NUC-21

Pair-production threshold

E_th = 2 m_e c² (1 + m_e/M) = 1.022 (1 + m_e/M) MeV. Photon in the field of mass M.

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NuclearPair production

Governing equation

E_{th}=2m_ec^2\,(1+m_e/M)

$M \in \left[5\times 10^{-4}, 240\right]$

where

M: Recoil mass (u)
E_{th}: Threshold (MeV)

The figure, part by part

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①

Governing law — red

E_{th}=2m_ec^2\,(1+m_e/M)

$M \in \left[5\times 10^{-4}, 240\right]$

$M = 5\times 10^{-4}$ $E_{th} = 2.143$

Along the curve

②

Domain of the variable — blue

M_{\min} = 5\times 10^{-4},\quad M_{\max} = 240

$M \in \left[5\times 10^{-4}, 240\right]$

$M = 5\times 10^{-4}$ $M = 5\times 10^{-4}$

Along the curve

③

Response curve — green

E_{th}=2m_ec^2\,(1+m_e/M)

$M \in \left[5\times 10^{-4}, 240\right]$

$M = 5\times 10^{-4}$ $E_{th} = 2.143$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E_{th}}{\mathrm{d}M}

$M \in \left[5\times 10^{-4}, 240\right]$

$M = 1.51$ $E_{th}^{\prime} = -0.371$

Along the curve

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-21 — Pair-production threshold) is the form associated with Pair production. Working symbols: $M$ $\rightarrow$ $E_{th}$ . In vacuum pair production is forbidden by four-momentum. A nucleus recoils and opens the channel.

Purpose

Purpose: compute

E_{th}

from

M

in Nuclear physics via

E_{th}=2m_ec^2\,(1+m_e/M)

E_th = 2 m_e c² (1 + m_e/M) = 1.022 (1 + m_e/M) MeV. Photon in the field of mass M. 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

M = 207.200\,\mathrm{u}

, the governing relation

E_{th}=2m_ec^2\,(1+m_e/M)

yields

E_{th} = 1.0220\,\mathrm{MeV}

. A photon, a nucleus, an e⁺e⁻ pair. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Recoil mass M207.2 u

Outputs

Threshold E_{th}1.0220 MeV

Reading speed

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Narration of this film

A photon, a nucleus, an e⁺e⁻ pair.

In vacuum pair production is forbidden by four-momentum. A nucleus recoils and opens the channel.

Reading speed

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