---
title: The Darwin Term and the Fine-Structure Formula
module: Fine Structure and the Dirac Atom
moduleNumber: 3
lessonNumber: 4
order: 304
summary: >
  The third fine-structure correction, the Darwin term, is a contact interaction
  proportional to ∇²V that acts only on s-states, physically a smearing of the
  electron over a Compton wavelength. Adding the relativistic, spin-orbit, and
  Darwin shifts, the ℓ-dependence cancels and the total collapses to a formula in n
  and j alone. The n=2 shell splits into 2S₁/₂, 2P₁/₂, 2P₃/₂, with the two j=½
  levels exactly degenerate, a coincidence the Dirac theory explains.
topics: [Fine Structure and the Dirac Atom]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 7 — Perturbation Theory; §7.3.3 The Darwin Term and Fine Structure"
  - book: Bransden & Joachain
    ref: "Ch. 5 — One-Electron Atoms in Fields; §5.1 Fine Structure"
  - book: Foot
    ref: "Ch. 2 — The Hydrogen Atom; fine-structure summary"
draft: false
---

Two of the three fine-structure corrections are now in hand: the
[relativistic kinetic term](/atomic-physics/fine-structure-and-the-dirac-atom/relativistic-kinetic-correction),
which acts on every state, and the
[spin-orbit term](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession),
which acts only on $\ell\ge1$ states and left an unresolved $0\times\infty$ for
$\ell=0$. The third correction, the Darwin term, fills exactly that gap: it acts
only on $\ell=0$ states, where $|\psi(0)|^2\ne0$. It has no classical analogue, and
its origin is only made fully transparent by the
[Dirac equation](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen);
the non-relativistic account describes it as a smearing of the electron's position
over a length of order its Compton wavelength. Once all three shifts are summed, a
striking simplification occurs: the separate $\ell$-dependences cancel, and the
total fine-structure shift depends only on $n$ and the total angular momentum $j$.

## The contact interaction

The reduction of the Dirac equation to a two-component non-relativistic theory (the
Foldy–Wouthuysen transformation) produces, beyond the mass-velocity and spin-orbit
terms, a third correction of the same order:[^gs-733][^bj-51]

$$
H_D = \frac{\hbar^2}{8m^2c^2}\,\nabla^2 V(r).
$$

For the Coulomb potential $V = -Ze^2/4\pi\epsilon_0 r$, the Laplacian of $1/r$ is a
delta function,

$$
\nabla^2\frac{1}{r} = -4\pi\,\delta^3(\vec r),
$$

so the Darwin term is a **contact interaction**, nonzero only at the origin:

$$
H_D = \frac{\hbar^2}{8m^2c^2}\frac{Ze^2}{4\pi\epsilon_0}\,4\pi\,\delta^3(\vec r)
= \frac{\pi\hbar^2}{2m^2c^2}\frac{Ze^2}{4\pi\epsilon_0}\,\delta^3(\vec r).
$$

A delta function at the origin sees only the value of the wave function there. The
first-order shift is

$$
E_D^{(1)} = \langle H_D\rangle
= \frac{\pi\hbar^2}{2m^2c^2}\frac{Ze^2}{4\pi\epsilon_0}\,|\psi(0)|^2.
$$

Only $\ell=0$ states have nonzero amplitude at the nucleus. Every $\ell\ge1$ wave
function carries a factor $r^\ell$ that vanishes at the origin, so $|\psi(0)|^2 = 0$
and the Darwin term contributes nothing there, precisely complementing the
spin-orbit term's domain.

> **Definition (Darwin term).** The contact correction $H_D = (\hbar^2/8m^2c^2)
> \nabla^2 V$, for a Coulomb field a delta function at the origin. It shifts only
> $\ell=0$ states, by an amount set by the probability density $|\psi(0)|^2$ at the
> nucleus, and has no counterpart in the classical or Bohr models.

## Zitterbewegung: why the electron is smeared

The physical picture behind the contact term is that the electron does not sit at a
sharp point but jitters over a region of order its reduced Compton wavelength
$\hbar/mc$. In the Dirac theory this trembling motion is called
**zitterbewegung**, an interference between the positive- and negative-energy
components of the spinor at the frequency $2mc^2/\hbar$. Averaged over the jitter,
the electron samples the potential not at $\vec r$ but over a small neighborhood,
and the potential it feels is the smeared average

$$
\langle V(\vec r + \delta\vec r)\rangle
\approx V(\vec r) + \tfrac12\langle\delta r_i\,\delta r_j\rangle\,\partial_i\partial_j V
\approx V(\vec r) + \tfrac16\langle(\delta r)^2\rangle\,\nabla^2 V.
$$

