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NUC-10

Half-life

T_{1/2} = ln 2 / λ. Time for half the nuclei to decay.

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DecayHalf-life

Governing equation

T1/2=ln⁡2/λT_{1/2}=\ln 2/\lambda

λ∈[1×10−8,10]\lambda \in \left[1\times 10^{-8}, 10\right]

where

\lambda
Decay constant (1/s)
T_{1/2}
Half-life (s)

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①

Governing law — red

T1/2=ln⁡2/λT_{1/2}=\ln 2/\lambda

λ∈[1×10−8,10]\lambda \in \left[1\times 10^{-8}, 10\right]

λ=1×10−8\lambda = 1\times 10^{-8}T1/2=6.93×107T_{1/2} = 6.93\times 10^{7}

②

Domain of the variable — blue

λmin⁡=1×10−8,λmax⁡=10\lambda_{\min} = 1\times 10^{-8},\quad \lambda_{\max} = 10

λ∈[1×10−8,10]\lambda \in \left[1\times 10^{-8}, 10\right]

λ=1×10−8\lambda = 1\times 10^{-8}λ=1×10−8\lambda = 1\times 10^{-8}

③

Response curve — green

T1/2=ln⁡2/λT_{1/2}=\ln 2/\lambda

λ∈[1×10−8,10]\lambda \in \left[1\times 10^{-8}, 10\right]

λ=1×10−8\lambda = 1\times 10^{-8}T1/2=6.93×107T_{1/2} = 6.93\times 10^{7}

④

Rate of change — amber

dT1/2dλ\frac{\mathrm{d}T_{1/2}}{\mathrm{d}\lambda}

λ∈[1×10−8,10]\lambda \in \left[1\times 10^{-8}, 10\right]

λ=0.063\lambda = 0.063T1/2′=−5.51×108T_{1/2}^{\prime} = -5.51\times 10^{8}

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-10 — Half-life) is the form associated with Half-life. Working symbols: λ\lambda →\rightarrow T1/2T_{1/2}. Independent of N. Related to mean life by T_{1/2} = τ ln 2.

Purpose

Purpose: compute T1/2T_{1/2} from λ\lambda in Nuclear physics via T1/2=ln⁡2/λT_{1/2}=\ln 2/\lambda T_{1/2} = ln 2 / λ. Time for half the nuclei to decay. 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 λ=0.010 1/s\lambda = 0.010\,\mathrm{1/s}, the governing relation T1/2=ln⁡2/λT_{1/2}=\ln 2/\lambda yields T1/2=69.315 sT_{1/2} = 69.315\,\mathrm{s}. A staircase of remaining nuclei, a half tick. Move a slider: the numbers are this situation, not a canned story.

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  • Half-life T_{1/2}69.315 s
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Narration of this film

A staircase of remaining nuclei, a half tick.

Independent of N. Related to mean life by T_{1/2} = τ ln 2.

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