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Cross section (physics)

From Wikipedia, the free encyclopedia
Probability of a given process occurring in a particle collision
For an object's radar signature, seeRadar cross section.
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In physics, thecross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, theRutherford cross-section is a measure of probability that analpha particle will be deflected by a given angle during an interaction with anatomic nucleus. Cross section is typically denotedσ (sigma) and is expressed in units of area, more specifically inbarns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of astochastic process.

When two discrete particles interact in classical physics, their mutualcross section is the areatransverse to their relative motion within which they must meet in order toscatter from each other. If the particles are hardinelasticspheres that interact only upon contact, their scattering cross section is related to their geometric size. If the particles interact through some action-at-a-distance force, such aselectromagnetism orgravity, their scattering cross section is generally larger than their geometric size.

When a cross section is specified as thedifferential limit of a function of some final-state variable, such as particle angle or energy, it is called adifferential cross section (see detailed discussion below). When a cross section is integrated over all scattering angles (and possibly other variables), it is called atotal cross section orintegrated total cross section. For example, inRayleigh scattering, the intensity scattered at the forward and backward angles is greater than the intensity scattered sideways, so the forward differential scattering cross section is greater than the perpendicular differential cross section, and by adding all of the infinitesimal cross sections over the whole range of angles with integral calculus, we can find the total cross section.

Scattering cross sections may be defined innuclear,atomic, andparticle physics for collisions of accelerated beams of one type of particle with targets (either stationary or moving) of a second type of particle. The probability for any given reaction to occur is in proportion to its cross section. Thus, specifying the cross section for a given reaction is a proxy for stating the probability that a given scattering process will occur.

The measuredreaction rate of a given process depends strongly on experimental variables such as the density of the target material, the intensity of the beam, the detection efficiency of the apparatus, or the angle setting of the detection apparatus. However, these quantities can be factored away, allowing measurement of the underlying two-particle collisional cross section.

Differential and total scattering cross sections are among the most important measurable quantities innuclear,atomic, andparticle physics.

With light scattering off of a particle, thecross section specifies the amount of optical power scattered from light of a given irradiance (power per area). Although the cross section has the same units as area, the cross section may not necessarily correspond to the actual physical size of the target given by other forms of measurement. It is not uncommon for the actual cross-sectional area of a scattering object to be much larger or smaller than the cross section relative to some physical process. For example,plasmonic nanoparticles can have light scattering cross sections for particular frequencies that are much larger than their actual cross-sectional areas.

Collision among gas particles

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Figure 1. In a gas of particles of individual diameter2r, the cross sectionσ, for collisions is related to the particle number densityn, and mean free path between collisionsλ.

In agas 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

σ=1nλ,{\displaystyle \sigma ={\frac {1}{n\lambda }},}

where

σ is the cross section of a two-particle collision (SI unit: m2),
λ is themean free path between collisions (SI unit: m),
n is thenumber density of the target particles (SI unit: m−3).

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

σ=π(2r)2{\displaystyle \sigma =\pi \left(2r\right)^{2}}

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, innuclear physics a "gas" of low-energyneutrons collides with nuclei in a reactor or other nuclear device, with across section that is energy-dependent and hence also with well-definedmean free path between collisions.

Attenuation of a beam of particles

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See also:Attenuation

If a beam of particles enters a thin layer of material of thicknessdz, thefluxΦ of the beam will decrease bydΦ according to

dΦdz=−nσΦ,{\displaystyle {\frac {\mathrm {d} \Phi }{\mathrm {d} z}}=-n\sigma \Phi ,}

whereσ is the total cross section ofall events, includingscattering,absorption, or transformation to another species. The volumetric number density of scattering centers is designated byn. Solving this equation exhibits the exponential attenuation of the beam intensity:

Φ=Φ0e−nσz,{\displaystyle \Phi =\Phi _{0}e^{-n\sigma z},}

whereΦ0 is the initial flux, andz is the total thickness of the material. For light, this is called theBeer–Lambert law.

Differential cross section

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Consider aclassical measurement where a single particle is scattered off a single stationary target particle. Conventionally, aspherical coordinate system is used, with the target placed at the origin and thez axis of this coordinate system aligned with the incident beam. The angleθ is thescattering angle, measured between the incident beam and the scattered beam, and theφ is theazimuthal angle.

