Thomson Scattering Cross Section

Equivalent forms

\sigma_T = \frac{8\pi}{3} r_e^2

A purely classical result that survives as the low-energy limit of QED's Compton formula.

Symbol	Name	SI unit	Dimension	Range
$σ_T$	Thomson cross sectionoutput Total scattering cross section for low-energy photons on free electrons	m²	L²	0 – 1e-27
$e$	Electron charge Elementary charge	C	I·T	1.602176634e-19 – 1.602176634e-19
$m_e$	Electron mass Rest mass of electron	kg	M	9.1093837015e-31 – 9.1093837015e-31
$c$	Speed of light Speed of light in vacuum	m/s	L·T⁻¹	299792458 – 299792458

Unit systems

Symbol	SI	CGS	Natural
$σ_T$	m²	cm²	(MeV)⁻²·(ℏc)²
$e$	C	esu	√(4πα)
$m_e$	kg	g	0.511 MeV
$c$	m/s	cm/s	1

Where it holds

Photon energy E ≪

m_e c^{2} (511\,\mathrm{keV})

; for higher energies use Klein-Nishina. Free electron assumed (no binding).

Dimensional analysis

\sigma _T \to [L^{2}]

(e^{2}/(4\pi \varepsilon _{0} m_e c^{2}))^{2} \to ([I\cdot T]^{2} \cdot [I^{-2}T^{-2}M^{-1}L^{3}T^{-2}\cdot F/m^{-1}]\cdot

...) reduces to

[L^{2}]

Cross section has units of area.

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Discovery

J. J. Thomson · 1906

Thomson derived the cross section while modeling the scattering of X-rays by atomic electrons, providing one of the first quantitative tests of electron count in atoms.

Try this

How effectively does a free electron scatter low-energy light?

Compute the classical Thomson cross section using e, m_e, c, ε₀ and compare with the high-energy Klein-Nishina limit.

Research status: stable

Real-world applications

Compton Gamma-Ray Observatory and X-ray polarimetry
Solar corona electron density mapping (K-corona is Thomson-scattered photospheric light)
Plasma diagnostics in fusion devices (laser Thomson scattering)
Cosmic Microwave Background polarization from electron scattering

Common misconceptions

$\sigma _T$ does not depend on photon frequency — a striking classical result.
Only valid in the long-wavelength limit; $at \gamma -ray$ energies recoil and spin effects matter.
Bound electrons scatter coherently with the atom (Rayleigh scattering), not as free electrons.

Experimental verification

Low-energy X-ray scattering by free electrons in plasmas; Thomson scattering diagnostics in tokamaks measure electron temperature/density via spectral broadening.

Derivation

Classical Larmor power radiated by an electron driven by E-field

E_{0} \cos (\omega t)

is

P = (e^{2}/6\pi \varepsilon _{0}c^{3})

⟨

a^{2}

⟩

= (e^{4} E_{0}^{2})/(12\pi \varepsilon _{0}^{2} m_e^{2} c^{3})

.

Incident intensity

I = \tfrac{1}{2} \varepsilon _{0} c E_{0}^{2}

.

Cross section

\sigma = P/I = (8\pi /3)(e^{2}/(4\pi \varepsilon _{0} m_e c^{2}))^{2} = (8\pi /3) r_e^{2}

.

Limiting cases

m_e \to \infty

⟶

\sigma _T \to 0

Heavy electrons would barely accelerate under the EM wave.

Photon E ≪

m_e c^{2}

⟶

\sigma _KN \to \sigma _T

Klein-Nishina reduces to Thomson at low photon energy.

Photon E ≫

m_e c^{2}

⟶

\sigma _KN

≪

\sigma _T

Quantum recoil suppresses scattering at high energies.

What if…

What if the electron were heavier (muon)?

$\sigma$ scales as $1/m^{2}$ — the muon cross section is $(207)^{-2} \approx 2.3e-5$ $of \sigma _T$ .

What if c were halved?

$\sigma _T \propto 1/c^{4}$ — would grow $16\times$ , making the universe even more opaque to early radiation.

What if photon energy approaches

m_e c^{2}

?

Klein-Nishina suppresses $\sigma$ ; the simple Thomson formula overestimates scattering.

1

Compute σ_T from constants

Given ·

e:: 1.602176634e-19
m e:: 9.1093837015e-31
c:: 299792458

Find ·

\sigma _T

Steps

Step 1: $r_e = (1.602176634e-19)^{2} / (4\pi \times 8.854e-12 \times 9.109e-31 \times (2.998e_{8})^{2}) \approx 2.818e-15\,\mathrm{m}$ .
Step 2: $r_e^{2} \approx 7.94e-30\,\mathrm{m}^{2}$ .
Step 3: $\sigma _T = (8\pi /3) \times 7.94e-30 = 6.65e-29\,\mathrm{m}^{2} = 0.665$ barn.

Result ·

\sigma _T = (8\pi /3)\cdot r_e^{2}

with

r_e = e^{2}/(4\pi \varepsilon _{0} m_e c^{2}) = 2.818e-15\,\mathrm{m} \to \sigma _T \approx 6.652e-29\,\mathrm{m}^{2} = 0.6652

barn

2

Mean free path of CMB photons before recombination

Given ·

σ T:: 6.6524e-29

Find ·

\lambda

Steps

Step 1: $n_e \approx 400\,\mathrm{cm}^{-3} = 4e_{8} m^{-3}$ .
Step 2: $n_e \sigma _T = 2.66e-20\,\mathrm{m}^{-1}$ .
Step 3: $\lambda \approx 3.8e19\,\mathrm{m}$ — short on cosmological scales, hence the universe was opaque.

Result · With electron density

n_e \approx 400\,\mathrm{cm}^{-3} pre

-recombination,

\lambda = 1/(n_e \sigma _T) \approx 1/(4e_{8} \times 6.65e-29) \approx 3.8e19\,\mathrm{m} \approx 1200\,\mathrm{pc}

.

klein nishina cross sectionCompton Scattering Wavelength Shift→rutherford scattering cross section