NUC-01

Exponential decay

N = N0 e^{−λt}, A = λ N. Half-life T½ = ln2 / λ.

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DecayRutherford

Governing equation

N=N_0 e^{-\lambda t},\quad T_{1/2}=\ln 2/\lambda

$t \in \left[0, 10000\right]$

where

N_0: Initial nuclei (—)
\lambda: Decay constant (1/s)
t: Time (s)
N: Remaining (—)
A: Activity (Bq)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

N=N_0 e^{-\lambda t},\quad T_{1/2}=\ln 2/\lambda

$t \in \left[0, 10000\right]$

$t = 0$ $N = 1\times 10^{6}$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0,\quad t_{\max} = 10000

$t \in \left[0, 10000\right]$

$t = 0$ $t = 0$

Along the curve

③

Response curve — green

N=N_0 e^{-\lambda t},\quad T_{1/2}=\ln 2/\lambda

$t \in \left[0, 10000\right]$

$t = 0$ $N = 1\times 10^{6}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}N}{\mathrm{d}t}

$t \in \left[0, 10000\right]$

$t = 62.89$ $N^{\prime} = -5690.1$

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-01 — Exponential decay) is the form associated with Rutherford. Working symbols: $N_0$ , $\lambda$ , $t$ $\rightarrow$ $N$ , $A$ . Each nucleus is memoryless. λ is a property of the species, not the sample size.

Purpose

Purpose: compute

N

A

from

N_0

\lambda

t

in Nuclear physics via

N=N_0 e^{-\lambda t},\quad T_{1/2}=\ln 2/\lambda

N = N0 e^{−λt}, A = λ N. Half-life T½ = ln2 / λ. 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

N_0 = 1.000e+6\,\mathrm{—}

\lambda = 0.010\,\mathrm{1/s}

t = 60.000\,\mathrm{s}

, the governing relation

N=N_0 e^{-\lambda t},\quad T_{1/2}=\ln 2/\lambda

yields

N = 548811.64\,\mathrm{—}

A = 5488.12\,\mathrm{Bq}

. A decaying staircase of counts. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Initial nuclei N_01000000 —Decay constant \lambda0.01 1/sTime t60 s

Outputs

Remaining N548811.64 —
Activity A5488.12 Bq

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Narration of this film

A decaying staircase of counts.

Each nucleus is memoryless. λ is a property of the species, not the sample size.

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