Theimpact parameterb is the perpendicular offset of the trajectory of the incoming particle, and the outgoing particle emerges at an angleθ. 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 angleθ is the solid angle elementdΩ = sinθ 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), theazimuthal angleφ is not changed by the scattering process, and the differential cross section can be written as

dσd(cos⁡θ)=∫02πdσdΩdφ{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} (\cos \theta )}}=\int _{0}^{2\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\,\mathrm {d} \varphi }.

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 fluxFinc off a stationary target consisting of many particles, the differential cross section⁠dσ/dΩ⁠ at an angle(θ,φ) is related to the flux of scattered particle detectionFout(θ,φ) in particles per unit time by

dσdΩ(θ,φ)=1ntΔΩFout(θ,φ)Finc.{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta ,\varphi )={\frac {1}{nt\Delta \Omega }}{\frac {F_{\text{out}}(\theta ,\varphi )}{F_{\text{inc}}}}.}

HereΔΩ is the finite angular size of the detector (SI unit:sr),n is thenumber density of the target particles (SI unit: m−3), andt is the thickness of the stationary target (SI unit: 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 sectionσ may be recovered by integrating the differential cross section⁠dσ/dΩ⁠ over the fullsolid angle (4π steradians):

σ=∮4πdσdΩdΩ=∫02π∫0πdσdΩsin⁡θdθdφ.{\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 .}

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 theintegral cross section ortotal 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 ofRutherford scattering provided strong evidence for the existence of the atomic nucleus.

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

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

Quantum scattering

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In thetime-independent formalism ofquantum scattering, the initialwave function (before scattering) is taken to be a plane wave with definitemomentumk:

ϕ−(r)⟶r→∞eikz,{\displaystyle \phi _{-}(\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;e^{ikz},}

wherez andr are therelative coordinates between the projectile and the target. The arrow indicates that this only describes theasymptotic 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:

ϕ+(r)⟶r→∞f(θ,ϕ)eikrr,{\displaystyle \phi _{+}(\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;f(\theta ,\phi ){\frac {e^{ikr}}{r}},}

wheref is some function of the angular coordinates known as thescattering 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

ϕ(r)⟶r→∞ϕ−(r)+ϕ+(r).{\displaystyle \phi (\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;\phi _{-}(\mathbf {r} )+\phi _{+}(\mathbf {r} ).}

The differential cross section is related to the scattering amplitude:

dσdΩ(θ,ϕ)=|f(θ,ϕ)|2.{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta ,\phi )={\bigl |}f(\theta ,\phi ){\bigr |}^{2}.}

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 twoparticles (i.e. observed when the two particles arecolliding 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 particlesIr) depends only on the number of incident particles per unit of time (current of incident particlesIi), the characteristics of target (for example the number of particles per unit of surfaceN), and the type of interaction. ForNσ ≪ 1 we have

Ir=IiNσ,σ=IrIi1N=probability of interaction×1N.{\displaystyle {\begin{aligned}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{aligned}}}

Relation to the S-matrix

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If thereduced masses andmomenta of the colliding system aremi,pi andmf,pf before and after the collision respectively, the differential cross section is given by[clarification needed]

dσdΩ=(2π)4mimfpfpi|Tfi|2,{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}=\left(2\pi \right)^{4}m_{\text{i}}m_{\text{f}}{\frac {p_{\text{f}}}{p_{\text{i}}}}{\bigl |}T_{{\text{f}}{\text{i}}}{\bigr |}^{2},}

where the on-shellT matrix is defined by

Sfi=δfi−2πiδ(Ef−Ei)δ(pi−pf)Tfi{\displaystyle S_{{\text{f}}{\text{i}}}=\delta _{{\text{f}}{\text{i}}}-2\pi i\delta \left(E_{\text{f}}-E_{\text{i}}\right)\delta \left(\mathbf {p} _{\text{i}}-\mathbf {p} _{\text{f}}\right)T_{{\text{f}}{\text{i}}}}

in terms of theS-matrix. Hereδ is theDirac delta function. The computation of the S-matrix is the main goal of thescattering theory.

Scattering off solid sphere and off spherical shell

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Consider spheres of radiusa{\displaystyle a}. Classically the cross section would beπa2{\displaystyle \pi a^{2}}. Quantum mechanically and for slow particles (i.e. for those whose de Broglie wave length is much larger than the dimensions of the scatterer) andS{\displaystyle S} waves the total cross section is4πa2{\displaystyle 4\pi a^{2}}. For fast particles higher angular momenta have to be taken into account and the total cross section is approximately2πa2{\displaystyle 2\pi a^{2}}. In the case of a spherical shell (potential a delta function) the total cross section allows resonances to appear.[1]

Units

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Although theSI unit of total cross sections ism2, a smaller unit is usually used in practice.

