---
title: Spin-Orbit Coupling and Thomas Precession
module: Fine Structure and the Dirac Atom
moduleNumber: 3
lessonNumber: 3
order: 303
summary: >
  In the electron's rest frame the nucleus orbits it, and the resulting current
  produces a magnetic field that couples to the electron's spin moment. The
  interaction is ξ(r) L·S, with ξ built from the Coulomb potential and the radial
  expectation ⟨1/r³⟩. A relativistic subtlety, Thomas precession, halves the naive
  coefficient because the electron's rest frame is accelerating. The result splits
  each ℓ≥1 level into a j=ℓ±½ doublet and makes (n, ℓ, j, mⱼ) the good quantum
  numbers.
topics: [Fine Structure and the Dirac Atom]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 7 — Perturbation Theory; §7.3.2 Spin-Orbit Coupling"
  - book: Foot
    ref: "Ch. 5 — The LS-Coupling Scheme; §5.1–5.3 Spin-Orbit Interaction"
  - book: Bransden & Joachain
    ref: "Ch. 5 — One-Electron Atoms in Fields; §5.1 Fine Structure"
draft: false
---

The [relativistic kinetic correction](/atomic-physics/fine-structure-and-the-dirac-atom/relativistic-kinetic-correction)
accounts for the electron moving fast; it says nothing about the electron's
[spin](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession).
Spin carries a magnetic moment, and a moving electron in the nuclear electric field
experiences, in its own frame, a magnetic field. The coupling of the spin moment to
that field is the spin-orbit interaction. It is the largest of the three
fine-structure terms for $\ell\ge1$ states and the one that splits a single
non-relativistic level into a doublet. Two ingredients need care: the magnitude of
the internal field, obtained by transforming the Coulomb field into the electron's
frame, and a factor of one-half that a naive frame transformation misses because
the electron's rest frame is not inertial. The second is Thomas precession, and
getting it right is what makes the spin-orbit energy agree with experiment and with
the exact [Dirac](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen)
result.

## The magnetic field in the electron's frame

In the laboratory frame the proton sits at rest and the electron orbits it. Boost
to the instantaneous rest frame of the electron: now the electron is momentarily at
rest and the proton circulates around it. A circulating charge is a current loop,
and a current loop makes a magnetic field at its center, where the electron sits.

Quantitatively, the leading-order transformation of the electromagnetic field to a
frame moving with velocity $\vec v$ relative to the lab gives, for a purely
electric lab field $\vec E$,[^gs-732]

$$
\vec B' = -\frac{1}{c^2}\,\vec v\times\vec E.
$$

The nuclear Coulomb field is radial, $\vec E = \dfrac{1}{e}\dfrac{\d V}{\d r}\,\hat r$,
where $V(r) = -Ze^2/4\pi\epsilon_0 r$ is the potential energy of the electron and
the factor $1/e$ converts it to the field seen by the charge. Using
$\vec v\times\hat r = \vec v\times\vec r/r$ and the orbital angular momentum
$\vec L = m\,\vec r\times\vec v$, so that $\vec r\times\vec v = -\vec L/m$,

$$
\vec B' = -\frac{1}{c^2}\vec v\times\frac{1}{e}\frac{\d V}{\d r}\frac{\vec r}{r}
= \frac{1}{m e c^2}\frac{1}{r}\frac{\d V}{\d r}\,\vec L.
$$

The internal field is parallel to $\vec L$ and its strength is governed by the
radial derivative of the potential. For hydrogen this field is enormous: at the
Bohr radius it reaches several tesla, which is why the spin-orbit splitting, though
tiny on the scale of the binding energy, is a real magnetic interaction.

