NUC-11

Mean life

τ = 1/λ = T_{1/2} / ln 2. The expected lifetime of one nucleus.

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DecayMean life

Governing equation

\tau=1/\lambda=T_{1/2}/\ln 2

$T_{1/2} \in \left[0.001, 1\times 10^{8}\right]$

where

T_{1/2}: Half-life (s)
\tau: Mean life (s)

The figure, part by part

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①

Governing law — red

\tau=1/\lambda=T_{1/2}/\ln 2

$T_{1/2} \in \left[0.001, 1\times 10^{8}\right]$

$T_{1/2} = 0.001$ $\tau = 0.001$

Along the curve

②

Domain of the variable — blue

T_{1/2}_{\min} = 0.001,\quad T_{1/2}_{\max} = 1\times 10^{8}

$T_{1/2} \in \left[0.001, 1\times 10^{8}\right]$

$T_{1/2} = 0.001$ $T_{1/2} = 0.001$

Along the curve

③

Response curve — green

\tau=1/\lambda=T_{1/2}/\ln 2

$T_{1/2} \in \left[0.001, 1\times 10^{8}\right]$

$T_{1/2} = 0.001$ $\tau = 0.001$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\tau}{\mathrm{d}T_{1/2}}

$T_{1/2} \in \left[0.001, 1\times 10^{8}\right]$

$T_{1/2} = 6.29\times 10^{5}$ $\tau^{\prime} = 1.443$

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-11 — Mean life) is the form associated with Mean life. Working symbols: $T_{1/2}$ $\rightarrow$ $\tau$ . ⟨t⟩ of the exponential. Particle-physics widths quote ħ/τ.

Purpose

Purpose: compute

\tau

from

T_{1/2}

in Nuclear physics via

\tau=1/\lambda=T_{1/2}/\ln 2

τ = 1/λ = T_{1/2} / ln 2. The expected lifetime of one nucleus. 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

T_{1/2} = 600.000\,\mathrm{s}

, the governing relation

\tau=1/\lambda=T_{1/2}/\ln 2

yields

\tau = 865.617\,\mathrm{s}

. An exponential, a 1/e mark. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Half-life T_{1/2}600 s

Outputs

Mean life \tau865.617 s

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Narration of this film

An exponential, a 1/e mark.

⟨t⟩ of the exponential. Particle-physics widths quote ħ/τ.

Reading speed

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