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In physics, a cross section $\sigma$ is an effective area that governs the probability of an event or reaction. The term is derived from the purely classical picture of (a large number of) point-like projectiles directed onto a target area that includes a solid target. Assuming that an event will occur if the projectile hits the solid target, and will not occur if it misses, the total event probability for the single projectile will be the ratio of the area of the section of the solid (the cross section, represented by $\sigma$ ) to the total targeted area. This basic concept is then extended to the cases where the reaction probability in the targeted area assumes intermediate values - because the target itself is not homogeneous, or because the interaction is mediated by a non-uniform field. scattering events which are commonly described by cross sections.

Units

The SI unit for 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 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 = 10⁻⁶ m⁻³). Scattering cross-sections are also widely used in the description of the scattering of neutrons by atomic nuclei. In this case, the conventional unit is the barn, b, where 1 b = 10⁻²⁸ m² = 100 fm².^[1] 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. Atomic and molecular cross sections often use units of Å² or a₀² where a₀ is the Bohr radius.

Rate

Assuming that the probability of an event occurring for a particle randomly projected at a target area, $A$ , is the ratio of the of the cross section of that event to the targeted area, ${\frac {\sigma }{A}}$ , then the rate of that event, $W$ , is given by:

W=J\sigma ={\frac {\text{no. of events}}{\text{unit time}}}

Where $n$ particles per unit volume in the beam (number density of particles) traveling with average velocity $v$ in the rest frame of the target combine to give the flux of the beam $J=nv$ . If $\sigma$ is unknown it can be experimentally determined using a known flux of particles and measuring the rate of the events.

\sigma ={\frac {W}{J}}

Partial cross section

For a particle beam incident on a several target particles, a rate of each type of reaction or event (labelled by an index r = 1, 2, 3...,) can be calculated from:^[2]

W_{r}=JN\sigma _{r}

Where $N$ is the number of target particles in the targeted area illuminated by the beam. The cross section of the event is $\sigma _{r}$ . Since the beam flux has dimensions of [length]⁻² [time]⁻¹ and $\sigma _{r}$ has dimensions of [length]² while $N$ is a dimensionless number, the rate $W$ has the dimensions of reciprocal time - which intuitively represents a frequency of recurring events.

The above rate formula assumes the following:

the beam particles all have the same kinetic energy,
the number density of the beam particles is sufficiently low: allowing the interactions between the particles within the beam to be neglected,
the number density of target particles is sufficiently low: so that only one scattering event per particle occurs as soon as the beam is incident with the target, and multiple scattering events within the target can be neglected,
the de Broglie wavelength of the beam is much smaller than the inter-particle separations within the target, so that diffraction effects through the target can be neglected,
the collision energy is sufficiently high allowing the binding energies in the target particles to be neglected.

These conditions are usually met in experiments, which allows for a very simple calculation of rate.

Sometimes the rate per unit target particle, or rate density, is more useful. For reaction r:^[3]

W_{r}/N=J\sigma _{r}

Beer-Lambert Law of Attenuation

For a beam of particles in x direction, the change in flux between positions $x$ and $x+dx$ due to impact events gives the following differential equation:

dJ=-J\sigma ndx

Integrating this equation gives the solution:

J=J_{0}e^{-\sigma nx}

When applied to photons, this is the Beer–Lambert Law of Attenuation. It provides another way to measure the cross section by diminution of beam projectiles as they travel through the target medium and can be written as:

J=J_{0}e^{\frac {-x}{\lambda }}

Where $\lambda ={\frac {1}{n\sigma }}$ is the ${\frac {1}{e}}$ “survival” distance of uncollided particles in the target medium. Thus $\lambda$ is a measure of distance between collisions.

Total cross section

The cross section $\sigma _{\text{r}}$ is specifically for one type of reaction, and is called the partial cross section. The total cross section, and corresponding total rate of the reaction, can be found by summing over the cross sections and rates for each reaction:^[2]

W=\sum _{r}W_{r}=JN\sum _{r}\sigma _{r}=JN\sigma _{\text{tot}}

Since collisions can be classified as elastic or inelastic, the total cross section $\sigma _{\text{tot}}$ can be written as the sum of the elastic scattering cross section $\sigma _{\text{el}}$ and the inelastic scattering cross section $\sigma _{\text{inel}}$ .