$$
% caption: In the electron's instantaneous rest frame the nucleus (charge +Ze)
% circulates, a current loop whose magnetic field B at the electron points along
% the orbital angular momentum L.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% orbit of nucleus around electron
\draw[black, thick] (0,0) ellipse (2.4 and 1.4);
% electron at center
\fill[black] (0,0) circle (2.6pt);
\node[black, anchor=north east] at (-0.05,-0.05) {electron};
% nucleus on the orbit
\fill[black] (2.4,0) circle (2.8pt);
\node[black, anchor=west] at (2.5,0) {nucleus $+Ze$};
% velocity arrow of nucleus
\draw[black, ->, thick] (2.4,0.25) -- (2.4,1.05) node[right] {$v$};
% B field out of center along L
\draw[acc, ->, very thick] (0,0) -- (0,1.7) node[above] {$B$ along $L$};
% current sense
\draw[black, ->] (-1.6,0.9) arc (150:210:1.4);
\node[black, anchor=east] at (-2.0,0) {current};
\end{tikzpicture}
$$

## The interaction Hamiltonian

The electron's [spin magnetic moment](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession)
is

$$
\vec\mu_s = -g_s\frac{e}{2m}\,\vec S,
\qquad g_s \approx 2,
$$

with $g_s$ the spin g-factor, very close to $2$. The energy of a magnetic moment in
a field is $-\vec\mu_s\cdot\vec B'$, so the naive spin-orbit energy is

$$
H_{\text{naive}} = -\vec\mu_s\cdot\vec B'
= g_s\frac{e}{2m}\,\vec S\cdot\frac{1}{mec^2}\frac{1}{r}\frac{\d V}{\d r}\vec L
= \frac{g_s}{2m^2c^2}\frac{1}{r}\frac{\d V}{\d r}\,\vec S\cdot\vec L.
$$

With $g_s = 2$ the prefactor is $1/m^2c^2$. This expression is wrong by a factor of
two. The error is not in the field transformation but in the assumption that the
energy of the moment can be read off in the electron's rest frame as though that
frame were inertial. It is not: the electron accelerates continuously toward the
nucleus, and a sequence of infinitesimal boosts along a curved path composes into a
net rotation, the Thomas precession, that the moment feels as an additional
effective field.

## Thomas precession

Two Lorentz boosts in different directions do not compose into a pure boost; their
product is a boost followed by a rotation, the Wigner rotation. An accelerating
particle is boosted successively along a turning velocity vector, so its rest frame
rotates relative to the lab even when no torque acts. The angular velocity of that
rotation, to lowest order in $v/c$, is[^gs-732][^foot-51]

$$
\vec\omega_T = -\frac{1}{2c^2}\,\vec a\times\vec v,
$$

where $\vec a$ is the electron's acceleration. For a bound electron the
acceleration is Coulombic, $\vec a = \vec F/m = -(1/m)(\d V/\d r)\hat r$, directed
toward the nucleus. The precession $\vec\omega_T$ is opposite to the orbital angular velocity
and is exactly half the magnitude that a naive co-rotating frame would
assign.

> **Definition (Thomas precession).** The kinematic precession of an accelerating
> body's rest frame relative to the lab, arising because successive non-collinear
> Lorentz boosts compose to a boost plus a rotation. For circular motion its rate is
> $\omega_T = (\gamma-1)\,\omega_{\text{orbit}} \approx \tfrac12 (v/c)^2\,
> \omega_{\text{orbit}}$, independent of any force law and present for any
> accelerated frame.

The moment precesses in the rotating rest frame at the rate set by the internal
field, but the frame itself precesses at $\vec\omega_T$; adding the two, the net
spin precession seen in the lab corresponds to an interaction energy reduced by the
factor $\tfrac12$. The corrected spin-orbit Hamiltonian is[^gs-732]

$$
H_{\text{so}} = \frac{1}{2m^2c^2}\frac{1}{r}\frac{\d V}{\d r}\,\vec S\cdot\vec L.
$$

The explicit factor $\tfrac12$ is the Thomas factor. It cancels the $g_s = 2$ that
would otherwise double the result, leaving an effective coefficient equivalent to
$g_s = 1$ for orbital motion. The same $\tfrac12$ emerges automatically, with no
frame gymnastics, from the non-relativistic reduction of the
[Dirac equation](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen);
its appearance there is the strongest evidence that the Thomas argument is correct.