With $\langle(\delta r)^2\rangle$ of order $(\hbar/mc)^2$, the correction is
$\sim(\hbar/mc)^2\nabla^2 V$, matching the Darwin form up to the numerical factor
that the careful reduction fixes at $\hbar^2/8m^2c^2$. Where the potential is
smooth (away from the origin) the smearing does nothing; where it is sharply curved
(at the Coulomb singularity) the average differs from the point value, and only
s-states, which visit the origin, register the difference.

$$
% caption: The point electron (top) feels V at one location; the smeared electron
% (bottom), jittering over a Compton-wavelength region, averages V over a small
% ball and so samples the curvature ∇²V near the origin.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% Coulomb well
\draw[->, black] (0,0) -- (6.2,0) node[right, black!70] {$r$};
\draw[->, black] (0,-2.6) -- (0,1.0) node[above, black!70] {$V$};
\draw[black, very thick] (0.35,-2.5) .. controls (0.8,-1.4) and (1.8,-0.75) .. (3.0,-0.5)
  .. controls (4.2,-0.32) and (5.4,-0.2) .. (6.0,-0.16);
% point electron
\fill[black] (0.85,-1.25) circle (2.4pt);
\node[black, anchor=west] at (1.0,-1.15) {point};
% smeared electron (a fuzzy ball)
\draw[acc, dashed] (0.85,-2.0) circle (0.32);
\fill[acc] (0.85,-2.0) circle (0.28);
\node[acc, anchor=west] at (1.3,-2.0) {smeared (Compton size)};
\end{tikzpicture}
$$

## Evaluating the Darwin shift

The s-state density at the origin follows from the hydrogenic
[wave functions](/atomic-physics/quantum-hydrogen-atom/hydrogen-wave-functions).
For $\ell=0$ the spherical harmonic is $Y_0^0 = 1/\sqrt{4\pi}$ and the radial
function at the origin gives

$$
|\psi_{n00}(0)|^2 = \frac{1}{\pi}\left(\frac{Z}{n a_0}\right)^3
= \frac{Z^3}{\pi n^3 a_0^3}.
$$

Substituting into $E_D^{(1)}$,

$$
E_D^{(1)} = \frac{\pi\hbar^2}{2m^2c^2}\frac{Ze^2}{4\pi\epsilon_0}
\frac{Z^3}{\pi n^3 a_0^3}
= \frac{\hbar^2}{2m^2c^2}\frac{Z^4 e^2}{4\pi\epsilon_0\,n^3 a_0^3}.
$$

Reducing the constants against $E_n$ and $a_0$ (using $a_0 = \hbar^2/mb$ with
$b = Ze^2/4\pi\epsilon_0$, so $m a_0 b = \hbar^2$) gives the compact form[^gs-733]

$$
E_D^{(1)} = \frac{E_n^2}{mc^2}\,2n
= \frac{(Z\alpha)^4 mc^2}{2n^3},
\qquad \ell = 0.
$$

The structural role of this value appears when it is combined with the
relativistic shift at $\ell=0$. Adding $E_r^{(1)}(\ell=0) =
-(E_n^2/2mc^2)(8n-3)$ to $E_D^{(1)} = (E_n^2/2mc^2)(4n)$ gives
$(E_n^2/2mc^2)(3-4n)$, reproducing the fine-structure formula evaluated at
$j=\tfrac12$. The Darwin term is precisely what makes the combined shift, once
written in terms of $j$, extend continuously to the $\ell=0$ end where the
spin-orbit expression alone is indeterminate.

## Summing the three corrections

For $\ell\ge1$ the fine-structure shift is $E_r^{(1)} + E_{\text{so}}^{(1)}$ (no
Darwin), and for $\ell=0$ it is $E_r^{(1)} + E_D^{(1)}$ (no spin-orbit). Both cases
reduce to the same expression. Take $\ell\ge1$ and add:

$$
E_{\text{fs}}^{(1)} = E_r^{(1)} + E_{\text{so}}^{(1)}
= -\frac{E_n^2}{2mc^2}\left[\frac{4n}{\ell+\tfrac12} - 3\right]
+ \frac{E_n^2}{mc^2}
\frac{n[\,j(j+1)-\ell(\ell+1)-\tfrac34\,]}{\ell(\ell+\tfrac12)(\ell+1)}.
$$

The algebra is a matter of putting both terms over the common denominator
$\ell(\ell+\tfrac12)(\ell+1)$ and using $j(j+1) = \ell(\ell+1) + \tfrac34 \pm
(\ell+\tfrac12)$ for $j = \ell\pm\tfrac12$. Every explicit $\ell$ cancels, leaving a
result that depends on $\ell$ only through $j$:[^gs-733]

$$
E_{\text{fs}}^{(1)} = \frac{E_n^2}{2mc^2}\left(3 - \frac{4n}{j+\tfrac12}\right).
$$

