1. Collision Among Gas Particles

 

Figure 1. In a gas of particles of individual diameter 2r, the cross section σ, for collisions is related to the particle number density n, and mean free path between collisions λ. https://handwiki.org/wiki/index.php?curid=2004355

In a gas of finite-sized particles there are collisions among particles that depend on their cross-sectional size. The average distance that a particle travels between collisions depends on the density of gas particles. These quantities are related by

[math]\displaystyle{ \sigma = \frac{1}{n \lambda}, }[/math]

where

σ is the cross section of a two-particle collision (SI units: m²),

λ is the mean free path between collisions (SI units: m),

n is the number density of the target particles (SI units: m⁻³).

If the particles in the gas can be treated as hard spheres of radius r that interact by direct contact, as illustrated in Figure 1, then the effective cross section for the collision of a pair is

[math]\displaystyle{ \sigma = \pi \left(2r\right)^2 }[/math]

If the particles in the gas interact by a force with a larger range than their physical size, then the cross section is a larger effective area that may depend on a variety of variables such as the energy of the particles.

Cross sections can be computed for atomic collisions but also are used in the subatomic realm. For example, in nuclear physics a "gas" of low-energy neutrons collides with nuclei in a reactor or other nuclear device, with a cross section that is energy-dependent and hence also with well-defined mean free path between collisions.

2. Attenuation of a Beam of Particles

If a beam of particles enters a thin layer of material of thickness dz, the flux Φ of the beam will decrease by dΦ according to

[math]\displaystyle{ \frac{\mathrm d \Phi}{\mathrm d z} = -n \sigma \Phi, }[/math]

where σ is the total cross section of all events, including scattering, absorption, or transformation to another species. The volumetric number density of scattering centers is designated by n. Solving this equation exhibits the exponential attenuation of the beam intensity:

[math]\displaystyle{ \Phi = \Phi_0 e^{-n \sigma z}, }[/math]

where Φ₀ is the initial flux, and z is the total thickness of the material. For light, this is called the Beer–Lambert law.

3. Differential Cross Section

Consider a classical measurement where a single particle is scattered off a single stationary target particle. Conventionally, a spherical coordinate system is used, with the target placed at the origin and the z axis of this coordinate system aligned with the incident beam. The angle θ is the scattering angle, measured between the incident beam and the scattered beam, and the φ is the azimuthal angle.

Differential cross section. https://handwiki.org/wiki/index.php?curid=2024972

The impact parameter b is the perpendicular offset of the trajectory of the incoming particle, and the outgoing particle emerges at an angle θ. For a given interaction (Coulombic, magnetic, gravitational, contact, etc.), the impact parameter and the scattering angle have a definite one-to-one functional dependence on each other. Generally the impact parameter can neither be controlled nor measured from event to event and is assumed to take all possible values when averaging over many scattering events. The differential size of the cross section is the area element in the plane of the impact parameter, i.e. dσ = b dφ db. The differential angular range of the scattered particle at angle θ is the solid angle element dΩ = sin θ dθ dφ. The differential cross section is the quotient of these quantities, dσ/dΩ.

It is a function of the scattering angle (and therefore also the impact parameter), plus other observables such as the momentum of the incoming particle. The differential cross section is always taken to be positive, even though larger impact parameters generally produce less deflection. In cylindrically symmetric situations (about the beam axis), the azimuthal angle φ is not changed by the scattering process, and the differential cross section can be written as

[math]\displaystyle{ \frac{\mathrm{d} \sigma}{\mathrm{d}(\cos \theta)} =\int_0^{2\pi} \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} \,\mathrm{d}\varphi }[/math].

In situations where the scattering process is not azimuthally symmetric, such as when the beam or target particles possess magnetic moments oriented perpendicular to the beam axis, the differential cross section must also be expressed as a function of the azimuthal angle.

For scattering of particles of incident flux F_inc off a stationary target consisting of many particles, the differential cross section dσ/dΩ at an angle (θ,φ) is related to the flux of scattered particle detection F_out(θ,φ) in particles per unit time by

[math]\displaystyle{ \frac{\mathrm d \sigma}{\mathrm d \Omega}(\theta,\varphi) = \frac{1}{n t \Delta\Omega} \frac{F_\text{out}(\theta,\varphi)}{F_\text{inc}}. }[/math]

Here ΔΩ is the finite angular size of the detector (SI unit: sr), n is the number density of the target particles (SI units: m⁻³), and t is the thickness of the stationary target (SI units: m). This formula assumes that the target is thin enough that each beam particle will interact with at most one target particle.

