Elastic Scattering as a Classical Oscillator

The ordinary frequency and angular frequency are related by

We often use \(\omega\) in this chapter because oscillators are cleaner in angular-frequency notation. The physics is the same; \(\omega\) is simply \(2\pi\) times the frequency \(\nu\).

The incoming wave drives an electron. The electron then accelerates and radiates. That emitted radiation is the scattered radiation. So the cross-section comes from the ratio

The incoming time-averaged Poynting flux is

where \(E_0\) is the amplitude of the incoming electric field. The emitted power from an accelerated charge follows from the Larmor formula. After time-averaging over one oscillation,

The factor \(1/3\) appears because the instantaneous Larmor formula has \(2e^2a^2/(3c^3)\), and the average of a squared sinusoidal oscillation contributes a factor \(1/2\). Therefore the main missing ingredient is the acceleration of the electron.

We now set up Newton's second law in the \(z\)-direction:

The dot notation means time derivative: \(\dot z=dz/dt\), and \(\ddot z=d^2z/dt^2\).

The driving force is the Lorentz force from the incoming electromagnetic wave. In the non-relativistic limit \(v/c\ll 1\), the magnetic part is small, so the electric field gives the dominant force:

Here the first \(e\) is the electron charge magnitude, while \(e^{i\omega t}\) is the complex exponential. We use the complex notation because it keeps the algebra clean. Physically we can take the real part at the end; it is equivalent to writing sines and cosines.

The restoring force is the usual harmonic-oscillator force:

The quantity \(\omega_0\) is the natural or eigen-frequency of the oscillator. It is the frequency the system would prefer if we removed the driving force.

The damping force is taken proportional to velocity:

In an ordinary mechanical oscillator, damping could come from friction or air resistance. Here the damping comes from the fact that the electron radiates energy away while it accelerates.

Putting the three forces together gives the driven, damped oscillator equation

Equivalently,

If we remove the driving term and damping term, this reduces to the usual harmonic oscillator. The incoming electromagnetic wave drives it; the damping prevents the response from becoming infinitely large at resonance.

Use the trial solution

Then

Solving for the amplitude gives

The scattered power depends on the square of the acceleration amplitude. Taking the absolute square removes the complex phase and gives

This is the key response of the electron. It says that the electron accelerates more strongly when the driving frequency approaches the oscillator frequency, and the damping term controls how sharp that response becomes.

The scattering cross-section is

Using

and inserting the acceleration response, the amplitude \(E_0\) cancels. That cancellation is important: the cross-section is a property of the interaction, not of how strong the incoming wave happens to be.

It is useful to write this as a constant times a frequency-dependent factor:

The quantity \(r_e=e^2/(m_ec^2)\) is called the classical electron radius. The name is historical. It is not a modern quantum-mechanical statement about the literal size of the electron, but it is a convenient way to remember the dimensions.

The Thomson cross-section \(\sigma_{\rm TH}\) is one of the most important constants in radiative astrophysics. It describes elastic scattering by free electrons in the low-energy limit. It becomes an important opacity source in ionized gases, in hot stellar interiors, and in the early universe before recombination.

The frequency-dependent factor in angular-frequency notation is

In ordinary frequency notation, using \(\omega=2\pi\nu\), the same expression can be written as

This one formula contains three useful physical limits: the Thomson limit, the Rayleigh limit, and the resonance limit.

Thomson limit: high frequency compared with the oscillator frequency

The Thomson limit is the high-frequency limit of the oscillator formula:

and we also assume weak damping, so the damping term does not dominate the denominator. Then

and the frequency-dependent factor tends to one. Therefore

This is the free-electron scattering result. A truly free electron has no bound oscillator frequency and no restoring force, so we can think of \(\nu_0=0\). In that case the cross-section becomes the Thomson cross-section directly.

This is important whenever the gas is ionized and many free electrons are present. It matters in hot stars, in ionized plasmas, and in the early universe before recombination. If the gas has very few free electrons, then Thomson scattering is not the dominant opacity.

The word elastic means that the photon energy does not change:

The incoming photon scatters, but it comes out with the same frequency. If the photon energy becomes high enough that the electron recoils significantly, this elastic approximation breaks down. Then we need Compton scattering. Thomson scattering is the low-energy limit of Compton scattering.

Rayleigh limit: low frequency compared with the oscillator frequency

The opposite limit is the Rayleigh limit:

again with weak damping. Now the denominator is dominated by \(\nu_0^4\), so

Because wavelength is inverse frequency, \(\nu\ll\nu_0\) means

So the same result in wavelength form is

This is the famous \(1/\lambda^4\) behavior. Shorter wavelengths scatter more efficiently. This is why Rayleigh scattering naturally pushes scattered light toward blue wavelengths.

Take a cloud or an atmosphere illuminated by a source that is roughly a blackbody at \(T\simeq5800\,{\rm K}\), like the Sun. Wien's law places the peak of the emission near

If the scattering particles are neutral hydrogen atoms, the first strong resonance is near

So the visible radiation satisfies \(\lambda\gg\lambda_0\). That places the scattering in the Rayleigh limit.

The source does not emit only at \(5000\,{\rm \AA}\). A blackbody emits a whole spectrum. The shorter-wavelength part of that spectrum is scattered more efficiently because of the \(1/\lambda^4\) factor. If some of that scattered light enters an observer's eye or telescope, the scattered light is biased toward shorter visible wavelengths.

