Nuclear decays

AS UNIT 3

Oscillations and Nuclei

5 Nuclear decays

Learners should be able to demonstrate and apply their knowledge and understanding of:
a) The spontaneous nature of nuclear decay; the nature of α, β and γ radiation, and equations to represent the nuclear transformations using the [math]{}_Z^A X[/math] notation
b) Different methods used to distinguish between α, β and γ radiation and the connections between the nature, penetration and range for ionizing particles
c) How to make allowance for background radiation in experimental measurements
d) The concept of the half-life, [math]T_{1⁄2}[/math]
e) The definition of the activity, A, and the Becquerel
f) The decay constant, λ, and the equation A = λ N
g) The exponential law of decay in graphical and algebraic form,

[math]N = N_0 e^{-\lambda t} \qquad \text{and} \qquad A = A_0 e^{-\lambda t} \\
N = \frac{N_0}{2^x} \qquad \text{and} \qquad A = \frac{A_0}{2^x}[/math]

Where x is the number of half-lives elapsed – not necessarily an integer
h) The derivation and use of [math]\lambda = \frac{\ln 2}{T_{1/2}}[/math]

Specified Practical Work

o   Investigation of radioactive decay – a dice analogy

o   Investigation of the variation of intensity of gamma radiation with distance

  • a)   Spontaneous Nuclear Decay

  • Nuclear decay is a random, spontaneous process by which an unstable nucleus transforms into a more stable nucleus.
  • During decay, the nucleus emits particles and/or electromagnetic radiation. The process occurs without any external trigger and is governed by the internal structure and energy configuration of the nucleus.
  • Statistical Nature:
  • – The decay of a large number of identical nuclei is best described statistically using the decay constant (λ) and half‐life ([math]T_{1⁄2}[/math]). Even though the time at which a particular nucleus decays is random, the overall decay rate follows an exponential law:
  • [math]N(t) = N_0 e^{-λt}[/math]
  • Where N(t) is the number of undecayed nuclei at time t and [math]N_o[/math]​ is the initial number.
  • Half-life:
  • The half-life ​[math]T_{1/2}[/math] is defined as the time required for half of the radioactive nuclei in a sample to decay:
  • [math]T_{1/2} = \frac{\ln(2)}{\lambda} \\
    T_{1/2} = \frac{0.693}{\lambda}[/math]

  • b) Types of Nuclear Radiation and Their Representations

  • i) Alpha (α) Radiation
  • Nature:
  • – An alpha particle is a helium nucleus, consisting of 2 protons and 2 neutrons. It is relatively heavy and carries a +2 charge.
  • Notation in Nuclear Transformations:
  • In the [math]{}_Z^A X[/math] notation:
  • Example:
    The alpha decay of uranium-238:
  • [math]{}_{92}^{238}\text{U} \rightarrow {}_{90}^{234}\text{Th} + {}_{2}^{4}\text{He}[/math]
  • Penetration and Range:
  • – Alpha particles have low penetration power; they can be stopped by a sheet of paper or a few centimeters of air. Despite their high ionization ability, their range in matter is very short.
  • ii) Beta (β) Radiation
  • Nature:
  • – Beta particles are high-speed electrons (β⁻) or positrons (β⁺) emitted from the nucleus. They have much lower mass than alpha particles and carry a charge of -1 (or +1 in the case of positrons).
  • Notation:
  • – Beta-minus ( [math]{}_{-1}^{0}\beta[/math]) decay:
  • [math]{}_{Z}^{A}X \rightarrow {}_{Z+1}^{A}Y + {}_{-1}^{0}\beta + {}_{0}^{0}\bar{\nu}_e[/math]
  • Example:
    Carbon-14 decaying into nitrogen-14:
  • [math]{}_{6}^{14}\text{C} \rightarrow {}_{7}^{14}\text{N} + {}_{-1}^{0}\beta + {}_{0}^{0}\bar{\nu}_e[/math]
  • Figure 1 Beta-minus and beta-plus decay
  • Penetration and Range:
  • – Beta particles are more penetrating than alpha particles, typically being stopped by a few millimeters to about a centimeter of aluminum. They can be deflected by magnetic fields due to their charge and lower mass.
  • iii) Gamma (γ) Radiation
  • Nature:
  • – Gamma rays are high-energy photons emitted from the nucleus during a transition from an excited state to a lower energy state. They carry no mass and no charge.
  • [math]{}_{Z}^{A}X^* \rightarrow {}_{Z}^{A}X + {}_{0}^{0}\gamma[/math]
  • Notation:
  • – Gamma emission is usually indicated by:
  • Figure 2 Gamma emission
  • Where the asterisk denotes an excited state.
  • Penetration and Range:
    Gamma rays are highly penetrating and require dense materials (such as lead or thick concrete) for shielding. Their interaction with matter occurs via processes such as the photoelectric effect, Compton scattering, and pair production.
  • ⇒ Distinguishing Between α, β, and γ Radiation
  • Different methods are used to distinguish these radiations based on their distinct properties:
  • Detection Techniques:
  • Alpha Particles:
  • – Use detectors with very thin windows (such as semiconductor detectors or scintillation counters). Their high ionization and low penetration make them easy to distinguish; a simple piece of paper can block them.
  • Beta Particles:
  • – Use Geiger-Müller tubes with thin walls, scintillation detectors, or magnetic deflection setups. Beta particles, being lighter and charged, are deflected by magnetic fields, and their penetration is intermediate.
  • Gamma Rays:
  • – Use scintillation detectors (e.g., NaI(Tl) crystals), high-purity germanium (HPGe) detectors, or ionization chambers. Gamma rays are identified by their high penetration and weak interaction with matter.
  • Connections Between Nature, Penetration, and Range:
  • Alpha Particles:
  • – Highly ionizing but have a short range due to their large mass and charge.
  • Beta Particles:
  • – Moderately ionizing with a longer range than alpha particles because of their smaller mass.
  • Gamma Rays:
  • – Least ionizing per interaction but extremely penetrating due to their electromagnetic nature.
  • Figure 3 Penetrating power of rays

