Scattering cross-section

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The scattering cross-section is an effective area that quantifies the intrinsic rate a given event(s) occurs during the scattering of two particle species (in quantum mechanics, this also includes waves). Conventionally, one of the species is treated as a projectile (incident beam), and the other species is treated as a target (scattering center).

Intuitively, an event is said to have a cross-section of σ if its rate is equal to the rate of collisions in an equivalent, idealized experiment where:

  • the projectiles are replaced by inert point-like particles, and
  • the targets are replaced by inert and inpenetratable disks of area σ (and hence the name "cross-section"),

with all other experimental parameters kept the same (assuming the the target sample is sufficiently thin). Mathematically, this is described as:

W = N I \sigma

where

  • W is the rate at which the event occurs (SI units: s−1),
  • N is the number of target particles within reach of the incident beam (dimensionless),
  • I is the particle flux (or intensity) of the incident beam (SI units: m−2 s−1), and
  • σ is the cross-section of this event (SI units: m−2).

Essentially, the cross-section is a measure of the rate that controls for the experimental parameters N and I.

Definition[edit]

Although it is possible to describe scattering in any reference frame, by convention we consider a reference frame in which one of the particle species at rest, which also simplifies the algebra. This frame is commonly known as the lab(oratory) frame. This stands in constrast to the center-of-mass frame, in which both particle species are moving towards each other with equal momentum.

The species at rest is referred to as the target species. The other moving species is the projectile species (or incident beam). The choice of the species is often not arbitrary as it depends on experimental feasibility. Usually, projectile is the less massive species. The reverse choice, where the projectile is the more massive species, is often called inverse kinematics.

The idealized scattering experiment involves the following setup:

  • A thin homogenous slab of the target species is placed perpendicular to the incident beam. This slab has thickness t and number density n.
  • A narrow, homegenous beam of the projectile species is sent towards the target at a particle flux (or intensity) of I. The beam overlaps the target with an area of A.

In this case, the cross section σ of an event occurring is related to its rate W by the following relation:

W = n t A I \sigma

This equation essentially defines the meaning of "cross section". The product n t A quantifies the number of target particles N that can be affected by projectiles.

Events[edit]

There are two major categories of events:

If the cross section of inelastic scattering is very small, the target is highly inert with respect to the projectiles. In constrast, if the cross section of inelastic scattering is very high, the target is is highly reactive with respect to the projectiles.

Differential cross section[edit]

The cross section is a scalar that only quantifies the intrinsic rate of an event. In constrast, the differential cross section dσ/dΩ is function that quantifies the intrinsic rate at which the scattered projectiles can be detected at a given angle. Note that the symbol dσ/dΩ is merely suggestive and not meant to be read literally as a derivative of σ with respect to "Ω".

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. Many types of scattering processes possess azimuthal symmetry and therefore do not depend on φ.

The differential cross section dσ/dΩ at an angle (θ, φ) is related to the rate of detection w at that angle by

w = N I \frac{d \sigma}{d \Omega} \Delta \Omega

where

  • ΔΩ is the angular span of the detector (SI unit: sr), which is assumed to be small and have perfect detection ratio,
  • N is the number of target particles within reach of the incident beam, and
  • I is the incident flux (or intensity).

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

\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 \phi

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.

Units[edit]

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

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 Å2 is a convenient unit: Å2 = 10−20 m2 = 104 pm2.

In nuclear and particle physics, the conventional unit is b, where 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−1.

Classical case[edit]

In a simple classical experiment where a single particle is scattered off a rigid target,

Differential cross section.svg

the impact parameter is the perpendicular offset of the trajectory of the incoming particle. The differential of the cross section is the area element in the plane of the impact parameter, i.e. d\sigma = d^2 b, where b is the impact parameter. The differential cross section is the differential quotient of this area element by the solid angle element in the direction of the particle exit trajectory:

\left| \frac{d \sigma}{d \Omega} \right|

It describes the change in the impact parameter necessary to cause a given change in the exit trajectory direction. The definition is slightly counterintuitive in that the independent variable (in the denominator) describes the effect and the dependent variable (in the numerator) the initial condition. The differential cross section is always taken to be positive, even though in the most frequent case of limited-range repulsive interactions, larger impact parameters cause less deflection. In rotationally symmetric problems, the azimuthal angle \varphi is not changed by the scattering process, and the differential cross section becomes

\frac{b}{\sin \vartheta} \left| \frac{d b}{d \vartheta}\right|,

where \vartheta is the angle between the incident and exit direction of the scattered particle, as shown in the figure.

Quantum case[edit]

In quantum mechanics, the wave function of the incident particle is a plane wave with amplitude 1, that is,  e^{ikz}. In general the scattered wave is of the form

f(\theta,\phi) \frac{e^{i k r} }{r}.

We then have as the definition of differential cross section

{d \sigma \over d \Omega} = |f|^2.

