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For reasons that may very well be nothing more than coincidences, accidental discoveries often further trigger more accidental discoveries.Or it simply is that most significant discoveries are accidental. In the year 1895, while Wilhelm Roentgen, a German physicist, experimented with his cathode ray tube privately in his lab, he observed that every time he turned on his apparatus, a flurosecent material glowed in the other corner of his lab. Fluroscent materials glow only when exposed to light. But the lab was dark. There must be some form of invisible radiation that his cathode ray tube was emitting and causing the material to glow. This was the accidental discovery of x-rays.

Widespread fascination followed this discovery across the globe. Rays invisible to the eyes, yet they allow one to see through a human body as if it were transparent. The curiosity of Henri Becquerel, a French physicist was intrigued by this discovery.

So far we’ve understood the rationale behind alpha decay, the resulting products and energies. But we never addressed the question of how quickly does the decay happen. If you’ve been assuming all the while that the decay is instantaneous, that is, as soon as there is an unstable nucleus, it decays into a stabler one, we’d never have discovered elements such as Uranium or Radium, for they’d have decayed long ago as soon as they formed.

Lord Rutherford discovered the law governing the rate of radioactive decay in 1900. Firstly it was observed that in a sample containing a large number of radioactive atoms of a single element, all of them do not decay at the same time. Instead, the atoms decay spontaneously (in a random fashion). Which means we can never be sure whether or not a specific atom will decay in the next second but we can speak in terms of the probability of that atom decaying in that second.

If $\mathrm{d}N$ is the number of atoms that decay in a time $\mathrm{d}t$, the rate of decay will be $\dfrac{\mathrm{d}N}{\mathrm{d}t}$. If decay happens such that one atom disintegrates every one second, we measure the decay rate in the unit $1\,\mathrm{becquerel}=1\,\mathrm{decay}/s$. We rarely have such low decay rates. Generally, more than a million decays happen every second. So a more convenient unit called the curie is used, $1\,\mathrm{curie}=3.7\times 10^{10}\,\mathrm{decays}/s$.

If we start out with $N$ undecayed nuclei of which $\mathrm{d}N$ nuclie decay in every interval of $\mathrm{d}t$, and $\dfrac{\mathrm{d}N}{\mathrm{d}t}$ is the rate of decay, we can be sure that this rate will decrease as time passes by, for the number of undecayed atoms decreases too. Note carefully that this decrease in the rate of decay is not because the sample itself modified in some way and the probability of decay of each atom reduced – the atoms have not begun delaying the decay longer. The probability of decay for each atom remains the same but the number that decay every second reduces due to the inevitable decrease in the number of undecayed atoms.

Think in terms of death rate of people across the globe. About a 100 people die every minute somehwere in the world. But if you were to count the number of people that die in your country alone and you find out that about 7 people die every minute, this reduced death rate doesn’t imply people in your country live longer but that there are lesser people in the country overall (compared with the global population) and hence fewer appear to die every minute.

As the atoms in the sample decay, the number of undecayed atoms ($N$) reduces and hence the number that may decay every second ($\mathrm{d}N$), of this reduced number of undecayed atoms, reduces too. In short, fewer total number of atoms means fewer atoms that decay every second. Mathematically, we may write this as,

${\mathrm{d}N\over\mathrm{d}t}\propto N$

If we represent the probability of each atom decaying per second by $\lambda$ (which we assume to be a constant), the probability of decay in a time $\mathrm{d}t$ will be $\lambda \mathrm{d}t$. Clearly the rate of decay does depend on this probability too. If each atom had a high probability of decaying per second, you would count a larger number of atoms decaying per second. Hence,