  • c)    Allowance for Background Radiation in Experimental Measurements

  • In any experiment involving radioactive materials or ionizing radiation, background radiation—which comes from cosmic rays, natural radioactivity in the environment, and even the detection equipment itself—must be accounted for:
  • Measurement of Background:
  • – Before introducing a radioactive source, measure the count rate (or intensity) with the detector in the experimental setup to determine the background level.
  • Subtraction of Background:
  • – When measurements are taken with the radioactive source present, the background count rate is subtracted from the total count rate to isolate the net radiation from the source.
  • Statistical Averaging:
  • – Since background radiation may fluctuate, multiple measurements and appropriate statistical analysis (averaging and error estimation) are necessary for accurate results.

  • d)   The Concept of Half-Life, [math]T_{1/2}[/math]

  • Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a characteristic property of a radioactive isotope.
  • ⇒  Mathematical Formulation:
  • The number of remaining nuclei N(t) after time t is given by the exponential decay law:
  • [math]N = N_0 e^{-λt}[/math]
  • Where:
  • ​ – [math]N_o[/math] is the initial number of nuclei.
  • – λ is the decay constant.
  • The half-life [math]T_{1/2}[/math]​ is defined by:
  • [math]N(T_{1/2}) = \frac{N_0}{2} \\
    \frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}[/math]
  • Taking the natural logarithm of both sides gives:
  • [math]\ln\left(\frac{1}{2}\right) = -\lambda T_{1/2} \\
    T_{1/2} = \frac{\ln(2)}{\lambda} \\
    T_{1/2} = \frac{0.693}{\lambda}[/math]
  • Figure 4 Half – life
  • Implications:
  • – The half-life is independent of the initial number of nuclei and depends solely on the intrinsic decay properties of the isotope.
  • – It is widely used in applications such as radiometric dating, nuclear medicine, and reactor physics.

  • e)    Activity and the Becquerel

  • ⇒  Definition of Activity, A
  • Activity is defined as the rate at which radioactive decays occur in a sample.
  • Mathematically, it is the number of decays per unit time.
  • The SI unit of activity is the Becquerel (Bq), Where
  • [math]1Bq = 1decay/second[/math]
  • In older literature, the Curie (Ci) is sometimes used
  • [math](1 Ci = 3.7 × 10^{10} Bq).[/math]

  • f)     Decay Constant, λ, and the Equation [math]A = λN[/math]

  • ⇒  Decay Constant (λ):
  • The decay constant λ is the probability per unit time that a given nucleus will decay.
  • It has units of inverse time (e.g.,[math]s^{-1}[/math] ).
  • A larger λ means the substance decays more rapidly.
  • –    Relationship Between Activity and Number of Nuclei
  • If N is the number of undecayed nuclei present in a sample, then the activity A (the number of decays per unit time) is given by:
  • [math]A = λN[/math]
  • This equation tells us that the rate of decay is proportional to the number of radioactive nuclei remaining.

