E.3.3 Quantitative analysis of decay (HL only) Notes

Radioactive Decay Law and Related Concepts

1. While holding a sample of radioactive material in your hand, you know it’s unstable at some point, its nuclei will decay, releasing particles or energy.
2. But how can you predict how much of it will remain after a certain time? How can the rate of decay be quantified, and how does this relate to the material's half-life?

These questions are at the heart of the radioactive decay law, a cornerstone of nuclear physics.

Evidence for the Strong Nuclear Force

1. While holding a sample of radioactive material, you might wonder what holds its nucleus together despite the repulsive forces between protons.
2. The answer lies in the strong nuclear force, which is responsible for binding protons and neutrons (nucleons) within the nucleus.
3. Unlike the electrostatic force, which decreases with distance according to an inverse-square law, the strong force is short-ranged and attractive, acting only at distances of about $10^{-15}$ m (1 femtometer).

The Role of the Neutron-to-Proton Ratio in Nuclear Stability

1. Not all combinations of protons and neutrons form stable nuclei.
2. The stability of a nuclide depends on the neutron-to-proton (N/Z) ratio.
3. Light elements (such as carbon and oxygen) tend to have nearly equal numbers of protons and neutrons ($N \approx Z$), while heavier elements require more neutrons than protons to remain stable.
4. This increasing neutron excess is necessary to counteract the growing electrostatic repulsion between protons.

Approximate Constancy of the Binding Energy Curve for $A > 60$

1. As discussed in the earlier sections, the binding energy per nucleon determines the stability of a nucleus.
2. For small nuclei, adding more nucleons significantly increases the binding energy per nucleon, making them more stable.
3. However, beyond a nucleon number of around 60 (approximately iron-56), the binding energy per nucleon remains nearly constant, meaning additional nucleons do not significantly increase nuclear stability.

This behavior is explained by the balance between the strong nuclear force (which is short-ranged) and the electrostatic repulsion (which acts over longer distances).

As the nucleus gets larger, nucleons interact mostly with their immediate neighbors rather than with all nucleons in the nucleus, leading to the plateau in the binding energy curve.

The Radioactive Decay Law

Radioactive decay is a random and spontaneous process.

1. However, for a large number of nuclei, the decay behavior can be described statistically.
2. The radioactive decay law is expressed mathematically as: $$\frac{\mathrm{d}N}{\mathrm{d}t} = -\lambda N$$ where:
   1. $N$ is the number of undecayed nuclei at time $t$,
   2. $\lambda$ is the decay constant, representing the probability of decay per unit time,
   3. $\frac{\mathrm{d}N}{\mathrm{d}t}$ is the rate of decay (negative because $N$ decreases over time).
3. Solving this differential equation yields the exponential decay formula: $$N = N_0 e^{-\lambda t}$$ where:
   1. $N_0$: The initial number of nuclei at $t = 0$,
   2. $t$: The elapsed time.

Decay Constant $\lambda$

Key Relationship: Decay Constant and Half-Life

The decay constant and half-life are related by:

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

where $\ln 2 \approx 0.693$.

Activity $A$

1. It is directly proportional to the number of undecayed nuclei: $$A = \lambda N$$ where:
   1. $A$ is the activity (measured in becquerels, where $1 \, \mathrm{Bq} = 1 \, \mathrm{decay/second}$),
   2. $\lambda$ is the decay constant,
   3. $N$ is the number of undecayed nuclei.
2. Since $N = N_0 e^{-\lambda t}$, activity also decreases exponentially: $$A = A_0 e^{-\lambda t}$$ Here, $A_0 = \lambda N_0$ is the initial activity.

Evidence for Discrete Nuclear Energy Levels from Alpha and Gamma Spectra

1. Unlike classical physics predictions, nuclear transitions do not produce a continuous spectrum of energy.
2. Instead, alpha and gamma radiation are observed to be emitted in specific, quantized energy levels, similar to the discrete spectral lines seen in atomic electron transitions.

The precise energy of emitted gamma photons provides direct evidence that nuclei have discrete energy levels, just as electrons do in atoms.

Half-Life $T_{1/2}$

1. The half-life is a practical way to quantify how quickly a radioactive substance decays.
2. After one half-life, half of the original nuclei remain undecayed.
3. After two half-lives, one-quarter remains, and so on.

Exponential Decay and Half-Life

1. Using the decay law $N = N_0 e^{-\lambda t}$, you can derive the concept of half-life.
2. At $t = T_{1/2}$, the number of nuclei is halved:$$\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$$
3. Dividing through by $N_0$ and taking the natural logarithm: $$\ln \frac{1}{2} = -\lambda T_{1/2}$$$$T_{1/2} = \frac{\ln 2}{\lambda}$$

Continuous Beta Decay Spectrum as Evidence for the Neutrino

1. When beta decay was first studied, scientists expected the emitted electrons (beta particles) to have a single energy value, corresponding to the difference in energy between the initial and final nuclear states.
2. However, experiments revealed a continuous spectrum of beta particle energies instead of a single sharp value.
3. This posed a major problem: if only the emitted electron carried energy away, energy conservation appeared to be violated.
4. To resolve this, physicist Wolfgang Pauli proposed the existence of an unseen, neutral particle, the neutrino, that carried away the missing energy.

Later experiments confirmed the existence of neutrinos, supporting the principle of energy conservation in nuclear reactions.