The $\ell=0$ case, with $E_r^{(1)}$ and $E_D^{(1)}$ and $j=\tfrac12$, reproduces the
same formula. The cancellation of $\ell$ is not an accident of the perturbative
bookkeeping; it reflects the deeper structure the
[Dirac equation](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen)
makes exact, in which $j$ and not $\ell$ is the natural label.

$$
% caption: The three corrections combine per level: relativistic (all ℓ),
% spin-orbit (ℓ≥1), Darwin (ℓ=0). Their sum, shown at right, depends only on n and
% j and is negative for both n=2 sublevels.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth,
  bx/.style={draw, minimum width=17mm, minimum height=8mm, align=center}]
\definecolor{acc}{HTML}{4A6FA5}
\node[bx, draw=black, text=black] (r) at (0,1.6) {relativistic};
\node[bx, draw=black, text=black] (so) at (0,0) {spin-orbit};
\node[bx, draw=black, text=black] (d) at (0,-1.6) {Darwin};
\node[bx, thick, draw=acc, text=acc] (sum) at (5.2,0) {$E^{(1)}_{\text{fs}}$};
\draw[black, ->] (r) -- (sum);
\draw[black, ->, dashed] (so) -- (sum);
\draw[black, ->, densely dotted] (d) -- (sum);
\node[black, anchor=south, font=\scriptsize] at (2.0,1.15) {$l$ cancels};
\end{tikzpicture}
$$

> **Theorem (Fine-structure formula).** To first order in perturbation theory the
> combined relativistic, spin-orbit, and Darwin corrections shift the hydrogenic
> level $E_n$ by
> $$
> E_{\text{fs}}^{(1)} = \frac{E_n^2}{2mc^2}\left(3 - \frac{4n}{j+\tfrac12}\right)
> = -\frac{E_n\,(Z\alpha)^2}{n^2}\left(\frac{n}{j+\tfrac12} - \frac34\right),
> $$
> so that
> $$
> E_{nj} = -\frac{Z^2\,\mathrm{Ry}}{n^2}
> \left[1 + \frac{(Z\alpha)^2}{n^2}\left(\frac{n}{j+\tfrac12} - \frac34\right)\right].
> $$
> The corrected energy depends only on $n$ and $j$; states of the same $n$ and $j$
> but different $\ell$ remain exactly degenerate at this order.

## The n=2 shell

The fine-structure formula turns the fourfold-degenerate (eightfold with spin)
$n=2$ level into a small ladder. The allowed states are $2s$ ($\ell=0$, $j=\tfrac12$),
$2p$ ($\ell=1$, $j=\tfrac12$ or $\tfrac32$). In spectroscopic notation
$n\,{}^{2s+1}\!L_j$ these are $2S_{1/2}$, $2P_{1/2}$, and $2P_{3/2}$. The formula
assigns

$$
E_{\text{fs}}(j=\tfrac12) = \frac{E_2^2}{2mc^2}\left(3 - \frac{8}{1}\right)
= -\frac{5E_2^2}{2mc^2},
\qquad
E_{\text{fs}}(j=\tfrac32) = \frac{E_2^2}{2mc^2}\left(3 - \frac{8}{2}\right)
= -\frac{E_2^2}{2mc^2}.
$$

So $2S_{1/2}$ and $2P_{1/2}$ share the lower energy (both $j=\tfrac12$), and
$2P_{3/2}$ sits above them by $4\,E_2^2/2mc^2 = 2E_2^2/mc^2$. Numerically the
$2P_{3/2}$–$2P_{1/2}$ separation is about $4.5\times10^{-5}$ eV, corresponding to
$10.9$ GHz, the fine-structure splitting of the Balmer-$\alpha$ line.

$$
% caption: The n=2 level of hydrogen split by fine structure. 2S₁/₂ and 2P₁/₂
% (both j=½) coincide and lie below 2P₃/₂ (j=3/2); the degeneracy of the two j=½
% levels is the Dirac prediction later broken by the Lamb shift.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% unperturbed n=2
\draw[black, dashed] (-0.2,2.6) -- (1.6,2.6);
\node[black, anchor=east] at (-0.3,2.6) {$n=2$};
% 2P3/2 above
\draw[acc, very thick] (3.0,2.9) -- (5.0,2.9);
\node[acc, anchor=west] at (5.1,2.9) {$2P_{\frac32}$};
% 2S1/2 and 2P1/2 degenerate below
\draw[acc, very thick] (3.0,1.4) -- (5.0,1.4);
\node[acc, anchor=west] at (5.1,1.55) {$2S_{\frac12}{=}2P_{\frac12}$};
\draw[black, dotted] (1.6,2.6) -- (3.0,2.9);
\draw[black, dotted] (1.6,2.6) -- (3.0,1.4);
% splitting arrow
\draw[black, <->] (2.4,2.9) -- (2.4,1.4);
\node[black, anchor=west, font=\scriptsize] at (2.55,2.15) {10.9 GHz};
\end{tikzpicture}
$$

