Radioactive decay: types, decay laws, and
half-life
Dr. Ahmad Redaa
2025-09-08
Radioactive Decay
- Definition of Radioactive Decay
- How radioactive decay is fundamental for radiometric dating?
- What is make radioavtive decay a usufel chronometer tool?
Types of Radioactive Decay
- Alpha Decay
- Beta Decay
- Gamma Decay
- Electron Capture
- Spontaneous Fission
Alpha Decay
Definition:
- Alpha decay occurs when an unstable nucleus emits an
alpha particle, composed of 2 protons and 2 neutrons, forming a new
element.
Key Characteristics:
- Alpha Particle: 2 protons and 2 neutrons (Helium-4
nucleus).
- Atomic Number: Decreases by 2 (e.g., \(^{238}_{92}\text{U}\) → \(^{234}_{90}\text{Th}\)).
- Mass Number: Decreases by 4.
Example:
- \(^{238}_{92}\text{U}\) →
\(^{234}_{90}\text{Th}\) + \(^{4}_{2}\text{He}\)
Beta Decay
Definition:
- Beta decay is a process in which a neutron in an
unstable nucleus is transformed into a proton, or a proton is
transformed into a neutron, accompanied by the emission of a beta
particle (electron or positron).
Key Characteristics:
- Beta-minus Decay: Neutron → Proton + Electron +
Antineutrino.
- Beta-plus Decay (Positron Emission): Proton → Neutron
+ Positron + Neutrino.
Beta-minus Decay (β⁻ Decay)
Process:
- A neutron in the nucleus is converted into a
proton.
- This results in the emission of an electron (beta
particle) and an antineutrino.
Effect on Atomic Number:
- The atomic number increases by 1.
- The mass number remains unchanged.
Example:
- Potassium-40 (K-40) → Calcium-40 (Ca-40) + Beta Particle
(e-)
\[
^{40}_{19}\text{K} \rightarrow ^{40}_{20}\text{Ca} + \text{e}^{-} +
\bar{\nu}_e
\]
Beta-plus Decay (β⁺ Decay) or Positron Emission
Process:
- A proton in the nucleus is converted into a
neutron.
- This results in the emission of a positron (positive
beta particle) and a neutrino.
Effect on Atomic Number:
- The atomic number decreases by 1.
- The mass number remains unchanged.
Example:
- Potassium-40 (K-40) → Argon-40 (Ar-40) + Positron
(e+)
\[
^{40}_{19}\text{K} \rightarrow ^{40}_{18}\text{Ar} + \text{e}^{+} +
\nu_e
\]
Gamma Decay
Process:
- Gamma decay occurs when an unstable nucleus releases
excess energy by emitting a gamma photon (γ).
- This process follows other types of decay, like alpha or beta decay,
when the nucleus is left in an excited state.
Effect on Nucleus:
- The atomic number and mass number of
the nucleus remain unchanged.
- The nucleus moves from a higher energy state to a lower, more stable
energy state.
Example:
- Cobalt-60 (Co-60) → Cobalt-60 (Co-60) + Gamma Photon
(γ)
- Cobalt-60 undergoes gamma decay to release energy after beta decay,
transitioning from an excited state to a more stable state.
\[
^{60}_{27}\text{Co}^* \rightarrow ^{60}_{27}\text{Co} + \gamma
\]
Note:
- Gamma rays are highly penetrating and require dense materials, like
lead, for shielding. They are a common form of high-energy radiation in
both natural and artificial radioactive processes.
Electron Capture
Process:
- Electron capture occurs when an
electron from the innermost orbital of an atom is
captured by the nucleus.
- This electron combines with a proton to form a
neutron and a neutrino.
Effect on Nucleus:
- The atomic number decreases by 1, while the
mass number remains unchanged.
- The nucleus transforms into a different element, but the overall mass
of the nucleus remains the same.
Example:
- Potassium-40 (K-40) + Electron → Argon-40 (Ar-40) + Neutrino
(νₑ)
- In this process, Potassium-40 captures an electron and becomes
Argon-40, emitting a neutrino.
\[
^{40}_{19}\text{K} + e^{-} \rightarrow ^{40}_{18}\text{Ar} + \nu_e
\]
Note:
- Electron capture often occurs in electron-rich environments and is an
alternative decay mode to beta-plus decay.
- The emitted neutrino carries away the excess energy and momentum from
the reaction.
Branching Decay
- Some isotopes can decay via two or more pathways.
- Each pathway has its own branching ratio (probability).
- The sum of branching probabilities = 100%.