In nuclear and particle physics, the conventional unit is the barnb, where 1 b = 10−28 m2 = 100 fm2.[2] Smallerprefixed units such asmb andμb 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 incentimetres. 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 asnephelometry, and is effective for particles of 2–50 μm in diameter: as such, it is widely used inmeteorology and in the measurement ofatmospheric pollution.

The scattering ofX-rays can also be described in terms of scattering cross sections, in which case the squareångström 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.[3]

Scattering of light

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For light, as in other settings, the scattering cross section for particles is generally different from thegeometrical cross section of the particle, and it depends upon thewavelength of light and thepermittivity, 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, includingabsorption,scattering, andphotoluminescence. The sum of the absorption and scattering cross sections is sometimes referred to as the attenuation or extinction cross section.

σ=σabs+σsc+σlum.{\displaystyle \sigma =\sigma _{\text{abs}}+\sigma _{\text{sc}}+\sigma _{\text{lum}}.}

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

Aλ=Clσ,{\displaystyle A_{\lambda }=Cl\sigma ,}

whereAλ is the attenuation at a givenwavelengthλ,C is the particle concentration as a number density, andl is thepath length. The absorbance of the radiation is thelogarithm (decadic or, more usually,natural) of the reciprocal of thetransmittanceT:[4]

Aλ=−log⁡T.{\displaystyle A_{\lambda }=-\log {\mathcal {T}}.}

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.

Cross section and Mie theory

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Cross sections commonly calculated usingMie theory include efficiency coefficients for extinctionQext{\textstyle Q_{\text{ext}}}, scatteringQsc{\textstyle Q_{\text{sc}}}, and AbsorptionQabs{\textstyle Q_{\text{abs}}} cross sections. These are normalized by the geometrical cross sections of the particleσgeom=πa2{\textstyle \sigma _{\text{geom}}=\pi a^{2}} asQα=σασgeom,α=ext,sc,abs.{\displaystyle Q_{\alpha }={\frac {\sigma _{\alpha }}{\sigma _{\text{geom}}}},\qquad \alpha ={\text{ext}},{\text{sc}},{\text{abs}}.}The cross section is defined by

σα=WαIinc{\displaystyle \sigma _{\alpha }={\frac {W_{\alpha }}{I_{\text{inc}}}}}

where[Wα]=[W]{\displaystyle \left[W_{\alpha }\right]=\left[{\text{W}}\right]} is the energy flow through the surrounding surface, and[Iinc]=[Wm2]{\displaystyle \left[I_{\text{inc}}\right]=\left[{\frac {\text{W}}{{\text{m}}^{2}}}\right]} is the intensity of the incident wave. For aplane wave the intensity is going to beIinc=|E|2/(2η){\displaystyle I_{\text{inc}}=|\mathbf {E} |^{2}/(2\eta )}, whereη=μμ0/(εε0){\displaystyle \eta ={\sqrt {\mu \mu _{0}/(\varepsilon \varepsilon _{0})}}} is theimpedance of the host medium.

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

Wa=−∮AΠ⋅r^dA{\displaystyle W_{\text{a}}=-\oint _{A}\mathbf {\Pi } \cdot {\hat {\mathbf {r} }}dA}

whereΠ=12Re⁡[E∗×H]{\textstyle \mathbf {\Pi } ={\frac {1}{2}}\operatorname {Re} \left[\mathbf {E} ^{*}\times \mathbf {H} \right]} is the time averagedPoynting vector. IfWa>0{\displaystyle W_{\text{a}}>0} energy is absorbed within the sphere, otherwise energy is being created within the sphere. We will not consider this case here. If the host medium is non-absorbing, the energy must be absorbed by the particle. We decompose the total field into incident and scattered partsE=Ei+Es{\displaystyle \mathbf {E} =\mathbf {E} _{\text{i}}+\mathbf {E} _{\text{s}}}, and the same for the magnetic fieldH{\displaystyle \mathbf {H} }. Thus, we can decomposeWa{\displaystyle W_{a}} into the three termsWa=Wi−Ws+Wext{\displaystyle W_{\text{a}}=W_{\text{i}}-W_{\text{s}}+W_{\text{ext}}}, where

