Radioactivity and the decay law
Nuclei that break up by themselves
Your NCERT book describes the three kinds of radioactive decay. The law of radioactive decay, in the next video, goes beyond the current syllabus, but it is useful, for example for competitive exams.
In 1896, Henri Becquerel wrapped a uranium compound in black paper and placed it near a photographic plate, with a piece of silver in between. Hours later the plate was blackened.
Something from the uranium had passed through both the paper and the silver. Nobody had shone anything on it. The uranium was giving out radiation on its own.
This is radioactivity. An unstable nucleus breaks up by itself, giving out radiation and changing into a different nucleus. We call it radioactive decay.
Three kinds of decay occur in nature. In alpha decay, the nucleus throws out an alpha particle: a helium nucleus, ⁴₂He.
In beta decay, it gives out an electron, or a positron: a particle with the mass of an electron but a positive charge.
In gamma decay, it gives out a gamma ray: a photon of very high energy, hundreds of keV or more, with a wavelength even shorter than X-rays.
Why are some nuclei unstable? Light stable nuclei have about as many neutrons as protons. Heavy ones need more neutrons, about 3 for every 2 protons, to dilute the proton repulsion.
A nucleus with too many neutrons, or too many protons, or simply too heavy, sooner or later decays towards stability. Only about 10 per cent of known isotopes are stable.
Decay is purely a matter of chance. We can never say when one particular nucleus will decay. But with billions of nuclei, the chances add up to a very exact law.
The law of radioactive decay
Take a sample of N nuclei. In a short time dt, the number that decay is proportional to N, and to dt. Twice the nuclei, twice the decays.
So dN equals minus λ N dt. The constant λ is the decay constant of that nuclide. The minus sign says N goes down.
Integrate from N₀ at t equals 0. ln N over N₀ equals minus λ t, which gives the law of radioactive decay: N equals N₀ e to the power minus λ t.
The number of decays per second is the activity, R. R equals minus dN by dt, which is λ N. So the activity also falls as R₀ e to the power minus λ t.
Activity is measured in becquerel: 1 Bq is one decay per second. An older unit is the curie: 1 Ci equals 3.7 × 10¹⁰ Bq, roughly the activity of one gram of radium.
The half-life, T½, is the time for half the nuclei to decay. Put N equals N₀ by 2: e to the minus λ T½ is one-half, so T½ equals ln 2 over λ, that is 0.693 over λ.
After one half-life, half are left. After two, a quarter. After three, an eighth. The curve never quite reaches zero.
The mean life τ is the average lifetime of a nucleus. It works out to τ equals 1 over λ. So T½ equals τ ln 2: the mean life is about 1.44 half-lives.
Example: tritium has a half-life of 12.5 years. How much is left after 50 years? 50 years is 4 half-lives, so one-half to the power 4: one-sixteenth, or 6.25 per cent.
Another: uranium-238 has a half-life of 4.5 × 10⁹ years, which is 1.42 × 10¹⁷ seconds. So λ is 0.693 divided by that: 4.88 × 10⁻¹⁸ per second.
One gram holds 6.023 × 10²³ over 238, about 2.53 × 10²¹ nuclei. R equals λ N: about 1.24 × 10⁴ becquerel. Twelve thousand decays every second, from a single gram.
Alpha, beta and gamma decay
In every decay, the mass number A and the charge are conserved. Use that to write the equations.
Alpha decay. Uranium-238 throws out an alpha particle, ⁴₂He. A drops by 4, Z drops by 2: ²³⁸₉₂U becomes ²³⁴₉₀Th, thorium, plus ⁴₂He.
In general, ᴬ_Z X goes to A minus 4, Z minus 2, Y, plus helium-4. Radium-226 decays the same way into radon-222.
The energy released is the Q value: the mass of X minus the masses of Y and helium, times c squared. For uranium-238: 238.05079 minus 234.04363 minus 4.00260 leaves 0.00456 u. Times 931.5: 4.25 MeV.
