Physics · Ch 13 — Nuclei

The Radioactive Decay Law

13.5

The Radioactive Decay Law

Radioactive decay is a purely random, statistical process: it is impossible to say exactly WHEN any one particular unstable nucleus will decay, but every nucleus of a given radioactive species has exactly the SAME fixed probability of decaying in any given small interval of time, regardless of how long it has already existed or of what is happening to the other nuclei around it. This physical fact leads directly to the RADIOACTIVE DECAY LAW: at any instant, the number of nuclei decaying per unit time (i.e. the rate at which the undecayed population shrinks) is directly proportional to the number of undecayed nuclei $N$ still present at that instant,

\frac{dN}{dt} = -\lambda N

where the constant of proportionality $\lambda$ , called the DECAY CONSTANT (unit: per second, or more generally per unit time), is a fixed property of the particular radioactive species, giving the fraction of the remaining nuclei that decays per unit time; the negative sign shows that $N$ decreases with time. Separating the variables and integrating,

\int_{N_0}^{N}\frac{dN}{N} = -\lambda\int_0^t dt \quad\Longrightarrow\quad \ln\!\left(\frac{N}{N_0}\right) = -\lambda t

gives the exponential form of the decay law,

N(t) = N_0\,e^{-\lambda t}

where $N_0$ is the number of undecayed nuclei present at $t=0$ . This single formula says that a radioactive sample never fully vanishes in a finite time -- it decays away exponentially, approaching (but mathematically never quite reaching) zero.

Two related time-quantities are used to describe how fast a given species decays. The HALF-LIFE $T_{1/2}$ is the time taken for exactly half of any sample of that species to decay, i.e. the time at which $N=N_0/2$ ; substituting into the decay law gives

T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}

The MEAN LIFE (or average life) $\tau$ is the average time for which a nucleus of the species survives before decaying, and works out to be simply the reciprocal of the decay constant, $\tau = 1/\lambda$ (derived in Exercise 5), giving the useful relation $T_{1/2} = 0.693\,\tau$ , so the mean life is always somewhat LONGER than the half-life. …

Figure 1Exponential radioactive decay curve

What this figure shows. A single graph with the number of undecayed nuclei $N$ on the vertical axis (starting from $N_0$ at the top) and time $t$ on the horizontal axis, starting at the origin. The curve starts at height $N_0$ when $t=0$ and falls away smoothly and continuously in the classic decaying-exponential shape: it drops steeply at first, then flattens out more and more as $t$ increases, getting closer and closer to the horizontal axis but never actually touching it, even far to the right of the graph. Three evenly spaced dashed horizontal lines are drawn at heights $N_0/2$ , $N_0/4$ and $N_0/8$ , each meeting the decay curve at one point; from each of these three points, a dashed vertical line drops down to the time axis, landing at $t=T_{1/2}$ , $t=2T_{1/2}$ and $t=3T_{1/2}$ respectively -- visually demonstrating that the SAME fixed interval of time, one half-life $T_{1/2}$ , always halves whatever population remains, however much or little is left. A sm …

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