${\mathrm{d}N\over\mathrm{d}t}=- N\lambda$

Or,

$\mathrm{d}N=-N\lambda\mathrm{d}t$

The negative sign here indicates that as the time $\mathrm{d}t$ progresses, the number of atoms that decay every second $\mathrm{d}N$ reduces. Now rearranging the above equation and integrating, assuming the number of (undecayed) atoms has reduced from $N_0$ to some $N$ in a time from $t=0$ to some $t$,

\begin{align*}

\int_{N_0}^N {\mathrm{d}N\over N}&=-\int_0^t\lambda\mathrm{d}t\\

\left[\ln{N}\right]_{N_0}^N&=-\left[\lambda t\right]_0^t\\

\ln{N}-\ln{N_0}&=-\lambda t\\

\ln{N\over N_0}&=-\lambda t\\

{N\over N_0}&=e^{-\lambda t}\\

N&=N_0e^{-\lambda t}

\end{align*}

This equation tells us the number of undecayed atoms at any instant of time $t$. For example, at $t=0$, we get $N=N_0$. Which is the definition of $N_0$ – the initial number of undecayed atoms. It is especially interesting to note that the number of undecayed atoms becomes zero only if $t\rightarrow \infty$. In other words, in any given sample, it will take a very very long time (perhaps even eternity) for all the atoms to decay.

To understand this last point better, let us think in terms of how long does it take for the number of undecayed atoms to reduce to half the initial number. In other words, what is $t=t_{1/2}$, for which $N=\dfrac{N_0}{2}$? Making these substitutions in the decay law,

\begin{align*}

{N_0\over 2}&=N_0e^{-\lambda t_{1/2}}\\

e^{-\lambda t_{1/2}}&={1\over 2}\\

-\lambda t_{1/2}&=\ln{2^{-1}}\\

t_{1/2}&={1\over \lambda}\ln{2}\\

t_{1/2}&={0.693\over \lambda}

\end{align*}

This is also called the half-life of the sample. As you can see, it depends only on the probability of decay per unit time, $\lambda$ (also known as the decay constant), which depends on the nature of the atoms in the sample. Therefore a different sample of the same element, differing only in the number of atoms, will have the same half-life.

The shortest half-life is that of Polonium-212 and is $0.3\mu s$. One normal blink of the eye is about 300,000 times longer than this. And the longest half-life is of Thorium-232, which is $1.3\times 10^{10}$ years, very close to the present age of the universe.

The half-life $t_{1/2}$ is therefore a finite quantity that can be computed according to the above equation. It tells us the time it takes for the atoms in a sample to reduce in number by $50\%$. This however certainly doesn’t mean that a $100\%$ of the atoms would have decayed in a time twice the half-life, $2\,t_{1/2}$. Once $50\%$ of the atoms have decayed, it is the $50\%$ of this remaining $50\%$ that will decay in the next interval of time $t_{1/2}$. This means only $25\%$ of the initial number $N_0$ will decay in the second interval. Adding up to a total of $75\%$ of $N_0$. Now, $25\%$ of the atoms still remian undecayed. Of this $25\%$, a $50\%$ of the atoms will decay in the third interval $t_{1/2}$. That is only $12.5\%$ of $N_0$ would decay in this interval, at the end of which a total of $87.5\%$ of the initial number $N_0$ would have decayed. Continuing this way, it may seem at first that you can arrive at a $100\%$ decay very soon. But as you can check by computing yourself for a few more intervals, achieving $100\%$ decay seems impossible. In principle, the decay process is eternal. We may very well reach $99.99\%$ decay in a reasonable time, but as time progresses, all you can do is tack on more decimal digits to obtain say $99.9997\%$ decay, but never a $100\%$

And this is the reason we still have elements such as uranium, thorium, radium and so on, today. If radioactive decay followed a different law that allowed all the atoms to decay in a short time, we might never have discovered them and our perioidc table would be short of 38 elements. Perhaps then the Curies wouldn’t have won the Nobel prize at all, the nuclear bomb would never have surfaced and we would be much farther in the race to unravel the grand unified theory, for the weak force would have remained shrouded. It is the weak force that is responsible for all radioactive decays.

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