$$
% caption: The spin axis (double arrow) precesses about the internal field while
% the electron rest frame itself precesses backward at ω_T; the two combine to
% halve the naive spin-orbit energy.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% orbit
\draw[black, thick] (0,0) circle (1.9);
\fill[black] (0,0) circle (2.2pt);
\node[black, anchor=north east] at (-0.05,-0.05) {nucleus};
% electron on orbit
\fill[black] (1.9,0) circle (2.6pt);
\node[black, anchor=west] at (2.05,0) {electron};
% orbital angular velocity
\draw[black, ->] (0.55,1.55) arc (70:110:1.6);
\node[black, anchor=south] at (0,1.75) {orbit};
% spin double arrow at electron
\draw[acc, ->, very thick] (1.9,0.15) -- (1.9,1.25);
\draw[acc, ->, very thick] (1.9,0.15) -- (2.7,0.9);
\node[acc, anchor=west] at (2.7,0.95) {spin};
% Thomas precession arrow (backward)
\draw[acc, ->] (1.35,0.55) arc (150:220:0.6);
\node[acc, anchor=east] at (1.25,0.2) {Thomas};
\end{tikzpicture}
$$

## Reducing L·S with the good quantum numbers

The Hamiltonian contains $\vec S\cdot\vec L$, which mixes the separate orientations
of spin and orbit. The individual projections $L_z$ and $S_z$ are no longer
conserved, because $\vec S\cdot\vec L$ does not commute with them. What is conserved
is the total angular momentum $\vec J = \vec L + \vec S$: since
$H_{\text{so}}\propto\vec L\cdot\vec S$ is a scalar built from $\vec L$ and $\vec S$,
it commutes with $J^2$, $J_z$, $L^2$, and $S^2$. Squaring $\vec J = \vec L + \vec S$,

$$
J^2 = L^2 + S^2 + 2\,\vec L\cdot\vec S
\quad\Longrightarrow\quad
\vec L\cdot\vec S = \tfrac12\big(J^2 - L^2 - S^2\big).
$$

On a simultaneous eigenstate $\lvert n\,\ell\,j\,m_j\rangle$ with $s=\tfrac12$,

$$
\langle\vec L\cdot\vec S\rangle
= \frac{\hbar^2}{2}\big[\,j(j+1) - \ell(\ell+1) - \tfrac34\,\big].
$$

For a given $\ell\ge1$ the total angular momentum takes two values, $j = \ell +
\tfrac12$ and $j = \ell - \tfrac12$, splitting the level into a doublet. The two
carry $\vec L\cdot\vec S$ of opposite sign:

$$
\langle\vec L\cdot\vec S\rangle =
\begin{cases}
+\tfrac12\ell\,\hbar^2, & j = \ell+\tfrac12,\\[4pt]
-\tfrac12(\ell+1)\,\hbar^2, & j = \ell-\tfrac12.
\end{cases}
$$

> **Definition (Good quantum numbers for fine structure).** The spin-orbit term
> breaks the separate conservation of $L_z$ and $S_z$ but preserves $J^2$ and
> $J_z$. The states that diagonalize $H_{\text{so}}$ within a degenerate shell are
> the coupled states $\lvert n\,\ell\,j\,m_j\rangle$, labelled by $(n,\ell,j,m_j)$
> rather than $(n,\ell,m_\ell,m_s)$. These are the "good" states for degenerate
> perturbation theory.

$$
% caption: Adding spin S (length √3/2 ℏ) to orbit L gives total J of length
% √(j(j+1)) ℏ in two ways: parallel-ish for j=ℓ+½ and antiparallel-ish for
% j=ℓ−½, the two rungs of the doublet.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% left: j = l + 1/2
\draw[black, ->, very thick] (0,0) -- (0,2.2) node[left] {$L$};
\draw[black, ->, very thick, densely dashed] (0,2.2) -- (0.9,3.0) node[right] {$S$};
\draw[acc, ->, thick] (0,0) -- (0.9,3.0);
\node[acc, anchor=west] at (0.55,1.5) {$J$};
\node[anchor=north] at (0.45,-0.15) {$j$ larger};
% right: j = l - 1/2
\begin{scope}[xshift=4.5cm]
\draw[black, ->, very thick] (0,0) -- (0,2.2) node[left] {$L$};
\draw[black, ->, very thick, densely dashed] (0,2.2) -- (0.9,1.4) node[right] {$S$};
\draw[acc, ->, thick] (0,0) -- (0.9,1.4);
\node[acc, anchor=west] at (0.55,0.6) {$J$};
\node[anchor=north] at (0.45,-0.15) {$j$ smaller};
\end{scope}
\end{tikzpicture}
$$

