Radioactive decay

• a) Isotopes

• Isotopes are atoms of the same element (i.e., same number of protons, Z) but with different numbers of neutrons, and hence different mass numbers (A).
• ⇒ Symbolic Representation:
• [math]{}^{A}_{Z}X[/math]
• – Z = proton number (atomic number)
• – A = nucleon number (protons + neutrons)
• – X = chemical symbol
• ⇒ Example:
• Hydrogen has three isotopes:
• – Protium: [math]{}^{1}_{1}\text{H}[/math](0 neutrons)
• – Deuterium: [math]{}^{1}_{2}\text{H}[/math](1 neutron)
• – Tritium: [math]{}^{1}_{3}\text{H}[/math](2 neutrons)
• 
• Figure 1 Isotopes
• Properties of Isotopes:
• – Chemically similar (same electron configuration)
• – Physically different (different mass, some may be radioactive)

• b) Nuclear Binding Energy and Mass Defect

• Mass Defect:
• When nucleons (protons and neutrons) combine to form a nucleus, the mass of the nucleus is less than the sum of the masses of the individual nucleons.
• This difference in mass is called the mass defect (Δm):
• [math]\Delta m = (Z m_p + N m_n) – m_{\text{nucleus}}[/math]
• Where:
• – Z: number of protons
• – N: number of neutrons
• – [math]m_p , m_n[/math]: mass of a proton and neutron
• – [math]m_{\text{nucleus}}[/math]: actual mass of the nucleus
• ⇒ Nuclear Binding Energy (BE):
• This “missing” mass is converted into energy that binds the nucleus together. According to Einstein’s mass-energy equivalence:
• [math]E = \Delta m . c^2[/math]
• – [math]\Delta m [/math]: mass defect in kg
• – [math]c\text{: speed of light} \approx 3 \times 10^8 \ \text{m/s}[/math]
• This energy is the nuclear binding energy.
• 
• Figure 2 Binding Energy
• ⇒ Significance:
• It represents the energy required to break the nucleus into its individual protons and neutrons.
• Higher binding energy means a more stable nucleus.

• c) Binding Energy per Nucleon vs Nucleon Number

• ⇒ Binding Energy per Nucleon:
• [math]\text{Binding energy per nucleon} = \frac{\text{Total binding energy}}{\text{Number of nucleons (A)}}[/math]
• It tells us how tightly each nucleon is bound within the nucleus.
• ⇒ Graph Description:
• When we plot binding energy per nucleon (y-axis) vs nucleon number A (x-axis), the graph shows:
• A sharp increase for light elements (like hydrogen to helium)
• A peak around iron (Fe, A ≈ 56): most stable nucleus
• A slow decrease for heavier nuclei beyond iron
• 
• Figure 3 Nuclear binding energy per nucleon Vs Mass Number
• ⇒ Interpretation:

• ⇒ Nuclear Reactions:
• Fusion: Light nuclei combine → form heavier nuclei → release energy.
• Fission: Heavy nucleus splits → smaller nuclei → release energy.
• Both fusion and fission work because the resulting products have higher binding energy per nucleon, meaning the system becomes more stable.
• 
• Figure 4 Nuclear fusion and fission

• d) Mass-Energy Equivalence (E = mc²) in Nuclear Reactions

• ⇒ Einstein’s Equation:
• [math]E = m c^2[/math]
• Where:
• – E = energy (in joules)
• – m = mass (in kilograms)
• – c = speed of light [math]\approx 3 \times 10^8 \ \text{m/s}[/math]
• Meaning:
• Mass and energy are interchangeable. Even a small amount of mass corresponds to a huge amount of energy due to the large value of [math]c^2[/math]
• In Nuclear Reactions:
• In processes like nuclear fission, fusion, or radioactive decay, the total mass of the products is slightly less than the mass of the reactants. This mass defect is converted into energy
• ⇒ Example:
• In fusion:
• [math]2H+3H→4He+n+Energy[/math]
• The missing mass in this reaction is what produces vast amounts of energy (as in stars or hydrogen bombs).

