Scattering of Electromagnetic Radiation

Definition of the Scattering Process

When an incident electromagnetic wave impinges upon a localized target (such as an electron, an atom, a molecule, or an aerosol particle), the electric and magnetic fields exert forces on the charges in the target. These accelerated charges oscillate and re-radiate electromagnetic waves in all directions. This redistribution of electromagnetic energy away from the forward propagation direction is called scattering.

Let the incident wave carry time-averaged intensity $I_{\text{inc}} = \langle S_{\text{inc}} \rangle$ (power per unit area). The scattered power $dP_{\text{scat}}$ emitted into a differential solid angle element $d\Omega = \sin\theta \, d\theta \, d\phi$ at distance $r$ is detected as scattered intensity $I_{\text{scat}}(r, \theta, \phi)$:

The Differential Scattering Cross-Section

The differential scattering cross-section $\frac{d\sigma}{d\Omega}$ is defined as the ratio of the power scattered per unit solid angle to the incident intensity:

Physically, $\frac{d\sigma}{d\Omega}$ represents the effective geometric area that the target presents to the incident beam for scattering radiation into direction $(\theta, \phi)$.

The Total Scattering Cross-Section

Integrating the differential cross-section over all $4\pi$ steradians yields the total scattering cross-section $\sigma_{\text{total}}$:

The total power extracted from the incident beam by scattering is simply:

Interaction with a Free Unbound Electron

Consider an unbound, free electron (mass $m_e$, charge $-e$) in a vacuum, illuminated by a linearly polarized monochromatic plane wave:

In the non-relativistic regime ($v \ll c$), the magnetic Lorentz force $\mathbf{F}_B = -e(\mathbf{v} \times \mathbf{B})$ is negligible compared to the electric force $\mathbf{F}_E = -e\mathbf{E}$ by a factor of $v/c$. Newton's second law for the electron motion is:

Re-Radiated Radiation Fields

The accelerating electron emits dipole radiation with an induced acceleration amplitude $a_0 = \frac{e E_0}{m_e}$. From Larmor's radiation formula, the electric field in the far radiation zone at distance $r$ is:

where $\theta$ is the angle between the acceleration axis $\hat{\mathbf{z}}$ and the observation direction $\hat{\mathbf{r}}$.

The Classical Electron Radius ($r_0$)

We define the fundamental physical constant known as the classical electron radius:

The scattered electric field amplitude simplifies to:

The scattered intensity is:

Differential and Total Thomson Cross-Sections

1. For Linearly Polarized Incident Light:

2. For Unpolarized Incident Light:

Averaging over all polarization angles yields the angular dependence in terms of the scattering angle $\Theta$ between incident ray $\mathbf{k}_i$ and scattered ray $\mathbf{k}_s$:

3. The Total Thomson Scattering Cross-Section ($\sigma_T$):

Integrating over the solid angle:

Substituting numerical values:

Profound Physical Properties of Thomson Scattering

1. Frequency Independence: The Thomson cross-section $\sigma_T$ is completely independent of the frequency $\omega$ of the incident wave. X-rays, microwaves, and radio waves are scattered by a free electron with the exact same cross-section!

2. Forward-Backward Symmetry: The angular factor $\frac{1 + \cos^2\Theta}{2}$ is symmetric between forward scattering ($\Theta = 0^\circ$) and back-scattering ($\Theta = 180^\circ$).

The Harmonically Bound Electron Model

In real gases, liquids, and solids, electrons are not free; they are bound to atomic nuclei by electrostatic restoring forces. We model a bound atomic electron as a classical damped harmonic oscillator with natural resonance frequency $\omega_0$ and radiative damping constant $\gamma$.