This has the simple interpretation of the probability of finding a scattered particle within a given solid 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_\text{r}) depends only on the number of incident particles per unit of time (current of incident particles I_\text{i}), the characteristics of target (for example the number of particles per unit of surface N), and the type of interaction. For N\sigma\ll 1 we have

I_\text{r}=I_\text{i}N\sigma\,
\sigma={{I_\text{r}}\over{I_\text{i}}}{{1}\over{N}}={\hbox{Probability of interaction}}\times{{1}\over{N}}

Relation to the S matrix[edit]

If the reduced masses and momenta of the colliding system are mi, pi and mf, pf before and after the collision respectively, the differential cross section is given by


{d\sigma \over d\Omega} = (2\pi)^4 m_i m_f {p_f \over p_i} |T_{fi}|^2,

where the on-shell T matrix is defined by


S_{fi} 
= 
\delta_{fi} 
- 
2\pi i \delta(E_f -E_i)
\delta(\mathbf{p}_i-\mathbf{p}_f) T_{fi}

in terms of the scattering matrix S. Here, \delta is the Dirac delta function. The computation of the S matrix is the main aim of the scattering theory.

Scattering of light on extended bodies[edit]

In the context of scattering light on extended bodies, the scattering cross-section, σscat, 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

\sigma = \sigma_\text{A} + \sigma_\text{S} + \sigma_\text{L}.\

The total cross-section is related to the absorbance of the light intensity through Beer-Lambert's law, which says absorbance is proportional to concentration: A_\lambda  = C l \sigma, where Aλ 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:[2]

A_\lambda = - \log \mathcal{T}.\

Relation to physical size[edit]

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 driving in foggy weather: the droplets of water (which form the fog) scatter red light less than they scatter the shorter wavelengths present in white light, and the red rear fog light can be distinguished more clearly than the white headlights of an approaching vehicle. That is to say that the scattering cross-section of the water droplets is smaller for red light than for light of shorter wavelengths, even though the physical size of the particles is the same.

Meteorological range[edit]

The scattering cross-section is related to the meteorological range, LV:

L_\text{V} = \frac{3.9}{C \sigma_\text{scat}}.\

The quantity C σscat is sometimes denoted bscat, the scattering coefficient per unit length.[3]

Applications[edit]

Differential and total scattering cross sections are among the most important measurable quantities in nuclear and particle physics. 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.

The total cross section in inelastic scattering is the sum of the total cross sections of all allowed individual processes. As a consequence, total cross sections of the creation of hadrons (i.e., strongly interacting particles) receive a factor of 3 from the quarks' color symmetry, allowing scientists to discover this symmetry.

Examples[edit]

Example 1: elastic collision of two hard spheres[edit]

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 the radii of the scattering center and scattered sphere, respectively, b the impact parameter and \vartheta the polar angle of the exit trajectory as above. Then the differential scattering cross section is

\left| \frac{d\sigma}{d\Omega} \right| = \frac14 (r+R)^2

The total cross section is

\sigma_\text{tot} = \pi \;(r+R)^2\ .

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.

Example 2: differential cross section for the geometric light scattering from the circle mirror[edit]

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.[4] The unit of cross section in one dimension is the unit of length, e. g. one meter. Let \alpha 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

dx =  r \cos \alpha d \alpha

the reflection angle of this ray with respect to the incoming ray is then 2 \alpha and the scattering angle is

\theta = \pi - 2 \alpha

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

I d \sigma = I dx(x) = I r \cos \alpha d \alpha = I \frac{r}{2} \sin (\theta/2)  d \theta = I \frac{d \sigma}{d \theta} d \theta

The differential cross section is therefore (d \Omega = d \theta)

\frac{d \sigma}{d \theta} = \frac{r}{2} \sin (\theta / 2)

As it is seen from the behaviour of the sine function this quantity has the maximum for the front backward scattering (\theta=\pi) (the light is reflected perpendicularly and it returns back) and the zero minimum for the scattering from the edge of the circle directly straight (\theta=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:

\sigma = \int_{0}^{2 \pi} \frac{d \sigma}{d \theta} d \theta = \int_{0}^{2 \pi} \frac{r}{2} (\sin \theta/2) d \theta =  - r \cos (\theta/2)  \bigg|_0^{2 \pi}  = 2 r

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 \sigma=\pi r^2.

Example 3: differential cross section for the geometric light scattering from the perfectly spherical mirror[edit]

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 parametrize the plane perpendicular to the incoming light beam by the cylindrical coordinates r and \phi. In any plane of the incoming and the reflected ray we can write now from the previous example:

r =  a \sin \alpha
dr =  a \cos \alpha d \alpha

while the impact area element is

 d \sigma =  d r(r) \times r d \phi  = \frac{a^2}{2} \sin (\theta/2) \cos (\theta/2)  d \theta d \phi

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

d\Omega=\sin (\theta) d \theta d \phi

and the trigonometric identity:

\sin (\theta)=2 \sin (\theta/2) \cos (\theta/2)

we obtain

\frac{d \sigma}{d \Omega} = \frac{a^2}{4}

while the total cross section as we expected is

\sigma = \oint_{4 \pi }^{} \frac{d \sigma}{d \Omega} d \Omega = \pi a^2

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

See also[edit]

References[edit]

  1. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), pp. 127–28, ISBN 92-822-2213-6 
  2. ^ Bajpai, P.K. "2. Spectrophotometry". Biological Instrumentation and Biology. ISBN 81-219-2633-5. 
  3. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Scattering cross-section, σscat".
  4. ^ M. Xu, R. R. Alfano (2003). "More on patterns in Mie scattering". Optics Communications 226: 1–5. Bibcode:2003OptCo.226....1X. doi:10.1016/j.optcom.2003.08.019.