• e) The Strong Nuclear Force

• The strong nuclear force is one of the four fundamental forces of nature. It is the force that holds the protons and neutrons together in the nucleus.
• 
• Figure 5 Strong Nuclear force
• ⇒ Characteristics:

• Importance:
• Without the strong nuclear force, atomic nuclei wouldn’t exist — they’d be torn apart by the repulsive electric force between protons.
• The Random and Spontaneous Nature of Radioactive Decay
• Radioactive decay is the process by which an unstable nucleus emits radiation to become more stable.
• ⇒ Spontaneous:
• It happens on its own—a nucleus cannot be forced to decay. No external trigger is needed.
• Random:
• We cannot predict when a specific atom will decay. We can only talk in terms of probabilities and half-lives for large numbers of atoms.
• Types of Radioactive Decay:

• ⇒ Decay Law:
• The rate of decay is proportional to the number of undecayed nuclei:
• [math]N(t) = N_0 e^{-\lambda t}[/math]
• – N(t) = number of undecayed nuclei at time
• ​- [math]N_0[/math] = initial number of nuclei
• – λ = decay constant

• g) Changes in the Nucleus After Radioactive Decay

• When a nucleus is unstable, it can become more stable by emitting radiation. This process alters the nuclear structure depending on the type of decay:
• ⇒ Alpha (α) Decay
• What is emitted?
• – An alpha particle [math]{}^{4}_{2}\text{He} \rightarrow 2 \text{ protons} + 2 \text{ neutrons}[/math]
• Effect on nucleus:
• – Mass number A decreases by 4
• – Atomic number Z decreases by 2
• Example:
• [math]{}^{238}_{92}\text{U} \rightarrow {}^{234}_{90}\text{Th} + {}^{4}_{2}\text{He}[/math]
• b) Beta-minus (β⁻) Decay
• What is emitted?
• An electron and an antineutrino ([math]\bar{v}_e[/math])
• – Cause: A neutron turns into a proton
• – Effect on nucleus:
• Mass number A stays the same
• Atomic number Z increases by 1
• Example:
• [math]{}^{14}_{6}\text{C} \rightarrow {}^{14}_{7}\text{N} + \beta^- + \bar{v}_e[/math]
• c) Beta-plus (β⁺) Decay
• What is emitted?
• A positron and a neutrino ([math]v_e[/math])
• – Cause: A proton turns into a neutron
• Effect on nucleus:
• – Mass number A stays the same
• – Atomic number Z decreases by 1
• Example:
• [math]{}^{11}_{6}\text{C} \rightarrow {}^{11}_{5}\text{B} + \beta^- + v_e[/math]
• d) Gamma (γ) Decay
• What is emitted?
• A high-energy photon (gamma ray)
• – Cause: The nucleus drops from an excited state to a lower energy state
• – Effect on nucleus:
• No change in A or Z
• Just a change in nuclear energy level
• Example:
• [math]{}^{60}_{27}\text{Co}^* \rightarrow {}^{60}_{27}\text{Co} + \gamma[/math]

• h) Radioactive Decay Equations Summary

• i) Neutrinos and Antineutrinos ([math]v[/math])

• When beta decay was first observed, scientists noticed that:
• – Energy and momentum didn’t seem to be conserved.
• – The emitted electron didn’t always carry away all expected energy.
• 
• Figure 6  Decay of neutron
• ⇒ The Solution: Neutrinos!
• a) Neutrinos ([math]v_e[/math]​):
• – Tiny, neutral (no electric charge)
• – Nearly massless
• – Hard to detect — barely interact with matter
• – Emitted in β⁺ decay to conserve energy, momentum, and angular momentum (spin)
• b) Antineutrinos ([math]\bar{v}_e[/math]):
• – The antimatter counterpart of neutrinos
• – Emitted in β⁻ decay

• j) Penetration and Ionizing Ability of Alpha, Beta, and Gamma Radiation

• – Ionizing ability refers to how easily the radiation knocks electrons off atoms, creating ions.
• – Alpha is most ionizing because it’s large and highly charged but cannot travel far.
• – Gamma rays are least ionizing but most penetrating due to their lack of mass and charge.