Under the driving force of an incident monochromatic plane wave $\mathbf{E}(t) = E_0 e^{-i\omega t} \hat{\mathbf{z}}$, the equation of motion is:

Assuming the steady-state sinusoidal response $x(t) = x_0 e^{-i\omega t}$:

The induced oscillating electric dipole moment is:

Radiation and Total Cross-Section Formula

Substituting the dipole moment amplitude $p_0$ into the Larmor power formula $P_{\text{scat}} = \frac{p_0^2 \omega^4}{12\pi \epsilon_0 c^3}$:

Dividing by the incident intensity $I_{\text{inc}} = \frac{1}{2} c \epsilon_0 E_0^2$:

In terms of the Thomson cross-section $\sigma_T$:

This master equation governs the entire spectrum of electromagnetic scattering across physics.

Derivation of the Low-Frequency Limit ($\omega \ll \omega_0$)

For atmospheric air molecules ($N_2, O_2$), electronic absorption transitions lie deep in the far-ultraviolet (resonance wavelengths $\lambda_0 \approx 100 - 150\text{ nm}$, so $\omega_0 \approx 1.5 \times 10^{16}\text{ rad/s}$). Visible light spans $\lambda \approx 400 - 700\text{ nm}$ (frequencies $\omega \approx 3 - 5 \times 10^{15}\text{ rad/s}$).

Therefore, for visible sunlight traversing the atmosphere:

In this low-frequency limit, the denominator $(\omega_0^2 - \omega^2)^2 + \gamma^2 \omega^2 \approx \omega_0^4$. The scattering cross-section reduces to Lord Rayleigh's celebrated scattering law (1871):

Since $\omega = \frac{2\pi c}{\lambda}$, the scattering cross-section is inversely proportional to the fourth power of wavelength:

Why the Daytime Sky is Blue

Compare blue sunlight ($\lambda_{\text{blue}} \approx 430\text{ nm}$) to red sunlight ($\lambda_{\text{red}} \approx 680\text{ nm}$):

Blue photons are scattered by atmospheric nitrogen and oxygen molecules over 6 times more intensely than red photons. When you look anywhere in the sky away from the direct disc of the Sun, you are viewing this scattered light, which is heavily dominated by short blue and violet wavelengths.

Why Sunsets are Deep Red

At sunset and sunrise, sunlight travels through a very long oblique path through Earth's atmosphere (air mass factor up to 40 times greater than at noon). By the Beer-Lambert transmission law:

The short blue and green wavelengths are almost entirely scattered out of the direct beam line of sight. Only the least-scattered long wavelengths—orange and deep red—penetrate the thick atmospheric column to reach the observer's eyes.

Polarization Mechanism of Skylight

Unpolarized incident sunlight traveling along $+z$ possesses electric field vibrations in all directions within the $xy$-plane.

Consider an observer viewing scattered light from an air molecule at an angle of $90^\circ$ relative to the incident solar beam (e.g., looking toward the horizon while the Sun is overhead):

1. Oscillations of the molecule's electrons parallel to the line of sight cannot radiate toward the observer (because oscillating dipoles emit zero power along their axis of oscillation).
2. Only oscillations perpendicular to both the solar beam and the line of sight produce radiation reaching the observer.

Consequently, sunlight scattered at an angle of $90^\circ$ from the Sun is nearly 100% linearly polarized. Bees and migratory birds utilize this celestial polarization pattern (the e-vector sky compass) for biological navigation even on overcast days.

Near-Resonance Behavior ($\omega \approx \omega_0$)

When the frequency of incident radiation closely approaches an atomic resonance ($\omega \to \omega_0$):

The cross-section becomes:

At exact resonance ($\omega = \omega_0$), the cross-section reaches a peak value:

Using the classical radiative damping constant $\gamma = \frac{2 r_0 \omega_0^2}{3 c}$:

Profound Physical Result: At atomic resonance, the effective scattering cross-section is proportional to the square of the wavelength $\lambda_0^2$, rather than the physical geometric size of the atom ($r_{\text{atom}}^2 \sim 10^{-20}\text{ m}^2$). For yellow sodium D-light ($\lambda_0 = 589\text{ nm}$), $\sigma_{\text{max}} \approx 1.6 \times 10^{-13}\text{ m}^2$, which is over seven orders of magnitude larger than the geometric cross-section of a sodium atom! This phenomenon is known as resonance fluorescence.