\sigma _{\text{tot}}=\sigma _{\text{el}}+\sigma _{\text{inel}}

The total cross section can be written as the sum of the cross-sections due to absorption, scattering and luminescence

\sigma _{\text{tot}}=\sigma _{\text{A}}+\sigma _{\text{S}}+\sigma _{\text{L}}.\

Differential cross section

A more detailed way of writing the collisional cross section comes from considering that scattered particles emerge at various angles. The differential cross section is defined as a ratio of two different kinds of fluxes: the outgoing particle flux, measured as particles per unit solid angle per unit area per unit time, divided by the incoming flux, measured in particles per unit area per unit time. It is written as $\sigma (\theta )$ or ${\frac {d\sigma }{d\Omega }}$

The differential cross section related to the total cross section by:

\sigma _{tot}=\int d\sigma =\int _{\Omega }{\frac {d\sigma }{d\Omega }}\,d\Omega =\int _{\varphi =0}^{2\pi }\int _{\vartheta =0}^{\pi }\,{\frac {d\sigma }{d\Omega }}\sin \vartheta \,d\vartheta \,d\varphi .

In terms of the differential cross section dσ_r(θ, φ) as a function of spherical polar angles θ and φ for reaction r, the differential rate is:^[2]

dW_{r}=JNd\sigma _{r}=JN{\frac {d\sigma _{r}}{d\Omega }}d\Omega

where dΩ = d(cosθ)dφ is the solid angle element in the vicinity of the event with vertex at the point of scattering. Integrating over θ and φ returns the rate for reaction r:

W_{r}=JN\int _{0}^{2\pi }d\varphi \int _{-1}^{+1}d(\cos \theta ){\frac {d\sigma _{r}}{d\Omega }}

Impact parameter

To calculate the differential cross section it is often useful introduce an impact parameter $b$ .

{\frac {d\sigma }{d\Omega }}={\frac {b}{\sin \vartheta }}\left|{\frac {db}{d\vartheta }}\right|

Deflection function

The scattering angle is the absolute value of the deflection function: $\theta =\left|\chi \right|$ The impact parameter is related to the scattering angle $\chi$ by^[4]

\chi =\pi -2b\int _{r_{\mathrm {min} }}^{\infty }{\frac {dr}{r^{2}{\sqrt {1-(b/r)^{2}-V(r)/E}}}}

$r_{\mathrm {min} }$ is the closest distance from the center in the center of mass frame and will make the root vanish. The differential cross section is related to the deflection angle by:

\sigma (\theta ,E)={\frac {b}{\sin \theta \left|{\frac {d\mathrm {X} }{db}}\right|}}

Hard Sphere

The hard sphere with radius $R$ has:

\chi =\pi -2b\int _{r_{\mathrm {min} }}^{\infty }{\frac {dr}{r^{2}{\sqrt {1-(b/r)^{2}}}}}

This, can be directly integrated to give:

\chi =\pi -2\sin ^{-1}({\frac {b}{R}})

The differential cross section for the hard sphere is:

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

As expected the total cross section for a hard sphere is:

\sigma =\pi R^{2}

Rutherford Scattering

See [Rutherford Scattering] For the [coulomb potential] of the form $V(r)={\frac {B}{r}}$ , $B>0$

\mathrm {X} =2sin^{-1}[{\frac {B/E}{(B^{2}/E^{2}+4b^{2})^{1/2}}}]

The partial cross section for Coulomb Scattering is:

{\frac {d\sigma }{d\Omega }}=({\frac {B/4E}{\sin ^{2}\theta /2}})^{2}

The total cross section for Coulomb scattering is infinite.

References

^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), pp. 127–28, ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16
^ ^a ^b ^c B.R. Martin, G. Shaw (2009). Particle Physics (3rd ed.). Manchester Physics Series, John Wiley & Sons. pp. 343–347. ISBN 978-0-470-03294-7.
^ G.F. Knoll (2010). Radiation detection and measurement (4th ed.). Wiley. p. 55. ISBN 978-0-470-13148-0.
^ Landau LD and Lifshitz EM (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).

J.D.Bjorken, S.D.Drell, Relativistic Quantum Mechanics, 1964
P.Roman, Introduction to Quantum Theory, 1969
W.Greiner, J.Reinhardt, Quantum Electrodynamics, 1994
R.G. Newton. Scattering Theory of Waves and Particles. McGraw Hill, 1966.
R.C. Fernow (1989). Introduction to Experimental Particle Physics. Cambridge University Press. ISBN 0-521-379-407.

External links

Category:Concepts in physics Category:Nuclear physics Category:Particle physics Category:Experimental particle physics Category:Scattering Category:Scattering theory Category:Dimensional analysis Category:Measurement

[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, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16

[Martin_Shaw_p_343_-_347-2] B.R. Martin, G. Shaw (2009). Particle Physics (3rd ed.). Manchester Physics Series, John Wiley & Sons. pp. 343–347. ISBN 978-0-470-03294-7.

[3] G.F. Knoll (2010). Radiation detection and measurement (4th ed.). Wiley. p. 55. ISBN 978-0-470-13148-0.

[4] Landau LD and Lifshitz EM (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).

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