Scattering cross-section

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The scattering cross-section is a hypothetical area which describes the likelihood of light or other radiation being scattered by a particle, the scattering center. It is a measure of the strength of the interaction between the scattered particle and one or several scattering centers.

Definition[edit]

In the simplest case, the differential scattering cross section refers to the way a single particle is scattered on a single scattering center:

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{db}{d\vartheta}\right|\ ,

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

The total scattering cross section can be defined (and computed) as the integral of the differential cross section over the whole solid angle:

\sigma_\text{tot} = \oint d\Omega \;\frac{d\sigma}{d\Omega}

It provides a measure of the strength of the interaction between the scattered particle and the scattering center.

When only considering a single collision between a particle beam and a sample at rest, an atom of the sample is usually taken as the scattering center. In collider experiments, both collision partners are accelerated, so either can be the scattered particle or the scattering center. When a particle is scattered by an extended body, multiple scattering centers may have to be taken into account.

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.[1] 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.

Units[edit]

The SI unit of total cross sections is the square meter, m2, although smaller units are usually used in practice. The name cross-section arises because it has the dimensions of area.

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 (1 cm2 = 10−4 m2) and the number concentration in cm−3 (1 cm−3 = 106 m−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, Å2, is a convenient unit: 1 Å2 = 10−20 m2 = 104 pm2.

In particle physics, where scattering processes between subatomic particles are investigated, the conventional unit is the barn, b, where 1 b = 10−28 m2 = 100 fm2.[2] Smaller prefixed units such as mb (millibarn), μb (microbarn) etc. are also widely used. The name of the unit barn originates from the fact that a scattering cross section of the order of a barn is unusually large, "big as a barn door".

The physics of scattering with a single scattering center[edit]

Classical Mechanics[edit]

In classical mechanics, the differential cross section is defined as follows: let a beam of intensity I_0 (measured in number of particles per area per time) be incident on a scattering center. In general, the angle at which a particle is scattered is a function of impact parameter. The number of scattered particles per solid angle per time (the radiant intensity), I_\text{s} is therefore well defined. We define the differential cross section to be

{d \sigma \over d \Omega} = \frac{I_\text{s}}{I_0}.

Note that this quantity has units of area. Furthermore, it depends only on the geometry of the scattering center, and not on the incident flux or distance of the detector from the scattering center. The geometric interpretation is as follows: consider particles that scatter through a solid angle d \Omega and ask what values of impact parameter produced them. These impact parameters form a differential area, d \sigma in space. The differential cross section is simply

{d \sigma \over d \Omega}.

Quantum Mechanics[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.

The integral cross section is the integral of the differential cross section on the whole sphere of observation (4π steradian):

\sigma=\int {d \sigma \over d \Omega} \, d\Omega.

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. Usual units are the cm2, the barn (1 b = 10−28 m2) and the corresponding submultiples: the millibarn (1 mb = 10−3 b), the microbarn (1 \mub = 10−6 b), the nanobarn ( 1 nb = 10−9 b), the picobarn (1 pb = 10−12 b), and the shed (1 shed = 10−24 b). 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_\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.

Particle physics[edit]

Differential and total scattering cross sections are among the most important measurable quantities in particle physics. Instead of the solid angle, the momentum transfer is often chosen as the independent variable of differential cross sections.

Differential cross sections in inelastic scattering contain peaks indicating the creation of particles, 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' colour symmetry, allowing scientists to discover this symmetry.

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:[3]

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.[4]

See also[edit]

References[edit]

  1. ^ 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. 
  2. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), pp. 127–28, ISBN 92-822-2213-6 
  3. ^ Bajpai, P.K. "2. Spectrophotometry". Biological Instrumentation and Biology. ISBN 81-219-2633-5. 
  4. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Scattering cross-section, σscat".