Classical models of the atom — electrons orbiting a nucleus like planets around a star — cannot explain the discrete energy levels observed in spectroscopy. A quantum mechanical treatment is required.

In quantum mechanics, electrons in an atom do not occupy fixed orbits. Instead, they occupy orbitals — regions of space described by wavefunctions — each associated with a specific, quantised energy. Electrons cannot exist between these energy levels. They occupy one level or another, and transitions between levels involve the absorption or emission of precisely defined amounts of energy.

This quantisation is the reason XPS peaks appear at sharp, reproducible binding energies rather than as a continuous smear. Each discrete peak corresponds to a discrete quantum mechanical energy level.

The energy and spatial character of each electron in an atom is completely described by four quantum numbers. These numbers are not arbitrary — they arise naturally from solving the quantum mechanical equations governing the atom.

Principal Quantum Number — n #

The principal quantum number n takes positive integer values: 1, 2, 3, 4, and so on.

It defines the electron shell and is the dominant factor determining the energy of the level. Higher n means the electron is, on average, further from the nucleus and less tightly bound — lower binding energy in XPS terms.

• n = 1 → K shell
• n = 2 → L shell
• n = 3 → M shell
• n = 4 → N shell

In XPS notation, the K shell corresponds to 1s electrons, the L shell to 2s and 2p electrons, the M shell to 3s, 3p, and 3d electrons, and so on.

Angular Momentum Quantum Number — l #

Within each shell defined by n, electrons can occupy subshells with different spatial shapes. The angular momentum quantum number l takes integer values from 0 to n−1.

The s subshell is spherically symmetric. p subshells have a dumbbell shape with directional character. d and f subshells are progressively more complex. These shapes matter for understanding bonding, but for XPS the key point is that s, p, d, and f subshells all have different energies, even within the same shell.

Magnetic Quantum Number — mₗ #

The magnetic quantum number mₗ takes integer values from −l to +l, giving 2l+1 possible values for each subshell.

For an s subshell (l = 0), there is only one value: mₗ = 0 — one orbital. For a p subshell (l = 1), mₗ = −1, 0, +1 — three orbitals. For a d subshell (l = 2), mₗ = −2, −1, 0, +1, +2 — five orbitals.

In an isolated atom with no external magnetic field, all orbitals within a subshell have the same energy — they are degenerate. In XPS of solids, this degeneracy is partially lifted by the crystal field and bonding environment, but for the purposes of understanding peak structure, the key consequence is the number of electrons each subshell can hold.

Spin Quantum Number — mₛ #

Every electron has an intrinsic angular momentum called spin. The spin quantum number mₛ takes only two values: +½ (spin up) or −½ (spin down).

The Pauli exclusion principle states that no two electrons in the same atom can share all four quantum numbers. Since mₗ and the other quantum numbers can be shared, the two electrons in any given orbital must have opposite spins. This limits each orbital to a maximum of two electrons.

Combining the quantum numbers gives the structure of each shell and subshell, along with the maximum number of electrons each can hold.

This structure directly maps onto the periodic table. The filling of successive shells and subshells as atomic number increases is what drives the periodic trends in binding energy that make XPS spectra predictable across elements.

Within a given atom, energy levels do not fill in a perfectly neat shell-by-shell sequence. The 4s orbital fills before 3d in most transition metals, for example — a consequence of the relative energies of these levels in many-electron atoms where electron-electron repulsion modifies the simple hydrogen-like picture.

For XPS, the important points are:

Binding energy increases as n decreases. Electrons closer to the nucleus are more tightly bound. The 1s level of any element has the highest binding energy of all its occupied levels.

Within a shell, s < p < d < f in binding energy. The 2s is more tightly bound than the 2p, the 3s more than 3p, and so on. This means that in a survey XPS spectrum, the 2s peak appears at higher binding energy than the 2p peak for the same element.

Binding energy increases across the periodic table. As atomic number increases, the nuclear charge increases, pulling all electrons closer and increasing their binding energies. The C 1s sits near 285 eV; the Fe 1s sits near 7112 eV — far beyond the range of standard XPS instruments, which is why iron is analysed via its 2p levels instead.

One of the most practically important consequences of quantum mechanics for XPS is spin-orbit coupling — the interaction between the orbital angular momentum of an electron and its spin angular momentum.

For electrons in subshells with l = 0 (s orbitals), there is no orbital angular momentum and therefore no spin-orbit coupling. The 1s, 2s, and 3s levels each produce a single XPS peak.

For electrons with l ≥ 1 — that is, p, d, and f subshells — the orbital and spin angular momenta can couple in two ways, defined by the total angular momentum quantum number j:

j = l + s or j = l − s

Where s = ½ for a single electron. This gives:

Each j value corresponds to a slightly different energy — and therefore a different binding energy in XPS. This is why p, d, and f core levels produce doublets: two peaks separated by the spin-orbit splitting energy.

The lower j value (j = l − ½) always lies at higher binding energy. The higher j value (j = l + ½) always lies at lower binding energy and is always more intense — because it corresponds to more possible electron states (a degeneracy of 2j+1).

The intensity ratio between the two components of a doublet is fixed by quantum mechanics:

• p doublets: 2:1 (p₃/₂ : p₁/₂)
• d doublets: 3:2 (d₅/₂ : d₃/₂)
• f doublets: 4:3 (f₇/₂ : f₅/₂)

These ratios are used as constraints during peak fitting, and deviations from them can indicate overlapping peaks, incorrect background subtraction, or genuine physical effects such as multiplet splitting.

An energy level diagram for an atom maps directly onto its XPS spectrum. Consider silicon (Si, Z = 14), with the electron configuration 1s² 2s² 2p⁶ 3s² 3p²:

• Si 1s — highest binding energy (~1839 eV), single peak, not routinely measured by standard XPS
• Si 2s — single peak (~149 eV)
• Si 2p — doublet: 2p₃/₂ (~99.3 eV) and 2p₁/₂ (~99.8 eV), small splitting of ~0.6 eV
• Si 3s, 3p — appear in the valence band region, low binding energy

A survey XPS spectrum of silicon shows exactly these features. Each peak traces back to a specific quantum mechanical energy level in the silicon atom. The correspondence is direct and unambiguous.

For heavier elements, the number of occupied subshells increases, more peaks appear in the spectrum, and the spin-orbit splittings become larger and more easily resolved. Gold (Au, Z = 79), for example, shows peaks from 4f, 4d, 4p, 4s, 5s, 5p, 5d, and valence levels — each pair of p, d, and f levels appearing as a doublet.