## The radial factor and the shift

Collecting the pieces, the first-order energy is the product of the angular factor
above and the radial expectation of the coefficient. For the Coulomb potential
$V = -Ze^2/4\pi\epsilon_0 r$, the derivative is $\d V/\d r = Ze^2/4\pi\epsilon_0
r^2$, so

$$
H_{\text{so}} = \frac{1}{2m^2c^2}\frac{Ze^2}{4\pi\epsilon_0}\frac{1}{r^3}\,\vec L\cdot\vec S
\equiv \xi(r)\,\vec L\cdot\vec S,
\qquad
\xi(r) = \frac{Ze^2}{8\pi\epsilon_0 m^2c^2}\frac{1}{r^3}.
$$

The energy requires the radial expectation $\langle 1/r^3\rangle$, the hydrogenic
[result](/atomic-physics/quantum-hydrogen-atom/expectation-values-virial)

$$
\left\langle\frac{1}{r^3}\right\rangle
= \frac{Z^3}{n^3\,\ell\left(\ell+\tfrac12\right)(\ell+1)\,a_0^3},
\qquad \ell \ge 1.
$$

The restriction $\ell\ge1$ is essential: $\langle 1/r^3\rangle$ diverges for $\ell=0$,
and indeed the $\vec L\cdot\vec S$ factor vanishes there ($\vec L = 0$ for an
s-state), an indeterminate $0\times\infty$ whose correct handling is the
[Darwin term](/atomic-physics/fine-structure-and-the-dirac-atom/darwin-term-fine-structure-formula).
Assembling the angular factor $\tfrac12[j(j+1)-\ell(\ell+1)-\tfrac34]$ with the
radial expectation gives the closed form:[^gs-732]

$$
E_{\text{so}}^{(1)}
= \frac{Ze^2}{8\pi\epsilon_0 m^2c^2}\frac{\hbar^2}{2}
\big[j(j+1)-\ell(\ell+1)-\tfrac34\big]
\frac{Z^3}{n^3\ell(\ell+\tfrac12)(\ell+1)a_0^3}.
$$

Simplifying against $E_n = -Z^2 e^2/8\pi\epsilon_0 a_0 n^2$ collapses the constants
to

> **Theorem (Spin-orbit shift for hydrogen).** For $\ell\ge1$ the first-order
> spin-orbit energy is
> $$
> E_{\text{so}}^{(1)}
> = \frac{E_n^2}{mc^2}\,
> \frac{n\big[\,j(j+1)-\ell(\ell+1)-\tfrac34\,\big]}
> {\ell\left(\ell+\tfrac12\right)(\ell+1)}.
> $$
> It is positive for $j=\ell+\tfrac12$ (spin aligned with the internal field
> raises the energy) and negative for $j=\ell-\tfrac12$, so the $j=\ell+\tfrac12$
> member of the doublet lies above the $j=\ell-\tfrac12$ member. The splitting is of
> order $(Z\alpha)^4 mc^2$, the same fine-structure scale as the relativistic
> term.