The total cross section σ may be recovered by integrating the differential cross section dσ/dΩ over the full solid angle (4π steradians):

[math]\displaystyle{ \sigma = \oint_{4\pi} \frac{\mathrm d \sigma}{\mathrm d \Omega} \, \mathrm d \Omega = \int_0^{2\pi} \int_0^\pi \frac{\mathrm d \sigma}{\mathrm d \Omega} \sin \theta \, \mathrm d \theta \, \mathrm d \varphi. }[/math]

It is common to omit the “differential” qualifier when the type of cross section can be inferred from context. In this case, σ may be referred to as the integral cross section or total cross section. The latter term may be confusing in contexts where multiple events are involved, since “total” can also refer to the sum of cross sections over all events.

The differential cross section is extremely useful quantity in many fields of physics, as measuring it can reveal a great amount of information about the internal structure of the target particles. For example, the differential cross section of Rutherford scattering provided strong evidence for the existence of the atomic nucleus.

Instead of the solid angle, the momentum transfer may be used as the independent variable of differential cross sections.

Differential cross sections in inelastic scattering contain resonance peaks that indicate the creation of metastable states and contain information about their energy and lifetime.

4. Quantum Scattering

In the time-independent formalism of quantum scattering, the initial wave function (before scattering) is taken to be a plane wave with definite momentum k:

[math]\displaystyle{ \phi_-(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; e^{i k z}, }[/math]

where z and r are the relative coordinates between the projectile and the target. The arrow indicates that this only describes the asymptotic behavior of the wave function when the projectile and target are too far apart for the interaction to have any effect.

After scattering takes place it is expected that the wave function takes on the following asymptotic form:

[math]\displaystyle{ \phi_+(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; f(\theta,\phi) \frac{e^{i k r}}{r}, }[/math]

where f is some function of the angular coordinates known as the scattering amplitude. This general form is valid for any short-ranged, energy-conserving interaction. It is not true for long-ranged interactions, so there are additional complications when dealing with electromagnetic interactions.

The full wave function of the system behaves asymptotically as the sum

[math]\displaystyle{ \phi(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; \phi_-(\mathbf r) + \phi_+(\mathbf r). }[/math]

The differential cross section is related to the scattering amplitude:

[math]\displaystyle{ \frac{\mathrm d \sigma}{\mathrm d \Omega}(\theta, \phi) = \bigl|f(\theta, \phi)\bigr|^2. }[/math]

This has the simple interpretation as the probability density for finding the scattered projectile at a given angle.

A cross section is therefore a measure of the effective surface area seen by the impinging particles, and as such is expressed in units of area. The cross section of two particles (i.e. observed when the two particles are colliding with each other) is a measure of the interaction event between the two particles. The cross section is proportional to the probability that an interaction will occur; for example in a simple scattering experiment the number of particles scattered per unit of time (current of scattered particles I_r) depends only on the number of incident particles per unit of time (current of incident particles I_i), the characteristics of target (for example the number of particles per unit of surface N), and the type of interaction. For Nσ ≪ 1 we have

[math]\displaystyle{ \begin{align} I_\text{r} &= I_\text{i}N\sigma, \\ \sigma &= \frac{I_\text{r}}{I_\text{i}} \frac{1}{N} \\ &= \text{probability of interaction} \times \frac{1}{N}. \end{align} }[/math]

4.1. Relation to the S-Matrix

If the reduced masses and momenta of the colliding system are m_i, p_i and m_f, p_f before and after the collision respectively, the differential cross section is given by^{[clarification needed]}

[math]\displaystyle{ \frac{\mathrm d\sigma}{\mathrm d\Omega} = \left(2\pi\right)^4 m_i m_f \frac{p_f}{p_i} \bigl|T_{fi}\bigr|^2, }[/math]

where the on-shell T matrix is defined by

[math]\displaystyle{ S_{fi} = \delta_{fi} - 2\pi i \delta\left(E_f - E_i\right) \delta\left(\mathbf{p}_i - \mathbf{p}_f\right) T_{fi} }[/math]

in terms of the S-matrix. Here δ is the Dirac delta function. The computation of the S-matrix is the main goal of the scattering theory.

5. Units

Although the SI unit of total cross sections is m², smaller units are usually used in practice.

