---
title: Quantum Defects and Alkali Spectra
module: The Quantum Hydrogen Atom
moduleNumber: 2
lessonNumber: 6
order: 206
summary: >
  An alkali atom is one valence electron outside a closed-shell core, and to a
  good approximation it is hydrogen with a modified quantum number. Core
  penetration makes low-ℓ states more bound than the Coulomb formula predicts,
  and the shortfall is captured by a single number per ℓ, the quantum defect δℓ.
  The spectrum then follows the Rydberg formula with n replaced by the effective
  n − δℓ, and the sodium D-line doublet is the worked case.
topics: [The Quantum Hydrogen Atom]
sources:
  - book: Foot
    ref: "Ch. 4 — The Alkalis; §4.5–4.6 Quantum Defect and the Central-Field Approximation"
  - book: Demtröder
    ref: "Ch. 5 — Atoms with More Than One Electron; alkali spectra and term diagrams"
  - book: Bransden & Joachain
    ref: "Ch. 8 — One-Electron Atoms with a Core; the quantum defect"
draft: false
---

The [ℓ-degeneracy of hydrogen](/atomic-physics/quantum-hydrogen-atom/symmetry-degeneracy-runge-lenz)
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 $V(r) \to -ke^2/r$;
- **deep inside the core** ($r \to 0$): the screening is undone and the electron
  sees the full nuclear charge, $V(r) \to -kZe^2/r$.

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 $E_n$.

$$
% caption: The valence electron sees the full nuclear charge Z inside the core
% and a screened charge of 1 outside it; the effective charge Z_eff(r) runs
% between the two, so the potential is central but not Coulombic.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.2,0) node[right, black!70] {$r$};
\draw[->, black] (0,0) -- (0,3.4) node[above, black!70] {net charge};
% core radius marker
\draw[black, dashed] (1.7,0) -- (1.7,3.2);
\node[black, anchor=south] at (1.7,3.2) {core edge};
% Z_eff: high (Z) inside, drops to 1 outside
\draw[acc, very thick] (0.15,3.0) .. controls (0.7,2.95) and (1.2,2.6) .. (1.7,1.6)
  .. controls (2.3,0.85) and (3.2,0.6) .. (6.0,0.5);
\draw[black, dashed] (0,0.5) -- (6.0,0.5);
\node[acc, anchor=east] at (-0.1,3.0) {$Z$};
\node[black, anchor=east] at (-0.1,0.5) {$1$};
\end{tikzpicture}
$$

## Penetration and the quantum defect

Whether the valence electron reaches the region of enhanced charge is decided by
the [centrifugal barrier](/atomic-physics/quantum-hydrogen-atom/radial-equation-in-full)
$\hbar^2\ell(\ell+1)/2\mu r^2$. 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 $-ke^2/r$ 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.

> **Definition (Quantum defect).** The bound energies of an alkali valence
> electron are
> $$
> E_{n\ell} = -\frac{R_M}{(n-\delta_\ell)^2} = -\frac{R_M}{(n^\ast)^2},
> $$
> where $R_M$ is the (reduced-mass) Rydberg energy, $\delta_\ell$ is the
> **quantum defect**, and $n^\ast = n-\delta_\ell$ is the **effective principal
> quantum number**. The defect is nearly independent of $n$ within a series and
> decreases rapidly with $\ell$.

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.

$$
% caption: A penetrating low-ℓ orbit dips inside the core and samples the
% enhanced nuclear attraction; a non-penetrating high-ℓ orbit stays outside and
% sees only the screened Coulomb tail.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% core
\fill[black] (0,0) circle (0.9);
\draw[black] (0,0) circle (0.9);
\fill[black!75] (0,0) circle (1.4pt);
\node[black, anchor=north] at (-1.1,-0.15) {core};
% penetrating eccentric orbit (dips into core)
\draw[acc, very thick, dashed] (0.55,0) ellipse (1.35 and 0.5);
\node[acc, anchor=west] at (2.0,0.2) {penetrating};
% non-penetrating circle (stays out)
\draw[black, very thick] (0,0) circle (1.9);
\node[black, anchor=south] at (0,1.95) {non-penetrating};
\end{tikzpicture}
$$

## 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 $-ke^2/r$, 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 $\pi\delta_\ell$.[^foot-45]

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

$$
\delta_\ell(n) = \delta_\ell^{(0)} + \frac{\delta_\ell^{(2)}}{(n-\delta_\ell^{(0)})^2} + \cdots,
$$

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.[^foot-46]

## 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.[^nist-asd]

| series | $\ell$ | $\delta_\ell$ (Na) | character |
| --- | --- | --- | --- |
| $ns$ | $0$ | $\approx 1.35$ | strongly penetrating |
| $np$ | $1$ | $\approx 0.86$ | penetrating |
| $nd$ | $2$ | $\approx 0.01$ | nearly hydrogenic |
| $nf$ | $3$ | $\approx 0.00$ | hydrogenic |

$$
% caption: The sodium quantum defect δℓ against ℓ, collapsing toward zero once
% the centrifugal barrier keeps the electron out of the core (ℓ ≥ 2).
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (5.6,0) node[right, black!70] {series};
\draw[->, black] (0,0) -- (0,3.0) node[above, black!70] {defect};
\foreach \x/\lab in {0.9/s, 2.1/p, 3.3/d, 4.5/f}
  \node[black, anchor=north] at (\x,0) {\lab};
% delta values scaled: 1.35->2.03, 0.86->1.29, 0.01->0.015, 0->0
\fill[acc] (0.9,2.03) circle (2.4pt);
\fill[acc] (2.1,1.29) circle (2.4pt);
\fill[acc] (3.3,0.05) circle (2.4pt);
\fill[acc] (4.5,0.02) circle (2.4pt);
\draw[acc, thick] (0.9,2.03) .. controls (1.5,1.7) and (1.8,1.5) .. (2.1,1.29)
  .. controls (2.6,0.9) and (2.9,0.15) .. (3.3,0.05) -- (4.5,0.02);
\node[acc, anchor=west] at (1.05,2.2) {1.35};
\end{tikzpicture}
$$