## The doublet and the Landé interval

The two rungs of a spin-orbit doublet are separated by an interval fixed by the
angular factors. The energy of the upper member minus the lower is

$$
\Delta E_{\text{so}} = E_{\text{so}}(\ell+\tfrac12) - E_{\text{so}}(\ell-\tfrac12)
= \frac{E_n^2}{mc^2}\frac{n(2\ell+1)}{\ell(\ell+\tfrac12)(\ell+1)}
= \frac{E_n^2}{mc^2}\frac{2n}{\ell(\ell+1)},
$$

using $\langle\vec L\cdot\vec S\rangle_{\ell+1/2} -
\langle\vec L\cdot\vec S\rangle_{\ell-1/2} = (2\ell+1)\hbar^2/2$. The classic case is
the sodium doublet: the $3p$ level splits into $3p_{1/2}$ and $3p_{3/2}$, and the
transitions to $3s_{1/2}$ produce the two yellow D-lines at $589.0$ and $589.6$ nm.

$$
% caption: A single p level (ℓ=1) splits under spin-orbit coupling into j=3/2
% above and j=1/2 below; the doublet spacing is the Landé interval, and the two
% transitions to an s level form the sodium D-line pair.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% unperturbed p
\draw[black, dashed] (0,2.4) -- (2.2,2.4);
\node[black, anchor=east] at (0,2.4) {$p$};
% split levels
\draw[acc, very thick] (2.6,2.75) -- (4.6,2.75);
\node[acc, anchor=west] at (4.7,2.75) {$p_{\frac32}$};
\draw[acc, very thick] (2.6,1.85) -- (4.6,1.85);
\node[acc, anchor=west] at (4.7,1.85) {$p_{\frac12}$};
\draw[black, dotted] (2.2,2.4) -- (2.6,2.75);
\draw[black, dotted] (2.2,2.4) -- (2.6,1.85);
% interval bracket
\draw[black, <->] (3.15,2.75) -- (3.15,1.85);
\node[black, anchor=east, font=\scriptsize] at (3.05,2.3) {interval};
% s level below
\draw[black, very thick] (2.6,-0.4) -- (4.6,-0.4);
\node[black, anchor=west] at (4.7,-0.4) {$s_{\frac12}$};
% two transitions
\draw[black, ->] (3.0,1.85) -- (3.0,-0.35);
\draw[black, ->] (4.2,2.75) -- (4.2,-0.35);
\node[black, anchor=north, font=\scriptsize] at (3.6,-0.5) {D lines};
\end{tikzpicture}
$$

The interval rule generalizes: within a fine-structure multiplet the spacing
between adjacent $j$ levels is proportional to the larger $j$, the
[Landé interval rule](/atomic-physics/many-electron-atoms/ls-jj-coupling-term-symbols),
because $\langle\vec L\cdot\vec S\rangle$ increases by $j\hbar^2$ from one level to
the next. For hydrogen the doublet is the whole multiplet, but in many-electron
atoms the same $\vec L\cdot\vec S$ structure produces multiplets of several
components, all governed by this rule.

The spin-orbit shift carries the "spin" content of fine structure, complementing
the "velocity" content of the relativistic term. Neither alone gives the observed
spectrum. Their sum, once the $\ell=0$ gap is filled by the
[Darwin term](/atomic-physics/fine-structure-and-the-dirac-atom/darwin-term-fine-structure-formula),
reorganizes into a formula depending only on $n$ and $j$, which the next lesson
derives and the [Dirac theory](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen)
confirms exactly.

[^gs-732]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_, 3rd ed., §7.3.2 — the internal magnetic field from the transformed Coulomb field, the Thomas factor of $\tfrac12$, the $\xi(r)\,\vec L\cdot\vec S$ Hamiltonian, the reduction $\vec L\cdot\vec S = \tfrac12(J^2 - L^2 - S^2)$, and the closed shift $E_{\text{so}}^{(1)} = (E_n^2/mc^2)\,n[j(j+1)-\ell(\ell+1)-\tfrac34]/[\ell(\ell+\tfrac12)(\ell+1)]$.
[^foot-51]: **Foot**, _Atomic Physics_, §5.1–5.3 — the spin-orbit interaction as the coupling of the spin moment to the motional magnetic field, the Thomas precession as a relativistic kinematic correction, and the resulting $\vec L\cdot\vec S$ splitting of alkali doublets. <https://global.oup.com/academic/product/atomic-physics-9780198506959>