Concise Notes on Radiation and Nuclear Decay

Nuclear Decay

Process by which an unstable nucleus transforms into a more stable nucleus.
Energetic particles or electromagnetic radiation are emitted.
Random process; probabilities can be determined but not exact timing.
When a nucleus decays the atomic number, Z may change, when this happens the atom transforms from one element to another.
All elements with Z > 82 (lead) are radioactive
Three main types of radiation: alpha, beta, and gamma.

Helium nuclei (two protons and two neutrons).
Positively charged (twice the electronic charge).
High kinetic energy (around 10% of the speed of light).
Strong ionizers, slowed rapidly by collisions.
When an atom emits alpha radiation:
- Mass number, A, decreases by 4.
- Atomic number, Z, decreases by 2.
Alpha decay equation: ^{A}{Z}X \rightarrow ^{A-4}{Z-2}Y + \alpha

Electrons (\beta^{-}, e^{-}$) or positrons (\beta^{+}, e^{+}$).
Positron is the antiparticle of the electron.
Can be negatively or positively charged.
High speed (around 90% the speed of light).
Low mass, lose energy rapidly during collisions.
Formed in nuclear reactions when a neutron transforms into a proton or vice versa.
In \beta^{-}$ decay, an antineutrino is produced; in \beta^{+}$ decay, a neutrino is produced.
When an atom emits beta radiation:
- Mass number stays the same.
- Atomic number increases or decreases by 1.
\beta^{-}$ decay equation: ^{A}{Z}X \rightarrow ^{A}{Z+1}Y + \beta^{-} + \text{antineutrino}
\beta^{+} decay equation: ^{A}{Z}X \rightarrow ^{A}{Z-1}Y + \beta^{+} + \text{neutrino}

Electromagnetic radiation (photons), no charge.
High end of the electromagnetic spectrum (high energy, high frequency).
When an atom emits gamma radiation:
- Atomic number and mass number do not change.
- The structure of the nucleus does not change, but its state of excitation does.
Gamma decay equation: ^{A}{Z}X^{*} \rightarrow ^{A}{Z}X + \gamma

Annihilation: When a particle and its antiparticle meet (e.g., an electron and a positron):
- They annihilate, and mass converts to energy (E = mc^2).
- Conservation laws are obeyed (charge, energy, momentum).
- Two photons of equal energy are created to conserve momentum (about 511 keV).

The number of atoms which decay in a given time depends on the number of atoms present at the beginning of that time:
- \frac{\Delta N}{\Delta t} \propto N
- \frac{\Delta N}{\Delta t} = -\lambda N (where \lambda is the decay constant).
Exponential decay: N(t) = N_0 e^{-\lambda t}
- N_0 is the number of radioactive atoms at the start.
- \lambda is the decay constant.
Nuclear activity: A = \lambda N
- A(t) = A_0 e^{-\lambda t}

Becquerel (Bq): 1 Bq = 1 disintegration per second (dps).
- 1 Bq = 2.7 \times 10^{-11} Ci
Curie (Ci): Based on the decay rate of 1 g of radium.
- 1 Ci = 3.7 \times 10^{10} dps

Time required for the amount of radionuclide to decrease to half the starting value.
The probability of decay is fixed, and the half-life is constant.
e^{-\lambda \tau_{1/2}} = \frac{1}{2}
\tau_{1/2} = \frac{\ln 2}{\lambda}
\lambda = \frac{0.693}{\tau_{1/2}}
N(t) = N0 \exp\left(-\frac{0.693t}{\tau{1/2}}\right)
A(t) = A0 \exp\left(-\frac{0.693t}{\tau{1/2}}\right)

Cosmic rays create neutrons, which interact with Nitrogen to create radioactive carbon-14 (^{14}C).
$^{14}C is radioactive with a half-life of 5730 years.
An equilibrium of 1 part $^{14}C to 10^{12} parts $^{12}C exists in living things.

Bremsstrahlung ("Braking radiation")
- Electrons decelerate and emit energy as photons.
- Maximum photon energy equals the kinetic energy gained by electrons.
Characteristic X-rays
- Generated via inner-shell electron transitions.
- Use Z_{eff} = (Z - 1) for energy calculations.
Crookes' Tube
- Ions accelerate towards cathode, liberate electrons that generate X-rays upon hitting the glass face.
- Tungsten is the best target material.
Coolidge or Thermionic Tubes
- Heated filament emits electrons.
- Electric field accelerates electrons, which strike the anode to generate X-rays.
- Heat is generated in the target; large heat capacity required.
- Independent control of current (beam intensity) and voltage (penetration ability).