Wi=−∮AΠi⋅r^dA≡0,Ws=∮AΠs⋅r^dA,Wext=∮AΠext⋅r^dA.{\displaystyle W_{\text{i}}=-\oint _{A}\mathbf {\Pi } _{\text{i}}\cdot {\hat {\mathbf {r} }}dA\equiv 0,\qquad W_{\text{s}}=\oint _{A}\mathbf {\Pi } _{\text{s}}\cdot {\hat {\mathbf {r} }}dA,\qquad W_{\text{ext}}=\oint _{A}\mathbf {\Pi } _{\text{ext}}\cdot {\hat {\mathbf {r} }}dA.}

whereΠi=12Re⁡[Ei∗×Hi]{\displaystyle \mathbf {\Pi } _{\text{i}}={\frac {1}{2}}\operatorname {Re} \left[\mathbf {E} _{\text{i}}^{*}\times \mathbf {H} _{\text{i}}\right]},Πs=12Re⁡[Es∗×Hs]{\displaystyle \mathbf {\Pi } _{\text{s}}={\frac {1}{2}}\operatorname {Re} \left[\mathbf {E} _{\text{s}}^{*}\times \mathbf {H} _{\text{s}}\right]}, andΠext=12Re⁡[Es∗×Hi+Ei∗×Hs]{\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]}.

All the field can be decomposed into the series ofvector spherical harmonics (VSH). After that, all the integrals can be taken.In the case of auniform sphere of radiusa{\displaystyle a}, permittivityε{\displaystyle \varepsilon }, and permeabilityμ{\displaystyle \mu }, the problem has a precise solution.[5] The scattering and extinction coefficients areQsc=2k2a2∑n=1∞(2n+1)(|an|2+|bn|2){\displaystyle Q_{\text{sc}}={\frac {2}{k^{2}a^{2}}}\sum _{n=1}^{\infty }(2n+1)(|a_{n}|^{2}+|b_{n}|^{2})}Qext=2k2a2∑n=1∞(2n+1)ℜ(an+bn){\displaystyle Q_{\text{ext}}={\frac {2}{k^{2}a^{2}}}\sum _{n=1}^{\infty }(2n+1)\Re (a_{n}+b_{n})} Wherek=nhostk0{\textstyle k=n_{\text{host}}k_{0}}. These are connected asσext=σsc+σabsorQext=Qsc+Qabs{\displaystyle \sigma _{\text{ext}}=\sigma _{\text{sc}}+\sigma _{\text{abs}}\qquad {\text{or}}\qquad Q_{\text{ext}}=Q_{\text{sc}}+Q_{\text{abs}}}

Dipole approximation for the scattering cross section

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Let us assume that a particle supports only electric andmagnetic dipole modes with polarizabilitiesp=αeE{\textstyle \mathbf {p} =\alpha ^{e}\mathbf {E} } andm=(μμ0)−1αmH{\textstyle \mathbf {m} =(\mu \mu _{0})^{-1}\alpha ^{m}\mathbf {H} } (here we use the notation of magnetic polarizability in the manner of Bekshaev et al.[6][7] rather than the notation of Nieto-Vesperinas et al.[8]) expressed through the Mie coefficients asαe=4πε0⋅i3ε2k3a1,αm=4πμ0⋅i3μ2k3b1.{\displaystyle \alpha ^{e}=4\pi \varepsilon _{0}\cdot i{\frac {3\varepsilon }{2k^{3}}}a_{1},\qquad \alpha ^{m}=4\pi \mu _{0}\cdot i{\frac {3\mu }{2k^{3}}}b_{1}.} Then the cross sections are given byσext=σext(e)+σext(m)=14πεε0⋅4πkℑ(αe)+14πμμ0⋅4πkℑ(αm){\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})}σsc=σsc(e)+σsc(m)=1(4πεε0)2⋅8π3k4|αe|2+1(4πμμ0)2⋅8π3k4|αm|2{\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}} and, finally, the electric and magnetic absorption cross sectionsσabs=σabs(e)+σabs(m){\textstyle \sigma _{\text{abs}}=\sigma _{\text{abs}}^{\text{(e)}}+\sigma _{\text{abs}}^{\text{(m)}}} areσabs(e)=14πεε0⋅4πk[ℑ(αe)−k36πεε0|αe|2]{\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]} andσabs(m)=14πμμ0⋅4πk[ℑ(αm)−k36πμμ0|αm|2]{\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]}