In nuclear and particle physics, the conventional unit is the barn b, where 1 b = 10⁻²⁸ m² = 100 fm².^[1] Smaller prefixed units such as mb and μb are also widely used. Correspondingly, the differential cross section can be measured in units such as mb/sr.

When the scattered radiation is visible light, it is conventional to measure the path length in centimetres. To avoid the need for conversion factors, the scattering cross section is expressed in cm², and the number concentration in cm⁻³. The measurement of the scattering of visible light is known as nephelometry, and is effective for particles of 2–50 µm in diameter: as such, it is widely used in meteorology and in the measurement of atmospheric pollution.

The scattering of X-rays can also be described in terms of scattering cross sections, in which case the square ångström is a convenient unit: 1 Å² = 10⁻²⁰ m² = 10000 pm² = 10⁸ b. The sum of the scattering, photoelectric, and pair-production cross-sections (in barns) is charted as the "atomic attenuation coefficient" (narrow-beam), in barns.^[2]

6. Scattering of Light

For light, as in other settings, the scattering cross section for particles is generally different from the geometrical cross section of the particle, and it depends upon the wavelength of light and the permittivity, shape, and size of the particle. The total amount of scattering in a sparse medium is proportional to the product of the scattering cross section and the number of particles present.

In the interaction of light with particles, many processes occur, each with their own cross sections, including absorption, scattering, and photoluminescence. The sum of the absorption and scattering cross sections is sometimes referred to as the attenuation or extinction cross section.

[math]\displaystyle{ \sigma = \sigma_\text{a} + \sigma_\text{s} + \sigma_\text{l}. }[/math]

The total extinction cross section is related to the attenuation of the light intensity through the Beer–Lambert law, which says that attenuation is proportional to particle concentration:

[math]\displaystyle{ A_\lambda = C l \sigma, }[/math]

where A_λ is the attenuation at a given wavelength λ, C is the particle concentration as a number density, and l is the path length. The absorbance of the radiation is the logarithm (decadic or, more usually, natural) of the reciprocal of the transmittance T:^[3]

[math]\displaystyle{ A_\lambda = -\log \mathcal{T}. }[/math]

Combining the scattering and absorption cross sections in this manner is often necessitated by the inability to distinguish them experimentally, and much research effort has been put into developing models that allow them to be distinguished, the Kubelka-Munk theory being one of the most important in this area.

6.1. Cross Section and Mie Theory

Cross sections commonly calculated using Mie theory include efficiency coefficients for extinction [math]\displaystyle{ Q_\text{ext} }[/math], scattering [math]\displaystyle{ Q_\text{sc} }[/math], and Absorption [math]\displaystyle{ Q_\text{abs} }[/math] cross sections. Those are normalized by the geometrical cross sections of the particle [math]\displaystyle{ \sigma_\text{geom} = \pi a^2 }[/math] as [math]\displaystyle{ Q_\alpha = \frac{\sigma_\alpha}{\sigma_\text{geom}}, \qquad \alpha = \text{ext}, \text{sc}, \text{abs}. }[/math] The cross section is defined by

[math]\displaystyle{ \sigma_\alpha = \frac{W_\alpha}{I_{\text{inc}}} }[/math]

where [math]\displaystyle{ \left[W_\alpha \right] = \left[ \text{W} \right] }[/math] is the energy flow through the surrounding surface, and [math]\displaystyle{ \left[I_{\text{inc}}\right] = \left[ \frac{\text{W}}{\text{m}^2} \right] }[/math] is the intensity of the incident wave. For a plane wave the intensity is going to be [math]\displaystyle{ I_{\text{inc}} = |\mathbf{E}|^2 / (2 \eta) }[/math], where [math]\displaystyle{ \eta = \sqrt{\mu \mu_0 / (\varepsilon \varepsilon_0)} }[/math] is the impedance of the host medium.