This is the basic reason the sky is blue. One should still remember the full perception problem: the Sun emits a spectrum, the atmosphere has its own transmission properties, and human eyes have wavelength-dependent sensitivity. Violet scatters strongly, but there is less violet in the solar spectrum and the eye is less sensitive there. The combined effect gives the familiar blue sky.

Earth's atmosphere is not a neutral hydrogen cloud, but the same Rayleigh idea applies to molecules in the air. In astrophysics, Rayleigh scattering is especially useful in cool gases, cool-star atmospheres, and exoplanet atmospheres. It is almost the opposite regime from Thomson scattering, which becomes important in hot ionized gas with many free electrons.

Now take the resonance case,

The important factor in the denominator is

Near resonance, \(\nu-\nu_0\) is small, but \(\nu+\nu_0\) is not small. So we approximate

This is why the line shape depends on \(\nu-\nu_0\), the distance from line center. The resonance approximation leads to a sharply peaked cross-section around \(\nu_0\).

To make this useful, we need an expression for the damping coefficient \(\gamma\). In this classical picture, damping comes from radiation reaction: the electron loses energy because it radiates.

The damping force was written as

The work done by the damping force per unit time is \(F_D\dot z\). Since damping removes energy from the oscillator, it must match the negative of the radiated power:

Using the instantaneous Larmor power,

and integrating over one period \(T\),

Now rewrite the integral as \(\int \ddot z\,\ddot z\,dt\) and integrate by parts:

Over a complete period the boundary term vanishes, because the oscillator returns to the same phase. Therefore

Putting this back into the work balance lets us identify the damping force as

But we also defined \(F_D=-m\gamma\dot z\). Hence

For an oscillator close to its natural frequency, \(\dddot z/\dot z\simeq-\omega_0^2\). This gives the classical damping coefficient

Using this damping coefficient in the resonance cross-section produces a Lorentzian line profile. The cross-section can be written as

The Lorentz profile is

This profile is normalized over the line:

Therefore the frequency-integrated cross-section is

The unit is important. This is not just \({\rm cm^2}\); it is \({\rm cm^2\,Hz}\), because we integrated over frequency. So it cannot be compared directly to the Thomson cross-section unless we keep track of the frequency scale.

The Lorentz profile has a sharp peak at \(\nu=\nu_0\). Its wings fall as

far from line center. This is steep, but not as steep as the exponential wings of a Gaussian Doppler profile. That difference becomes important when line wings are measured carefully.

At line center, the Lorentz profile has peak value

The half-maximum value is \(2/\gamma\). Solving for the offset from line center gives the half-width

The full width at half maximum is twice this:

In angular-frequency units this is especially simple:

For the classical oscillator, \(\gamma=\gamma_{\rm cl}\), so the classical natural line width is set by the radiation damping coefficient.

If we convert to wavelength using \(\nu\lambda=c\), then for small changes

Using the classical damping coefficient gives a classical natural width in wavelength of roughly

This is extremely narrow. For an ultraviolet line around \(10^3\,{\rm \AA}\), the fractional width is of order

The classical result is useful, but it also shows where the purely classical description becomes inadequate. It predicts that all spectral lines have the same integrated strength and the same kind of natural width. Real atoms do not behave that way. Hydrogen, carbon, and other atoms have different line strengths, and different transitions within the same atom also differ. To describe that, we need quantum mechanics and experimental oscillator strengths.

The quality factor of the resonance is

For atomic resonances, this can be enormous:

Mechanical oscillators can have much smaller quality factors. A swing, a whistle, or a bell can have a resonance, but atomic electrons behave like extremely high-\(Q\) electric oscillators. They are very efficient antennas for emitting and absorbing light at their resonance frequencies.

A useful exercise is to show that

The factor \(\nu_0\) is needed because \(\sigma_{\rm cl}\) is frequency-integrated and has units \({\rm cm^2\,Hz}\), while \(\sigma_{\rm TH}\) has units \({\rm cm^2}\). This relation is a compact way to see that a line near resonance can be enhanced by a factor of order \(10^7\) to \(10^8\) relative to the Thomson scale.

This is why spectral lines are powerful in astrophysical radiation forces. A generic ionized gas always has a free-electron scattering baseline set by \(\sigma_{\rm TH}\). But if photons overlap a strong line transition, the effective interaction can be much larger. That produces line forces.

Line forces are central in line-driven winds from massive stars and in outflows from luminous accretion discs around supermassive black holes. The light pushes strongly on spectral lines, and the gas can be accelerated outward. In practice the enhancement is not literally unlimited, because a line can saturate: once all available light in that frequency interval is already blocked, adding more opacity cannot extract more force from that same light.

The quantum correction is usually introduced through an oscillator strength \(f\). In a more realistic line cross-section, the classical strength is multiplied by this factor:

The Lorentzian form is still useful for many cases, but the damping coefficient and the oscillator strength become line-specific. That is how real atomic transitions get different strengths and different widths.

Turn off the driving force. The oscillator equation becomes

For weak damping, the solution is approximately

The amplitude decays as \(e^{-\gamma t/2}\). The energy is proportional to amplitude squared, so the energy decays as \(e^{-\gamma t}\). That gives the characteristic lifetime

For a classical atomic oscillator, this time scale is very short, of order

So when the driving stops, the excited oscillation dies very quickly. Strong allowed transitions in quantum atomic physics have lifetimes of a similar order of magnitude, which is why resonance lines are so common and so important in spectra.