• k) Activity, Count Rate, and Half-Life in Radioactive Decay

• ⇒ Activity (A):
• Definition:
• The number of decays per second.
• Unit: Becquerel (Bq) = 1 decay per second.
• Equation:
• [math]A = \lambda N[/math]
• Where:
• – λ = decay constant,
• – N = number of undecayed nuclei.
• 
• Figure 7 Half-Life of substances
• ⇒ Count Rate:
• What a Geiger-Müller counter or detector records—counts per second.
• Proportional to activity but can be affected by distance, shielding, and detector efficiency.
• Half-life (T₁/₂):
• Definition:
• Time taken for half of the radioactive nuclei in a sample to decay.
• Constant for a given isotope.
• Related to the decay constant by:
• [math]T_{\frac{1}{2}} = \frac{\ln(2)}{\lambda}[/math]
• Changes in Activity and Count Rate Using Integer Values of Half-life

• Example:
• If you start with an activity of 800 Bq:
• – After 1 half-life → 400 Bq
• – After 2 half-lives → 200 Bq
• – After 3 half-lives → 100 Bq
• This exponential decay means the activity never truly reaches zero but becomes negligible over time.

• m) Effect of Background Radiation on Count Rate

• Radiation always presents in our environment.
• Sources:
• – Cosmic rays from space
• – Radioactive rocks (like granite)
• – Radon gas
• – Medical imaging
• – Fallout from nuclear tests
• ⇒ Impact on Measurements:
• Background radiation adds to the measured count rate, so it must be accounted for.
• To get the true count rate of a radioactive source:
• [math]\text{Corrected Count Rate} = \text{Measured Count Rate} – \text{Background Count Rate}[/math]
• ⇒ Example:
• Background = 20 counts/min
• Measured = 70 counts/min
• Actual from source = 70 – 20 = 50 counts/min

Additional higher level: 5 hours

• a) The Evidence for the Strong Nuclear Force

• ⇒ The Strong Nuclear Force:
• – The strong nuclear force is one of the four fundamental forces of nature.
• – It is an attractive force that acts between protons and neutrons (nucleons) in the nucleus.
• – It operates at very short ranges (about 1–3 femtometers, or 10⁻¹⁵ meters).
• – It is stronger than the electrostatic repulsion between positively charged protons, but only at very small distances.
• ⇒ Evidence for Its Existence:
• 1. Stability of the nucleus:
• – Protons repel each other due to their positive charges (Coulomb repulsion).
• – Despite this, nuclei remain bound—suggesting another, stronger force is at work.
• 2. Alpha particle scattering experiments (Rutherford):
• – Showed that atoms have small, dense nuclei.
• – The observed deflection patterns indicated forces stronger than electrostatic repulsion at short range.
• 
• Figure 8 Alpha particle scattering
• 3. Nuclear binding energy:
• – The energy required to break a nucleus into its nucleons implies a strong attractive force holding them together.
• – This energy can be calculated from the mass defect using Einstein’s equation [math]E = mc^2[/math].