> **Worked example.** Evaluate the $n=2$ fine-structure splitting numerically. The
> overall scale is $E_2^2/2mc^2$ with $E_2 = -3.40$ eV and $mc^2 = 511$ keV, so
> $$
> \frac{E_2^2}{2mc^2} = \frac{(3.40~\text{eV})^2}{2\times 5.11\times10^{5}~\text{eV}}
> = 1.13\times10^{-5}~\text{eV}.
> $$
> The $2P_{3/2}$–$2P_{1/2}$ gap is $2E_2^2/mc^2 = 4\times(1.13\times10^{-5}) =
> 4.5\times10^{-5}$ eV. Converting to frequency with $E = h\nu$,
> $$
> \nu = \frac{4.5\times10^{-5}~\text{eV}}{4.14\times10^{-15}~\text{eV\,s}}
> = 1.1\times10^{10}~\text{Hz} \approx 11~\text{GHz},
> $$
> matching the measured $10.969$ GHz fine-structure interval of hydrogen $n=2$. The
> same scale controls the alkali doublets: the sodium D-line splitting is larger
> because the effective $Z$ seen by the penetrating $3p$ electron near the core is
> well above unity.

## The level ordering and the surviving degeneracy

The fine structure lowers all levels (the bracket $n/(j+\tfrac12) - \tfrac34$ is
positive for every allowed state) and orders them by $j$: at fixed $n$, larger $j$
means a smaller downward shift, so the levels rise with $j$. The dependence on $j$
alone, not $\ell$, is the signature feature.

$$
% caption: Within a fixed n, the fine-structure shift rises monotonically with j
% (smaller downward shift); the horizontal axis is j and levels of different ℓ but
% equal j land on the same point.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.2,0) node[right, black!70] {$j$};
\draw[->, black] (0,-3.0) -- (0,0.7) node[above, black!70] {$E_{\text{fs}}$};
\foreach \x/\lab in {1/{\tfrac12}, 2.5/{\tfrac32}, 4/{\tfrac52}, 5.5/{\tfrac72}} \node[black, anchor=south] at (\x,0.08) {$\lab$};
% shift ~ -(n/(j+1/2)-3/4); take n=4: j=1/2 -> -(4-.75)=-3.25; j=3/2->-(2-.75)=-1.25; j=5/2->-(1.333-.75)=-0.583; j=7/2->-(1-.75)=-0.25
% scale by 0.85
\foreach \x/\y in {1/-2.76, 2.5/-1.06, 4/-0.50, 5.5/-0.21}{
  \draw[black, very thick] (\x-0.45,\y) -- (\x+0.45,\y);
  \fill[black] (\x,\y) circle (1.4pt);
}
\draw[black, dotted] (1,-2.76) .. controls (1.8,-1.7) and (2.5,-1.06) .. (2.5,-1.06);
\node[black, anchor=north west, font=\scriptsize] at (3.4,-1.5) {rises toward $E=0$};
\end{tikzpicture}
$$

The degeneracy of $2S_{1/2}$ and $2P_{1/2}$ is the sharpest prediction of the
formula and the most consequential. Perturbation theory gives it, and the exact
[Dirac equation](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen)
gives it too: states of equal $n$ and $j$ are degenerate to all orders in the Dirac
Coulomb problem. Experiment disagrees. In 1947 Lamb and Retherford measured a
splitting of about $1$ GHz between $2S_{1/2}$ and $2P_{1/2}$, the
[Lamb shift](/atomic-physics/qed-corrections-and-hyperfine-structure/lamb-shift-qed),
which no theory built on a single-particle wave equation can produce. Its
explanation requires the quantized electromagnetic field, and it marks the boundary
where atomic fine structure meets quantum electrodynamics. The fine-structure
formula is exact within its domain; the domain simply does not include the vacuum
fluctuations that lift the last degeneracy.

[^gs-733]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_, 3rd ed., §7.3.3 — the Darwin term $H_D = (\hbar^2/8m^2c^2)\nabla^2 V$ as a contact interaction on s-states, the sum of relativistic, spin-orbit, and Darwin shifts, and the closed fine-structure formula $E_{\text{fs}}^{(1)} = (E_n^2/2mc^2)(3 - 4n/(j+\tfrac12))$.
[^bj-51]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §5.1 — the Darwin term from the Foldy–Wouthuysen reduction, its interpretation as zitterbewegung smearing, and the assembled fine-structure spectrum. <https://www.pearson.com/en-gb/subject-catalog/p/physics-of-atoms-and-molecules/P200000005386>