Beta-minus decay. Inside the nucleus, a neutron turns into a proton, an electron and an antineutrino. A stays the same; Z goes up by 1.
For example, ³²₁₅P becomes ³²₁₆S, sulphur, plus an electron and an antineutrino. Its half-life is 14.3 days.
Beta-plus decay is the reverse. A proton inside the nucleus turns into a neutron, a positron and a neutrino. Z goes down by 1: ²²₁₁Na becomes ²²₁₀Ne, with a half-life of 2.6 years.
The neutrino and antineutrino are neutral and almost massless. They share the energy with the electron or positron, so beta particles come out with a whole range of energies.
Gamma decay. After an alpha or beta decay, the new nucleus is often left in an excited state. Like an excited atom, it drops to a lower level by giving out a photon, but now of MeV energy.
Cobalt-60 beta-decays to an excited nickel-60 nucleus, which then gives out two gamma rays, of 1.17 and 1.33 MeV. In gamma decay, neither A nor Z changes.
Try it: watch a sample decay
- Each dot is a nucleus. Every instant, each one has the same small chance of decaying; it turns white, then dark.
- Compare the orange simulated curve with the blue law N₀e^(−λt). At each T½ line, about half of what was left has gone.
- Try N₀ = 100 and N₀ = 900. With more nuclei the simulation follows the law more closely.
- Change the half-life: λ, τ and the activity readout change, but the shape of the curve, plotted against T½, does not.
Tritium after 50 years
The half-life of tritium is 12.5 years. What fraction of a sample of pure tritium remains undecayed after 50 years?
- 1/4
- 1/8
- 1/16
- None of it
Answer
C. 50 years is 50/12.5 = 4 half-lives. Each halves what is left: N/N₀ = (½)⁴ = 1/16 (6.25 %). 1/4 confuses the number of half-lives with the fraction. The sample never completely disappears: the decay is exponential.
Alpha decay of uranium
²³⁸₉₂U undergoes alpha decay. What is the daughter nucleus?
- ²³⁴₉₂U
- ²³⁴₉₀Th
- ²³⁸₉₀Th
- ²³⁸₉₃Np
Answer
B. An alpha particle is ⁴₂He. A and Z are conserved: A = 238 − 4 = 234, Z = 92 − 2 = 90, which is thorium: ²³⁴₉₀Th. Keeping Z = 92 forgets the two protons the alpha carries away. ²³⁸₉₃Np would come from β⁻ decay (Z up by 1, A unchanged).
Mean life
A radioactive nuclide has a half-life of 10 minutes. What is its mean life?
- 6.93 min
- 10 min
- 14.4 min
- 20 min
Answer
C. λ = 0.693/T½ = 0.693/10 = 0.0693 per minute, and τ = 1/λ = 1/0.0693 = 14.4 min. Equivalently τ = T½/ln 2 = 1.44 T½. 6.93 min comes from multiplying by 0.693 instead of dividing.
Summary
- Decay law:
dN/dt = −λN, N = N₀e^(−λt) - Activity:
R = λN = R₀e^(−λt); 1 Bq = 1 decay/s, 1 Ci = 3.7 × 10¹⁰ Bq - Half-life and mean life:
T½ = ln 2/λ = 0.693/λ, τ = 1/λ, T½ = τ ln 2 - Decays:
α: A − 4, Z − 2; β⁻: Z + 1; β⁺: Z − 1; γ: no change
- Radioactivity (Becquerel, 1896): unstable nuclei decay by themselves, giving out α particles (⁴He nuclei), β particles (electrons or positrons) or γ rays (high-energy photons).
- Each nucleus decays at random, but a large sample follows N = N₀e^(−λt) exactly. After n half-lives, (½)ⁿ is left.
- In β⁻ decay n → p + e⁻ + ν̄; in β⁺ decay p → n + e⁺ + ν inside the nucleus. The (anti)neutrino shares the energy.
- γ rays come from a nucleus left in an excited state after α or β decay, e.g. ⁶⁰Ni after ⁶⁰Co decays (1.17 and 1.33 MeV).