22.3.4 Spontaneous and Random Nature of Nuclear Decay

Overview

Radioactive decay has two fundamental characteristics that distinguish it from ordinary chemical or physical processes: it is spontaneous and it is random.

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Spontaneous Nature of Decay

Nuclear decay is said to be spontaneous because it occurs entirely on its own, without any external trigger or influence. The decay of a nucleus is an internal nuclear event — it depends only on the internal structure of the nucleus itself.

Key Implication

The rate of decay is not affected by external physical or chemical conditions such as:

• Temperature
• Pressure
• Chemical bonding or environment
• Electric or magnetic fields
• Physical state (solid, liquid, gas)

This is because nuclear forces operate at a scale ($\sim 10^{-15}$ m) far smaller than atomic or molecular interactions. Heating a sample, for example, only adds energy to the electron shells — it cannot penetrate the nucleus to alter its stability.

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Random Nature of Decay

Nuclear decay is also random: it is impossible to predict which specific nucleus in a sample will decay at any given moment, or exactly when a particular nucleus will decay.

Key Implication

For any single nucleus, we can only state a probability of decay per unit time — this probability is characterised by the decay constant $\lambda$.

$\text{Probability of decay per unit time}=\lambda$

The decay constant $\lambda$ is a fixed property of the isotope. It does not change as the sample ages, and it does not depend on how many nuclei have already decayed.

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Decay as a Statistical Process

Although individual decays are unpredictable, a large collection of nuclei follows a well-defined exponential decay law:

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

This equation is derived from the assumption that each nucleus has the same constant probability $\lambda$ of decaying per unit time, independent of all others. The result is the exponential decay equation:

$N=N_0{e}^{-\lambda t}$

This statistical predictability emerges from the law of large numbers — with millions of nuclei, the average behaviour is highly regular even though each individual decay is random.

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Fluctuations in Count Rate

Because decay is random, the count rate measured by a Geiger-Müller (GM) tube or similar detector will fluctuate from moment to moment, even for a source with a constant average activity.

• If you measure the count rate repeatedly over equal time intervals, you will get slightly different values each time.
• These fluctuations are not due to instrument error — they are a direct consequence of the random nature of decay.
• The fluctuations follow a Poisson distribution. For a mean count of $\bar{N}$, the standard deviation is $\sqrt{\bar{N}}$.

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Summary Table