For the case of a no-inside-gain particle, i.e. no energy is emitted by the particle internally (σabs>0{\textstyle \sigma _{\text{abs}}>0}), we have a particular case of theOptical theorem14πεε0ℑ(αe)+14πμμ0ℑ(αm)≥2k33[|αe|2(4πεε0)2+|αm|2(4πμμ0)2]{\displaystyle {\frac {1}{4\pi \varepsilon \varepsilon _{0}}}\Im (\alpha ^{e})+{\frac {1}{4\pi \mu \mu _{0}}}\Im (\alpha ^{m})\geq {\frac {2k^{3}}{3}}\left[{\frac {|\alpha ^{e}|^{2}}{(4\pi \varepsilon \varepsilon _{0})^{2}}}+{\frac {|\alpha ^{m}|^{2}}{(4\pi \mu \mu _{0})^{2}}}\right]} Equality occurs for non-absorbing particles, i.e. forℑ(ε)=ℑ(μ)=0{\textstyle \Im (\varepsilon )=\Im (\mu )=0}.

Scattering of light on extended bodies

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In the context of scattering light on extended bodies, the scattering cross section,σsc, describes the likelihood of light being scattered by a macroscopic particle. In general, the scattering cross section is different from thegeometrical cross section of a particle, as it depends upon the wavelength of light and thepermittivity 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, thetotal cross section (σ) is the sum of the cross sections due toabsorption, scattering, andluminescence:

σ=σabs+σsc+σlum.{\displaystyle \sigma =\sigma _{\text{abs}}+\sigma _{\text{sc}}+\sigma _{\text{lum}}.}

The total cross section is related to theabsorbance of the light intensity through theBeer–Lambert law, which says that absorbance is proportional to concentration:Aλ =Clσ, whereAλ is the absorbance at a givenwavelengthλ,C is the concentration as anumber density, andl is thepath length. The extinction orabsorbance of the radiation is thelogarithm (decadic or, more usually,natural) of the reciprocal of thetransmittanceT:[4]

Aλ=−log⁡T.{\displaystyle A_{\lambda }=-\log {\mathcal {T}}.}

Relation to physical size

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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. 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.

Meteorological range

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The scattering cross section is related to themeteorological rangeLV:

LV=3.9Cσscat.{\displaystyle L_{\text{V}}={\frac {3.9}{C\sigma _{\text{scat}}}}.}

The quantityCσscat is sometimes denotedbscat, the scattering coefficient per unit length.[9]

Examples

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Elastic collision of two hard spheres

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The following equations apply to two hard spheres that undergo a perfectly elastic collision.[10] LetR andr denote the radii of the scattering center and scattered sphere, respectively. The differential cross section is

dσdΩ=R24,{\displaystyle {\frac {d\sigma }{d\Omega }}={\frac {R^{2}}{4}},}

and the total cross section is

σtot=π(r+R)2.{\displaystyle \sigma _{\text{tot}}=\pi \left(r+R\right)^{2}.}

In other words, the total scattering cross section is equal to the area of the circle (with radiusr +R) within which the center of mass of the incoming sphere has to arrive for it to be deflected.

Rutherford scattering

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InRutherford scattering, an incident particle with chargeq and energyE scatters off a fixed particle with chargeQ. The differential cross section is

dσdΩ=(qQ16πε0Esin2⁡(θ/2))2{\displaystyle {\frac {d\sigma }{d\Omega }}=\left({\frac {q\,Q}{16\pi \varepsilon _{0}E\sin ^{2}(\theta /2)}}\right)^{2}}

whereε0{\displaystyle \varepsilon _{0}} is thevacuum permittivity.[11] The total cross section is infinite unless a cutoff for small scattering anglesθ{\displaystyle \theta } is applied.[12] This is due to the long range of the1/r{\displaystyle 1/r} Coulomb potential.

Scattering from a 2D circular mirror

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The following example deals with a beam of light scattering off a circle with radiusr and a perfectly reflecting boundary. The beam consists of a uniform density of parallel rays, and the beam-circle interaction is modeled within the framework ofgeometric optics. Because the problem is genuinely two-dimensional, the cross section has unit of length (e.g., metre). Letα be the angle between thelight ray and theradius joining the reflection point of the ray with the center point of the mirror. Then the increase of the length element perpendicular to the beam is

dx=rcos⁡αdα.{\displaystyle \mathrm {d} x=r\cos \alpha \,\mathrm {d} \alpha .}

The reflection angle of this ray with respect to the incoming ray is2α, and the scattering angle is

