Editing Cross section (physics) (section)

== Differential cross section ==

Consider a [[classical mechanics|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 {{math|''z''}} axis of this coordinate system aligned with the incident beam. The angle {{math|''θ''}} is the '''scattering angle''', measured between the incident beam and the scattered beam, and the {{math|''φ''}} is the [[azimuthal angle]].
: [[File:Differential cross section.svg|none|600px]]

The [[impact parameter]] {{math|''b''}} is the perpendicular offset of the trajectory of the incoming particle, and the outgoing particle emerges at an angle {{math|''θ''}}. For a given interaction ([[Coulomb's law|coulombic]], [[magnetism|magnetic]], [[gravitation]]al, 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. {{math|d''σ'' {{=}} ''b'' d''φ'' d''b''}}. The differential angular range of the scattered particle at angle {{math|''θ''}} is the solid angle element {{math|dΩ {{=}} sin ''θ'' d''θ'' d''φ''}}. The differential cross section is the quotient of these quantities, {{math|{{sfrac|d''σ''|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]] {{math|''φ''}} is not changed by the scattering process, and the differential cross section can be written as
: <math> \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 {{math|''F''<sub>inc</sub>}} off a stationary target consisting of many particles, the differential cross section {{math|{{sfrac|d''σ''|dΩ}}}} at an angle {{math|(''θ'',''φ'')}} is related to the flux of scattered particle detection {{math|''F''<sub>out</sub>(''θ'',''φ'')}} in particles per unit time by
: <math>\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 {{math|ΔΩ}} is the finite angular size of the detector (SI unit: [[steradian|sr]]), {{math|''n''}} is the [[number density]] of the target particles (SI unit: m<sup>−3</sup>), and {{math|''t''}} 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 section {{math|''σ''}} may be recovered by integrating the differential cross section {{math|{{sfrac|d''σ''|dΩ}}}} over the full [[solid angle]] ({{math|4π}} steradians):
: <math>\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" [[Grammatical modifier|qualifier]] when the type of cross section can be inferred from context. In this case, {{math|''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 (particle physics)|resonance peaks]] that indicate the creation of metastable states and contain information about their energy and lifetime.