• b) The Role of the Neutron-to-Proton Ratio (n/p) in Nuclide Stability

• ⇒ Neutron-to-Proton Ratio:
• Protons repel each other via the electromagnetic force.
• Neutrons do not experience this repulsion but contribute to the strong nuclear force.
• An appropriate balance of neutrons and protons is needed to ensure the nucleus is stable.
• ⇒ Stable Nuclides:
• For light elements (Z < 20), stability is achieved when n ≈ p (i.e., n/p ≈ 1).
• For heavier elements, more neutrons are needed to offset the growing electrostatic repulsion between protons, so n/p > 1.
• Instability:
• Nuclei with too many or too few neutrons become unstable and undergo radioactive decay to achieve a more stable n/p ratio:
• – Beta-minus (β⁻) decay: Converts a neutron into a proton to decrease n/p.
• – Beta-plus (β⁺) decay or electron capture: Converts a proton into a neutron to increase n/p.
• Stability Line:
• When you plot all stable nuclei on a neutron vs. proton graph, they lie along a curved line of stability.
• Radioactive nuclides lie off this line and decay toward it.

• c) The Approximate Constancy of the Binding Energy per Nucleon for A > 60

• ⇒ Binding Energy per Nucleon:
• It is the average energy required to remove a nucleon from the nucleus.
• Indicates how tightly bound a nucleus is.
• Calculated as:
• [math]\text{Binding energy per nucleon} = \frac{\text{Total binding energy}}{\text{Number of nucleons (A)}}[/math]
• ​ Binding Energy Curve:
• The curve of binding energy per nucleon vs. nucleon number (A):
• – Rises sharply for small A (e.g., Hydrogen to Iron).
• – Peaks around A ≈ 56 (Iron-56 has the highest binding energy per nucleon ~8.8 MeV).
• – Remains nearly constant (around 8 MeV) for A > 60.
• – Slightly decreases for very heavy elements (e.g., Uranium-238).
• 
• Figure 9 Binding energy curve
• ⇒ Implications:
• Nuclei with A ≈ 56 (like iron and nickel) are the most stable.
• Fusion (light nuclei combining) releases energy because binding energy per nucleon increases.
• Fission (heavy nuclei splitting) also releases energy for the same reason.
• Explains why both fusion and fission can be energy sources.

• d) The Spectrum of Alpha and Gamma Radiations as Evidence for Discrete Nuclear Energy Levels

• ⇒ Discrete Energy Levels in the Nucleus:
• Just like electrons in atoms, nucleons (protons and neutrons) in the nucleus also occupy quantized (discrete) energy levels.
• When a nucleus transitions from a higher energy state to a lower one, it emits radiation.
• 
• Figure 10 Penetration of radiations
• Alpha Radiation:
• Alpha particles (⁴He nuclei) are emitted by heavy, unstable nuclei.
• The energies of emitted alpha particles are not continuous, but appear at specific, fixed values.
• This indicates that:
• – The parent nucleus has discrete energy levels.
• – The daughter nucleus is left in a specific energy state after the emission.
• Gamma Radiation:
• Gamma rays are electromagnetic photons emitted when a nucleus in an excited state drops to a lower energy state.
• Gamma ray emissions have sharp, well-defined energy peaks, providing direct evidence of quantized nuclear energy levels.
• Conclusion:
• The alpha and gamma spectra show specific energy values, confirming that the nucleus has discrete energy states, just like atomic electrons.

• e) The Continuous Spectrum of Beta Decay as Evidence for the Neutrino

• The Problem:
• In beta decay, a nucleus emits a beta particle (electron or positron).
• Unlike alpha and gamma decay, the energy spectrum of beta particles is continuous, not discrete.
• Early Observations:
• If only the beta particle and the nucleus were involved, energy and momentum would not appear to be conserved in some decays.
• Beta particles were found to have a range of energies, with a maximum energy, rather than a fixed value.
• ⇒ The Solution: The Neutrino
• Wolfgang Pauli proposed the existence of a third particle to conserve energy, momentum, and angular momentum: the neutrino (or antineutrino).
• Later, Enrico Fermi incorporated the neutrino into a complete beta decay theory.
• The neutrino is:
• – Neutral
• – Very light (nearly massless)
• – Weakly interacting (hard to detect)
• ⇒ Beta Decay Process:
• Example of β⁻ decay:
• [math]n \rightarrow p + e^- + \bar{v}_e[/math]
• Conclusion:
• The continuous energy distribution of beta particles is explained by the sharing of energy between the beta particle and the neutrino. This is strong evidence for the existence of the neutrino.