The main approach is based on the following. Firstly, we construct an imaginary sphere of radius [math]\displaystyle{ r }[/math] (surface [math]\displaystyle{ A }[/math]) around the particle (the scatterer). The net rate of electromagnetic energy crosses the surface [math]\displaystyle{ A }[/math] is

[math]\displaystyle{ W_a = - \oint_A \mathbf{\Pi} \cdot \hat{\mathbf{r}} dA }[/math]

where [math]\displaystyle{ \mathbf{\Pi} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}^* \times \mathbf{H} \right] }[/math] is the time averaged Poynting vector. If [math]\displaystyle{ W_a \gt 0 }[/math] energy is absorbed within the sphere, otherwise energy is being created within the sphere. We exclude from consideration the later one. Once host medium is non-absorbing, the energy is absorbed by the particle. We decompose the total field into incident and scattered parts [math]\displaystyle{ \mathbf{E} = \mathbf{E}_i + \mathbf{E}_s }[/math], and the same for the magnetic field [math]\displaystyle{ \mathbf{H} }[/math]. Thus, we can decompose [math]\displaystyle{ W_a }[/math] into the three terms [math]\displaystyle{ W_a = W_i - W_s + W_{\text{ext}} }[/math], where

[math]\displaystyle{ W_i = - \oint_A \mathbf{\Pi}_i \cdot \hat{\mathbf{r}} dA \equiv 0, \qquad W_s = \oint_A \mathbf{\Pi}_s \cdot \hat{\mathbf{r}} dA, \qquad W_{\text{ext}} = \oint_A \mathbf{\Pi}_{\text{ext}} \cdot \hat{\mathbf{r}} dA. }[/math]

Where [math]\displaystyle{ \mathbf{\Pi}_i = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_i^* \times \mathbf{H}_i \right] }[/math], [math]\displaystyle{ \mathbf{\Pi}_s = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_s^* \times \mathbf{H}_s \right] }[/math], and [math]\displaystyle{ \mathbf{\Pi}_{\text{ext}} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_s^* \times \mathbf{H}_i + \mathbf{E}_i^* \times \mathbf{H}_s \right] }[/math].

All the field can be decomposed into the series of vector spherical harmonics (VSH). After that, all the integrals can be taken. In the case of a uniform sphere of radius [math]\displaystyle{ a }[/math], permittivity [math]\displaystyle{ \varepsilon }[/math], and permeability [math]\displaystyle{ \mu }[/math] the problem has a precise solution.^[4] The scattering and extinction coefficients are [math]\displaystyle{ Q_\text{sc} = \frac{2}{k^2a^2}\sum_{n=1}^\infty (2n+1)(|a_{n}|^2+|b_{n}|^2) }[/math] [math]\displaystyle{ Q_\text{ext} = \frac{2}{k^2a^2}\sum_{n=1}^\infty (2n+1)\Re(a_{n}+b_{n}) }[/math] Where [math]\displaystyle{ k = n_\text{host} k_0 }[/math]. Those are connected as [math]\displaystyle{ \sigma_\text{ext} = \sigma_\text{sc} + \sigma_\text{abs} \qquad \text{or} \qquad Q_\text{ext} = Q_\text{sc} + Q_\text{abs} }[/math]

6.2. Dipole Approximation for the Scattering Cross Section

Let us assume that particle support only electric and magnetic dipole modes with polarizabilities [math]\displaystyle{ \mathbf{p} = \alpha^e \mathbf{E} }[/math] and [math]\displaystyle{ \mathbf{m} = (\mu \mu_0)^{-1}\alpha^m \mathbf{H} }[/math] (here we use the notation of magnetic polarizability in the manner of Bekshaev et al.^[5][6] rather than the notation of Nieto-Vesperians et al.^[7]) expressed through the Mie coefficients as [math]\displaystyle{ \alpha^e = 4 \pi \varepsilon_0 \cdot i \frac{3 \varepsilon}{2 k^3} a_1, \qquad \alpha^m = 4 \pi \mu_0 \cdot i \frac{3 \mu}{2 k^3} b_1. }[/math] Then the cross sections are going to be [math]\displaystyle{ \sigma_{\text{ext}} = \sigma_{\text{ext}}^{\text{(e)}} + \sigma_{\text{ext}}^{\text{(m)}} = \frac{1}{4\pi \varepsilon \varepsilon_0} \cdot 4\pi k \Im(\alpha^e) + \frac{1}{4\pi \mu \mu_0} \cdot 4\pi k \Im(\alpha^m) }[/math] [math]\displaystyle{ \sigma_{\text{sc}} = \sigma_{\text{sc}}^{\text{(e)}} + \sigma_{\text{sc}}^{\text{(m)}} = \frac{1}{(4\pi \varepsilon \varepsilon_0)^2} \cdot \frac{8\pi}{3} k^4 |\alpha^e|^2 + \frac{1}{(4\pi \mu \mu_0)^2} \cdot \frac{8\pi}{3} k^4 |\alpha^m|^2 }[/math] and, finally, the electric and magnetic absorption cross sections [math]\displaystyle{ \sigma_{\text{abs}} = \sigma_{\text{abs}}^{\text{(e)}} + \sigma_{\text{abs}}^{\text{(m)}} }[/math] are [math]\displaystyle{ \sigma_{\text{abs}}^{\text{(e)}} = \frac{1}{4 \pi \varepsilon \varepsilon_0} \cdot 4\pi k \left[ \Im(\alpha^e) - \frac{k^3}{6 \pi \varepsilon \varepsilon_0} |\alpha^e|^2\right] }[/math] and [math]\displaystyle{ \sigma_{\text{abs}}^{\text{(m)}} = \frac{1}{4 \pi \mu \mu_0} \cdot 4\pi k \left[ \Im(\alpha^m) - \frac{k^3}{6 \pi \mu \mu_0} |\alpha^m|^2\right] }[/math]