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.

$$
% caption: The sodium term diagram: the s, p, and d series shifted below the
% hydrogenic level of equal n by their quantum defects, with the principal
% series np → 3s marked.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% ionization limit
\draw[black, thick] (-0.3,3.6) -- (6.5,3.6);
\node[black!70, anchor=west] at (6.5,3.6) {ionization};
% columns: s at x=0.6, p at 2.4, d at 4.2, hydrogen ref at 6.0
\node[black!70] at (0.6,-0.4) {ns};
\node[black!70] at (2.4,-0.4) {np};
\node[black!70] at (4.2,-0.4) {nd};
\node[black] at (6.0,-0.4) {H};
% s levels (strongly depressed): 3s low, 4s, 5s
\draw[black, thick] (0.1,0.2) -- (1.1,0.2); \node[black, anchor=east] at (0.05,0.2) {3s};
\draw[black, thick] (0.1,2.2) -- (1.1,2.2); \node[black, anchor=east] at (0.05,2.2) {4s};
\draw[black, thick] (0.1,2.9) -- (1.1,2.9);
% p levels
\draw[black, thick] (1.9,1.5) -- (2.9,1.5); \node[black, anchor=east] at (1.85,1.5) {3p};
\draw[black, thick] (1.9,2.7) -- (2.9,2.7); \node[black, anchor=east] at (1.85,2.7) {4p};
\draw[black, thick] (1.9,3.1) -- (2.9,3.1);
% d levels (near hydrogenic)
\draw[black, thick] (3.7,2.75) -- (4.7,2.75); \node[black, anchor=east] at (3.65,2.75) {3d};
\draw[black, thick] (3.7,3.15) -- (4.7,3.15);
% hydrogen ref levels
\draw[black, dashed] (5.5,2.75) -- (6.5,2.75); \node[black, anchor=west] at (6.5,2.75) {n=3};
\draw[black, dashed] (5.5,3.15) -- (6.5,3.15); \node[black, anchor=west] at (6.5,3.15) {n=4};
% principal series transition 3p -> 3s
\draw[acc, ->] (2.4,1.5) -- (0.6,0.25);
\node[acc, anchor=west] at (0.15,1.05) {D lines};
\end{tikzpicture}
$$

## 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](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession)
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:[^nist-asd]

$$
\begin{aligned}
\text{D}_2:\quad 3p_{3/2}\to 3s_{1/2}, &\qquad \lambda = 588.995~\text{nm},\\
\text{D}_1:\quad 3p_{1/2}\to 3s_{1/2}, &\qquad \lambda = 589.592~\text{nm}.
\end{aligned}
$$

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](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession)
takes apart in detail.

$$
% caption: The sodium D doublet: the 3p level split by spin-orbit coupling into
% two fine-structure levels, each decaying to the 3s ground level, giving the D1
% and D2 lines near 589 nm.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% upper split levels
\draw[acc, thick] (1.5,3.0) -- (3.5,3.0); \node[acc, anchor=west] at (3.55,3.0) {$3p_{\frac{3}{2}}$};
\draw[acc, thick] (1.5,2.5) -- (3.5,2.5); \node[acc, anchor=west] at (3.55,2.5) {$3p_{\frac{1}{2}}$};
% ground level
\draw[black, thick] (1.5,0.0) -- (3.5,0.0); \node[black, anchor=west] at (3.55,0.0) {$3s_{\frac{1}{2}}$};
% transitions
\draw[black, ->] (2.1,3.0) -- (2.1,0.05);
\draw[black, ->] (2.9,2.5) -- (2.9,0.05);
\node[black!70, anchor=east] at (2.05,1.5) {D2};
\node[black!70, anchor=west] at (2.95,1.5) {D1};
\node[black, anchor=south] at (2.5,3.05) {split by spin-orbit};
\node[black, anchor=north] at (2.1,-0.1) {589.0};
\node[black, anchor=north] at (2.95,-0.1) {589.6};
\end{tikzpicture}
$$

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](/atomic-physics/quantum-hydrogen-atom/rydberg-atoms)
of the next lesson.

[^foot-45]: **Foot**, _Atomic Physics_, §4.5 — the quantum defect, its origin in core penetration, and the interpretation of $\pi\delta_\ell$ as the phase shift of the valence wave function relative to the pure-Coulomb solution. <https://global.oup.com/academic/product/atomic-physics-9780198506959>
[^foot-46]: **Foot**, _Atomic Physics_, §4.6 — the Rydberg-Ritz expansion of $\delta_\ell(n)$ and the connection to low-energy electron scattering (Seaton's theorem) underlying quantum-defect theory. <https://global.oup.com/academic/product/atomic-physics-9780198506959>
[^nist-asd]: **NIST Atomic Spectra Database**, sodium (Na I) levels and lines — quantum defects inferred from the tabulated term values and the D-line wavelengths $\lambda(\text{D}_2)=588.995$ nm, $\lambda(\text{D}_1)=589.592$ nm. <https://www.nist.gov/pml/atomic-spectra-database>