• f) The Decay Constant λ and the Radioactive Decay Law

• ⇒ Radioactive Decay:
• Radioactive substances decay over time in a probabilistic way.
• The rate of decay is proportional to the number of undecayed nuclei.
• ⇒ The Decay Law:
• The number of undecayed nuclei at time t is given by:
• [math]N = N_0 e^{-\lambda t}[/math]
• Where:
• – N = number of undecayed nuclei at time t
• ​- [math]N_0[/math]= initial number of nuclei at t=0
• – λ = decay constant (probability of decay per unit time)
• – e = Euler’s number (~2.718)
• Meaning of the Decay Constant λ:
• – Has units of s⁻¹.
• – A higher value of λ means faster decay.
• – Related to the half-life [math]T_{\frac{1}{2}}[/math]of the substance by:
• – [math]T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}[/math]
• ⇒ Graphical Representation:
• The decay curve is an exponential drop.
• After each half-life, the number of undecayed nuclei is halved.

• g) The Decay Constant Approximates the Probability of Decay in Unit Time (Only for Small Intervals)

• ⇒ The Decay Constant (λ):
• The decay constant, denoted by λ, is a measure of how quickly a radioactive substance decays.
• It represents the probability per unit time that a given nucleus will decay.
• Important Clarification:
• The decay constant is not an exact probability, but a rate.
• It becomes approximately equal to the actual probability of decay only over a very small time interval [math]\Delta t[/math] .
•  In simple terms:
• For small time intervals [math]\Delta t[/math], the probability that a nucleus decays in that interval is roughly:
• [math]\text{Probability of decay} \approx \lambda \Delta t[/math]
• This approximation is only valid if [math]\Delta t \ll \frac{1}{\lambda}[/math](i.e., time is small enough that the chance of multiple decays is negligible).

• h) Activity as the Rate of Decay:

• Activity (A)
• – Activity is defined as the rate at which radioactive nuclei decay in a sample.
• – It tells us how many decays occur per second.
• Measured in becquerels (Bq):
• [math]1\ \text{Bq} = 1\ \text{decay/second}[/math]
• ⇒ Mathematical Definition:
• [math]A = -\frac{dN}{dt} = \lambda N[/math]
• Where:
• – A = activity
• – λ = decay constant
• – N = number of undecayed nuclei at time t
• Using the decay law [math]N = N_0 e^{-\lambda t}[/math], we can also write:
• [math]A = \lambda N_0 e^{-\lambda t}[/math]
• Activity decreases exponentially with time, just like the number of undecayed nuclei.

• i) Half-Life and its Relationship with the Decay Constant

• Half-Life (​[math]T_{\frac{1}{2}} [/math])
• The half-life of a radioactive substance is the time required for half of the nuclei in a sample to decay.
• That is:
• [math]N = \frac{N_0}{2} \quad \text{when} \quad t = T_{\frac{1}{2}}[/math]
• ⇒ Deriving the Formula:
• Start from the decay law:
• [math]N = N_0 e^{-\lambda t}[/math]
• Set [math]N = \frac{N_0}{2} \quad \text{When} \quad t = T_{\frac{1}{2}}[/math]:
• [math]\frac{N_0}{2} = N_0 e^{-\lambda T_{\frac{1}{2}}}[/math]
• Divide both sides by ​[math]N_0[/math]:
• [math]\frac{1}{2} = e^{-\lambda T_{\frac{1}{2}}}[/math]
• Take natural log on both sides:
• [math]\ln\left(\frac{1}{2}\right) = -\lambda T_{\frac{1}{2}} \\
  -\ln 2 = -\lambda T_{\frac{1}{2}} \\
  T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}[/math]