Example
- Potassium-40 ( \(^{40}_{19}K\) ) is
unstable and decays through two main modes:
1. Beta-minus decay
(89%)
\[
^{40}_{19}K \;\;\rightarrow\;\; ^{40}_{20}Ca + e^- + \bar{\nu}_e
\]
- Produces stable Calcium-40
- Increases atomic number by 1
2. Electron capture
(11%)
(with minor positron emission)
\[
^{40}_{19}K + e^- \;\;\rightarrow\;\; ^{40}_{18}Ar + \nu_e
\]
- Produces stable Argon-40
- Decreases atomic number by 1
Spontaneous Fission
Spontaneous fission is a type of radioactive decay
where an atomic nucleus splits into two or more smaller nuclei, along
with the release of energy and neutrons, without the influence of an
external cause
Process:
- Unprompted Division:
- An unstable heavy nucleus spontaneously splits into two or more
smaller nuclei.
- Occurs without external influence, driven by the nucleus’s intrinsic
instability.
- Energy Release:
- Significant amount of energy is released as kinetic energy of the
fission fragments and radiation.
- Energy is derived from the conversion of mass into energy (\(E = mc^2\)).
General Nuclear Reaction:
\[
^{A}_{Z}\text{X} \rightarrow ^{A_1}_{Z_1}\text{Y} + ^{A_2}_{Z_2}\text{Z}
+ n_1 + n_2 + \ldots
\]
where:
- \(^{A}_{Z}\text{X}\) is the parent
nucleus.
- \(^{A_1}_{Z_1}\text{Y}\) and \(^{A_2}_{Z_2}\text{Z}\) are the fission
fragments.
- \(n_1, n_2, \ldots\) are the neutrons
released.
Characteristics:
- Less Common:
- Compared to alpha or beta decay, spontaneous fission is rarer due to
the high energy barrier.
- Heavy Elements:
- Common in very heavy elements like Uranium-238 (\(^{238}\text{U}\)) and Plutonium-240 (\(^{240}\text{Pu}\)).
Example:
- Plutonium-240 (\(^{240}\text{Pu}\))
\[
^{240}_{94}\text{Pu} \rightarrow ^{140}_{56}\text{Ba} +
^{94}_{38}\text{Kr} + 2n
\]
- Plutonium-240 splits into Barium-140 and Krypton-94, with the
emission of two neutrons.
Deriving the Law of Radioactive Decay
1. Assumption
- The rate of decay of a radioactive substance is directly
proportional to the number of undecayed nuclei \(n\) at any time \(t\).
\[
\frac{dn}{dt} \propto \frac{1}{n}
\]
- Expressed mathematically: \[
\frac{dn}{dt} = -\lambda n
\] where \(\lambda\) is the
decay constant.
2. Separation of Variables
- Rearrange the equation to separate variables \(n\) and \(t\): \[
\frac{dn}{n} = -\lambda \, dt
\]
3. Definite Integration
- Integrate both sides with limits \(t =
0\) to \(t\) for time, and \(n = n_0\) to \(n\) for the number of nuclei: \[
\int_{n_0}^{n} \frac{1}{n} \, dn = \int_{0}^{t} -\lambda \, dt
\]
- This yields: \[
\ln \left( \frac{n}{n_0} \right) = -\lambda t
\]
4. Solving for \(n\)
- Exponentiate both sides to solve for \(n\): \[
n = n_0 e^{-\lambda t}
\]
5. Daughter Nuclei and Age
Calculation
Assume that at \(t = 0\), the
number of parent nuclei is \(n_0\).
The total number of daughter nuclei at time \(t\) is \(n\).
The number of daughter nuclei produced by decay is: \[
D^* = n_0 - n
\]
as \[
n_0 = n e^{\lambda t}
\]
Then \[
D^* = n e^{\lambda t} - n
\]
If the the number of daughter atoms at time \(t\) = 0 is \(D_0\), then the total number of daughter
atoms after time \(t\) is given as
\[
D = D_0 + n \left( e^{\lambda t} - 1 \right)
\]
Half-Life
Definition:
- Half-Life (\(t_{1/2}\)) is the time required
for half of the radioactive nuclei in a sample to decay.
Key Points:
Exponential Decay:
- Radioactive decay follows an exponential decay law, meaning the rate
of decay is proportional to the number of remaining radioactive
nuclei.
Formula:
\[
N(t) = N_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}
\]
where:
- \(N(t)\) is the number of
radioactive nuclei at time \(t\).
- \(N_0\) is the initial number of
radioactive nuclei.
- \(t_{1/2}\) is the half-life of the
radioactive substance.
- \(t\) is the elapsed time.
Decay Constant (\(\lambda\)):
The relationship between the half-life and the decay constant is:
\[
\lambda = \frac{\ln(2)}{t_{1/2}}
\]
Example:
