Quantum Defects and Alkali Spectra

The ℓ-degeneracy of hydrogen was a consequence of the exact $1/r$ potential and disappears the moment the potential deviates from it. The alkali atoms — lithium, sodium, potassium, rubidium, caesium — are the cleanest place to watch that happen. Each has a single electron outside a closed noble-gas core, so it is nearly a one-electron atom, and its spectrum is nearly hydrogenic. What separates it from hydrogen is that the valence electron, on the inner part of its orbit, penetrates the core and feels more than the single net charge it sees from far away. The extra binding is captured, series by series, by one number.

The core and the effective potential

Sodium is the representative case: $Z=11$, with the configuration $1s^{2}2s^{2}2p^{6}3s$. Ten electrons fill the neon core; the eleventh, the valence $3s$ electron, moves in the field of the nucleus plus the core. Two limits fix the potential it feels:

• far outside the core ($r\to \infty$): the ten core electrons screen ten of the eleven protons, and the valence electron sees a net charge $+1$, a pure Coulomb tail $V(r)\to -k{e}^2/r$;
• deep inside the core ($r\to 0$): the screening is undone and the electron sees the full nuclear charge, $V(r)\to -kZ{e}^2/r$.

Between the two the effective charge $Z_{\text{eff}}(r)$ runs from $Z$ at small $r$ down to $1$ at large $r$. The potential is central but no longer $1/r$, so the energies acquire an $\ell$-dependence: $E_{n\ell }$, not $E_n$.

Penetration and the quantum defect

Whether the valence electron reaches the region of enhanced charge is decided by the centrifugal barrier${\hslash }^2\ell (\ell +1)/2\mu r^2$. A low-$\ell$ state has a small barrier and a radial function $R_{n\ell }\sim r^{\ell }$ that is appreciable near the origin, so it penetrates the core and gains binding energy. A high-$\ell$ state is held outside by the barrier, samples only the $-k{e}^2/r$ tail, and stays hydrogenic.

The energies are still well described by a Rydberg formula, but with the principal quantum number reduced by an $\ell$-dependent shift.

The defect is positive because penetration deepens the binding: $n^{\ast }<n$ makes $E_{n\ell }$ more negative than the hydrogenic $-R_M/n^2$. A penetrating $s$ electron of sodium behaves as though its principal quantum number were reduced by more than one full unit.

The quantum defect as a phase shift

The near-constancy of ${\delta }_{\ell }$ in $n$ is not an accident of fitting; it follows from the structure of the wave function outside the core. Beyond the core radius the potential is exactly $-k{e}^2/r$, so the valence wave function is a Coulomb function whose phase is shifted, relative to the hydrogenic one, by the part of the orbit spent inside the core. Matching the inside and outside solutions at the core boundary sets a phase that is the same at every energy near threshold, because the electron crosses the small core quickly and picks up an energy-independent phase there. That phase is $\pi {\delta }_{\ell }$.¹

The empirical $n$-dependence, when needed, is a small correction organized by the Rydberg-Ritz expansion

in which the leading constant ${\delta }_{\ell }^{(0)}$ dominates and the higher terms account for the weak variation across a series. The connection between the bound-state defect and the low-energy electron-core scattering phase shift is Seaton's theorem, the foundation of quantum-defect theory.²

Sodium: defects and the term diagram

The measured quantum defects of sodium show the collapse with $\ell$ directly: the $s$ series is shifted by more than one unit, the $p$ series by nearly one, and the $d$ and $f$ series are almost hydrogenic.³

The term diagram is the hydrogen ladder with each series pulled down by its defect. The $s$ terms drop far below the hydrogenic level of the same $n$; the $p$ terms less; the $d$ and $f$ terms sit almost on the hydrogen lines. Optical transitions obey the dipole rule $\Delta \ell =\pm 1$, so the strong lines run between adjacent columns.

The sodium D-line doublet

The strongest line of the principal series is the $3p\to 3s$ transition, the familiar yellow of a sodium lamp. It is not a single line: the $3p$ level is split by the spin-orbit interaction into $3p_{3/2}$ and $3p_{1/2}$, so the transition to the single $3s_{1/2}$ ground level is a doublet, the D lines:³

The separation, $\Delta \lambda \approx 0.6$ nm, corresponds to a fine-structure splitting of the $3p$ level of about $17.2{\text{cm}}^{-1}$, roughly $2.1\text{meV}$. The doublet structure is the everyday fingerprint of the spin-orbit coupling that the next module takes apart in detail.

The quantum defect turns the hydrogen solution into a working model of every alkali. It reduces the entire valence spectrum to a Rydberg formula plus one number per series, and it isolates the two pieces of physics that hydrogen hid: the $\ell$-dependence that core penetration restores, and the spin-orbit splitting that resolves each line into its fine structure. The same effective-quantum-number description, pushed to very large $n$, produces the exaggerated Rydberg atoms of the next lesson.