For the case of no-inside-gain particle, i.e. there no energy is emitted by the particle internally ([math]\displaystyle{ \sigma_{\text{abs}} \gt 0 }[/math]), we have a particular case of the Optical theorem [math]\displaystyle{ \frac{1}{4\pi \varepsilon \varepsilon_0} \Im(\alpha^e) + \frac{1}{4\pi \mu \mu_0} \Im(\alpha^m) \geq \frac{2 k^3}{3} \left[ \frac{|\alpha^e|^2}{(4\pi \varepsilon \varepsilon_0)^2} + \frac{|\alpha^m|^2}{(4\pi \mu \mu_0)^2} \right] }[/math] The sign of equality [math]\displaystyle{ \geq \to = }[/math] is achieved for non-absorbing particles, i.e. for [math]\displaystyle{ \Im(\varepsilon) = \Im(\mu) = 0 }[/math].

6.3. Scattering of Light on Extended Bodies

In the context of scattering light on extended bodies, the scattering cross section, σ_scat, describes the likelihood of light being scattered by a macroscopic particle. In general, the scattering cross section is different from the geometrical cross section of a particle, as it depends upon the wavelength of light and the permittivity in addition to the shape and size of the particle. The total amount of scattering in a sparse medium is determined by the product of the scattering cross section and the number of particles present. In terms of area, the total cross section (σ) is the sum of the cross sections due to absorption, scattering, and luminescence:

[math]\displaystyle{ \sigma = \sigma_\text{a} + \sigma_\text{s} + \sigma_\text{l}. }[/math]

The total cross section is related to the absorbance of the light intensity through the Beer–Lambert law, which says that absorbance is proportional to concentration: A_λ = Clσ, where A_λ is the absorbance at a given wavelength λ, C is the concentration as a number density, and l is the path length. The extinction or absorbance of the radiation is the logarithm (decadic or, more usually, natural) of the reciprocal of the transmittance T:^[3]

[math]\displaystyle{ A_\lambda = - \log \mathcal{T}. }[/math]

Relation to physical size

There is no simple relationship between the scattering cross section and the physical size of the particles, as the scattering cross section depends on the wavelength of radiation used. This can be seen when looking at a halo surrounding the moon on a decently foggy evening: Red light photons experience a larger cross sectional area of water droplets than photons of higher energy do. The halo around the moon thus has a perimeter of red light due to lower energy photons being scattering further from the center of the moon. Photons from the rest of the visible spectrum are left within the center of the halo and perceived as white light.

6.4. Meteorological Range

The scattering cross section is related to the meteorological range L_V:

[math]\displaystyle{ L_\text{V} = \frac{3.9}{C \sigma_\text{scat}}. }[/math]

The quantity Cσ_scat is sometimes denoted b_scat, the scattering coefficient per unit length.^[8]

7. Examples

7.1. Example 1: Elastic Collision of Two Hard Spheres

The elastic collision of two hard spheres is an instructive example that demonstrates the sense of calling this quantity a cross section. R and r are respectively the radii of the scattering center and scattered sphere. The total cross section is

[math]\displaystyle{ \sigma_\text{tot} = \pi \left(r + R\right)^2. }[/math]

So in this case the total scattering cross section is equal to the area of the circle (with radius r + R) within which the center of mass of the incoming sphere has to arrive for it to be deflected, and outside which it passes by the stationary scattering center. When the radius of the incoming sphere is approaching zero, the cross section is just the area of a circle with radius R.