θ=π−2α.{\displaystyle \theta =\pi -2\alpha .}

The differential relationship between incident and reflected intensityI is

Idσ=Idx(x)=Ircos⁡αdα=Ir2sin⁡(θ2)dθ=Idσdθdθ.{\displaystyle I\,\mathrm {d} \sigma =I\,\mathrm {d} x(x)=Ir\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 .}

The differential cross section is therefore (dΩ = dθ)

dσdθ=r2sin⁡(θ2).{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \theta }}={\frac {r}{2}}\sin \left({\frac {\theta }{2}}\right).}

Its maximum atθ = π corresponds to backward scattering, and its minimum atθ = 0 corresponds to scattering from the edge of the circle directly forward. This expression confirms the intuitive expectations that the mirror circle acts like a diverginglens. The total cross section is equal to the diameter of the circle:

σ=∫02πdσdθdθ=∫02πr2sin⁡(θ2)dθ=2r.{\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 =2r.}

Scattering from a 3D spherical mirror

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The result from the previous example can be used to solve the analogous problem in three dimensions, i.e., scattering from a perfectly reflecting sphere of radiusa.

The plane perpendicular to the incoming light beam can be parameterized by cylindrical coordinatesr andφ. In any plane of the incoming and the reflected ray we can write (from the previous example):

r=asin⁡α,dr=acos⁡αdα,{\displaystyle {\begin{aligned}r&=a\sin \alpha ,\\\mathrm {d} r&=a\cos \alpha \,\mathrm {d} \alpha ,\end{aligned}}}

while the impact area element is

dσ=dr(r)×rdφ=a22sin⁡(θ2)cos⁡(θ2)dθdφ.{\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 .}

In spherical coordinates,

dΩ=sin⁡θdθdφ.{\displaystyle \mathrm {d} \Omega =\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi .}

Together with the trigonometric identity

sin⁡θ=2sin⁡(θ2)cos⁡(θ2),{\displaystyle \sin \theta =2\sin \left({\frac {\theta }{2}}\right)\cos \left({\frac {\theta }{2}}\right),}

we obtain

dσdΩ=a24.{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}={\frac {a^{2}}{4}}.}

The total cross section is

σ=∮4πdσdΩdΩ=πa2.{\displaystyle \sigma =\oint _{4\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\,\mathrm {d} \Omega =\pi a^{2}.}

See also

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References

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  1. ^ H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd edition (2012) World Scientific; pp. 271 - 272, 311 - 314.
  2. ^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, retrieved2021-12-16
  3. ^Nondestructive Testing Handbook Volume 4 Radiographic Testing, ASNT, 2002, chapter 22.
  4. ^abBajpai, P. K. (2008).Biological instrumentation and methodology (Revised 2nd ed.). Ram Nagar, New Delhi: S. Chand & Company Ltd.ISBN 978-81-219-2633-1.OCLC 943495167.
  5. ^Bohren, Craig F., and Donald R. Huffman. Absorption and scattering of light by small particles. John Wiley & Sons, 2008.
  6. ^Bekshaev, A Ya (2013-04-01). "Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows".Journal of Optics.15 (4) 044004.arXiv:1210.5730.Bibcode:2013JOpt...15d4004B.doi:10.1088/2040-8978/15/4/044004.ISSN 2040-8978.S2CID 119234614.
  7. ^Bliokh, Konstantin Y.; Bekshaev, Aleksandr Y.; Nori, Franco (2014-03-06)."Extraordinary momentum and spin in evanescent waves".Nature Communications.5 (1). Springer Science and Business Media LLC: 3300.arXiv:1308.0547.Bibcode:2014NatCo...5.3300B.doi:10.1038/ncomms4300.ISSN 2041-1723.PMID 24598730.S2CID 15832637.
  8. ^Nieto-Vesperinas, M.; Sáenz, J. J.; Gómez-Medina, R.; Chantada, L. (2010-05-14)."Optical forces on small magnetodielectric particle".Optics Express.18 (11). The Optical Society:11428–11443.Bibcode:2010OExpr..1811428N.doi:10.1364/oe.18.011428.ISSN 1094-4087.PMID 20589003.
  9. ^IUPAC,Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "Scattering cross section,σscat".doi:10.1351/goldbook.S05490
  10. ^Taylor 2005, pp. 564, 574.
  11. ^Taylor 2005, p. 576.
  12. ^Griffiths 2005, p. 409.

Bibliography

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External links

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