficult to interpret, they just behave a little different, but as long as you know what this behaviour is and should be, you can absolutely analyse them quite easily. In fact they are probably the two easiest first row transition metals to analyse… but more on that later.
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