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NUC-08

Klein–Nishina snapshot

dσ/dΩ = (r_e²/2) (ε′/ε)² (ε′/ε + ε/ε′ − sin²θ). Unpolarised Compton differential.

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NuclearKlein–Nishina

Governing equation

dσdΩ=re22(ε′ε)2(ε′ε+εε′−sin⁡2θ)\dfrac{d\sigma}{d\Omega}=\dfrac{r_e^2}{2}\left(\dfrac{\varepsilon'}{\varepsilon}\right)^2\left(\dfrac{\varepsilon'}{\varepsilon}+\dfrac{\varepsilon}{\varepsilon'}-\sin^2\theta\right)

θ∈[0,180]\theta \in \left[0, 180\right]

where

E
Photon energy (keV)
\theta
Angle (°)
\varepsilon'/\varepsilon
Energy ratio (—)
d\sigma/d\Omega
Differential KN (mb/sr)

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NUC-08 · spectrum
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①

Governing law — red

dσdΩ=re22(ε′ε)2(ε′ε+εε′−sin⁡2θ)\dfrac{d\sigma}{d\Omega}=\dfrac{r_e^2}{2}\left(\dfrac{\varepsilon'}{\varepsilon}\right)^2\left(\dfrac{\varepsilon'}{\varepsilon}+\dfrac{\varepsilon}{\varepsilon'}-\sin^2\theta\right)

θ∈[0,180]\theta \in \left[0, 180\right]

θ=0\theta = 0ε′/ε=79.4\varepsilon'/\varepsilon = 79.4

②

Domain of the variable — blue

θmin⁡=0,θmax⁡=180\theta_{\min} = 0,\quad \theta_{\max} = 180

θ∈[0,180]\theta \in \left[0, 180\right]

θ=0\theta = 0θ=0\theta = 0

③

Response curve — green

dσdΩ=re22(ε′ε)2(ε′ε+εε′−sin⁡2θ)\dfrac{d\sigma}{d\Omega}=\dfrac{r_e^2}{2}\left(\dfrac{\varepsilon'}{\varepsilon}\right)^2\left(\dfrac{\varepsilon'}{\varepsilon}+\dfrac{\varepsilon}{\varepsilon'}-\sin^2\theta\right)

θ∈[0,180]\theta \in \left[0, 180\right]

θ=0\theta = 0ε′/ε=79.4\varepsilon'/\varepsilon = 79.4

④

Rate of change — amber

dε′/εdθ\frac{\mathrm{d}\varepsilon'/\varepsilon}{\mathrm{d}\theta}

θ∈[0,180]\theta \in \left[0, 180\right]

θ=1.132\theta = 1.132ε′/ε′=−0.082\varepsilon'/\varepsilon^{\prime} = -0.082

Lecture brief

Historical brief

Rutherford, Chadwick, the semi-empirical mass formula, fission (Hahn–Strassmann 1938) and fusion Q-values, then Compton and Bethe–Bloch stopping, are the nuclear toolkit. Decay, binding and dose start here. This sheet (NUC-08 — Klein–Nishina snapshot) is the form associated with Klein–Nishina. Working symbols: EE, θ\theta →\rightarrow ε′/ε\varepsilon'/\varepsilon, dσ/dΩd\sigma/d\Omega. Reduces to Thomson (r_e²/2)(1+cos²θ) as ε → 0. Forward-peaked at high energy.

Purpose

Purpose: compute ε′/ε\varepsilon'/\varepsilon, dσ/dΩd\sigma/d\Omega from EE, θ\theta in Nuclear physics via dσdΩ=re22(ε′ε)2(ε′ε+εε′−sin⁡2θ)\dfrac{d\sigma}{d\Omega}=\dfrac{r_e^2}{2}\left(\dfrac{\varepsilon'}{\varepsilon}\right)^2\left(\dfrac{\varepsilon'}{\varepsilon}+\dfrac{\varepsilon}{\varepsilon'}-\sin^2\theta\right) dσ/dΩ = (r_e²/2) (ε′/ε)² (ε′/ε + ε/ε′ − sin²θ). Unpolarised Compton differential. Use it when a real nuclear physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given E=511.000 keVE = 511.000\,\mathrm{keV}, θ=60.000 ∘\theta = 60.000\,\mathrm{^{\circ}}, the governing relation dσdΩ=re22(ε′ε)2(ε′ε+εε′−sin⁡2θ)\dfrac{d\sigma}{d\Omega}=\dfrac{r_e^2}{2}\left(\dfrac{\varepsilon'}{\varepsilon}\right)^2\left(\dfrac{\varepsilon'}{\varepsilon}+\dfrac{\varepsilon}{\varepsilon'}-\sin^2\theta\right) yields ε′/ε=0.6667 —\varepsilon'/\varepsilon = 0.6667\,\mathrm{—}, dσ/dΩ=24.9963 mb/srd\sigma/d\Omega = 24.9963\,\mathrm{mb/sr}. A polar plot of dσ/dΩ. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Energy ratio \varepsilon'/\varepsilon0.6667 —
  • Differential KN d\sigma/d\Omega24.9963 mb/sr
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NUC-08 · spectrum
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Narration of this film

A polar plot of dσ/dΩ.

Reduces to Thomson (r_e²/2)(1+cos²θ) as ε → 0. Forward-peaked at high energy.

Reading speed

Watch on YouTube