7.2. Example 2: Scattering Light from a 2D Circular Mirror

Another example illustrates the details of the calculation of a simple light scattering model obtained by a reduction of the dimension. For simplicity, we will consider the scattering of a beam of light on a plane treated as a uniform density of parallel rays and within the framework of geometrical optics from a circle with radius r with a perfectly reflecting boundary. Its three-dimensional equivalent is therefore the more difficult problem of a laser or flashlight light scattering from the mirror sphere, for example, from the mechanical bearing ball.^[9] The unit of cross section in one dimension is the unit of length, for example 1 m. Let α be the angle between the light ray and the radius joining the reflection point of the light ray with the center point of the circle mirror. Then the increase of the length element perpendicular to the light beam is expressed by this angle as

[math]\displaystyle{ \mathrm dx = r \cos \alpha \,\mathrm d \alpha, }[/math]

the reflection angle of this ray with respect to the incoming ray is then 2α, and the scattering angle is

[math]\displaystyle{ \theta = \pi - 2 \alpha. }[/math]

The energy or the number of photons reflected from the light beam with the intensity or density of photons I on the length dx is

[math]\displaystyle{ I \,\mathrm d \sigma = I \,\mathrm dx(x) = I r \cos \alpha \,\mathrm d \alpha = I \frac{r}{2} \sin \left(\frac{\theta}{2}\right) \,\mathrm d \theta = I \frac{\mathrm d \sigma}{\mathrm d \theta} \,\mathrm d \theta. }[/math]

The differential cross section is therefore (dΩ = dθ)

[math]\displaystyle{ \frac{\mathrm d \sigma}{\mathrm d \theta} = \frac{r}{2} \sin \left(\frac{\theta}{2}\right). }[/math]

As it is seen from the behaviour of the sine function, this quantity has the maximum for the backward scattering (θ = π; the light is reflected perpendicularly and returns), and the zero minimum for the scattering from the edge of the circle directly forward (θ = 0). It confirms the intuitive expectations that the mirror circle acts like a diverging lens, and a thin beam is more diluted the closer it is from the edge defined with respect to the incoming direction. The total cross section can be obtained by summing (integrating) the differential section of the entire range of angles:

[math]\displaystyle{ \sigma = \int_0^{2 \pi} \frac{\mathrm d \sigma}{\mathrm d \theta} \,\mathrm d \theta = \int_0^{2 \pi} \frac{r}{2} \sin \left(\frac{\theta}{2}\right) \,\mathrm d \theta = \left. -r \cos \left(\frac{\theta}{2}\right) \right|_0^{2 \pi} = 2 r, }[/math]

so it is equal as much as the circular mirror is totally screening the two-dimensional space for the beam of light. In three dimensions for the mirror ball with the radius r it is therefore equal σ = πr².

7.3. Example 3: Scattering Light from a 3D Spherical Mirror

We can now use the result from the Example 2 to calculate the differential cross section for the light scattering from the perfectly reflecting sphere in three dimensions. Let us denote now the radius of the sphere as a. Let us parameterize the plane perpendicular to the incoming light beam by the cylindrical coordinates r and φ. In any plane of the incoming and the reflected ray we can write now from the previous example:

[math]\displaystyle{ \begin{align} r &= a \sin \alpha,\\ \mathrm dr &= a \cos \alpha \,\mathrm d \alpha, \end{align} }[/math]

while the impact area element is

[math]\displaystyle{ \mathrm d \sigma = \mathrm d r(r) \times r \,\mathrm d \varphi = \frac{a^2}{2} \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right) \,\mathrm d \theta \,\mathrm d \varphi. }[/math]

Using the relation for the solid angle in the spherical coordinates:

[math]\displaystyle{ \mathrm d\Omega = \sin \theta \,\mathrm d \theta \,\mathrm d \varphi }[/math]

and the trigonometric identity

[math]\displaystyle{ \sin \theta = 2 \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right), }[/math]

we obtain

[math]\displaystyle{ \frac{\mathrm d \sigma}{\mathrm d \Omega} = \frac{a^2}{4}, }[/math]

while the total cross section as we expected is

[math]\displaystyle{ \sigma = \oint_{4 \pi} \frac{\mathrm d \sigma}{\mathrm d \Omega} \,\mathrm d \Omega = \pi a^2. }[/math]

As one can see, it also agrees with the result from the Example 1 if the photon is assumed to be a rigid sphere of zero radius.