  • g)   Exponential Law of Decay

  • ⇒  Algebraic Form
  • For the Number of Nuclei:
  • The decay of a radioactive sample follows an exponential law:
  • [math]N(t) = N_0 e^{-λt}[/math]
  • Where:
  • ​- [math]N_o[/math] is the initial number of nuclei at t=0,
  • – λ is the decay constant,
  • – t is the time elapsed.
  • For the Activity:
  • Since [math]A = λN[/math], the activity at time ttt is:
  • [math]A(t) = A_0 e^{-λt}[/math]
  • Where [math]A_0 = λN_0[/math] ​ is the initial activity.
  • Graphical Representation
  • ⇒ Plot of N(t) t:
  • A graph of [math]N(t) = N_0 e^{-λt}[/math] is a decreasing exponential curve. At t=0, N= [math]N_o[/math] ​, and as t increases, N(t) approaches zero asymptotically.
  • ⇒ Plot of A(t) t:
  • Similarly, the activity A(t) decreases exponentially with time and follows the same shape as N(t).
  • ⇒  Alternate Representation Using Half-Lives
  • If x represents the number of half-lives elapsed (which may not be an integer), then:
  • – The number of nuclei remaining can also be written as:
  • [math]N = \frac{N_0}{2x}[/math]
  • – Similarly, the activity becomes:
  • [math]A = \frac{A_0}{2x}[/math]
  • Here, ​[math]x = \frac{t}{T_{1/2}}[/math] where [math]\frac{t}{T_{1/2}}[/math] is the half-life.

  • h)   Derivation and Use of [math]\lambda = \frac{\ln 2}{T_{1/2}}[/math]

  • ⇒  Derivation:
  • 1. Definition of Half-Life:
  • The half-life [math]T_{1/2}[/math] is the time required for half of the radioactive nuclei to decay. By definition, when
  • [math]t = T_{1/2} \\
    N(T_{1/2}) = \frac{N_0}{2}[/math]
  • 2. Using the Exponential Decay Law:
  • [math]N(T_{1/2}) = N_0 e^{-\lambda T_{1/2}}[/math]
  • 3. Solving for λ:
  • Divide both sides by ​[math]N_o[/math]:
  • [math]e^{-\lambda T_{1/2}} = \frac{1}{2}[/math]
  • – Take the natural logarithm of both sides:
  • [math]-\lambda T_{1/2} = \ln \left(\frac{1}{2}\right) \\
    -\lambda T_{1/2} = -\ln 2[/math]
  • – Solve for λ:
  • [math]\lambda = \frac{\ln 2}{T_{1/2}}[/math]
  • ⇒  Use of the Formula:
  • This relation allows you to determine the decay constant if you know the half-life of a radioactive substance.
  • Conversely, if you measure the decay constant λ experimentally, you can compute the half-life as:
  • [math]T_{1/2} = \frac{\ln 2}{\lambda}[/math]

  • Specified Practical Work

  • I) Investigation of Radioactive Decay – A Dice Analogy

  • ⇒ Purpose:
  • The dice analogy is used to illustrate the random, spontaneous nature of radioactive decay and to demonstrate how the number of undecayed nuclei decreases exponentially over time.
  • ⇒  Concept:
  • In a radioactive sample, each nucleus has a fixed probability of decaying in a given time interval.
  • Similarly, when you roll a dice, each face has an equal chance of showing up. By assigning one face (e.g., the “6”) to represent a decay event, you can simulate decay using dice.
  • With many dice (representing many nuclei), you will observe that, over repeated trials (time intervals), the number of dice that have not “decayed” (i.e., did not show the chosen number) decreases
  • exponentially.
  • ⇒  Apparatus and Materials:
  • A large number of identical dice (e.g., 50 or 100 dice) to represent a sample of radioactive nuclei.
  • A container or tray to roll the dice.
  • A recording sheet to note the number of dice that have not “decayed” after each roll.
  • A timer or stopwatch to simulate equal time intervals (each roll represents one time unit).
  • Figure 5 Investigation of radioactive decay
  • ⇒  Procedure:
  • 1. Initial Setup:
  • Start with [math]N_o[/math]​ dice (e.g., 100 dice) representing the initial number of radioactive nuclei.
  • Decide on a rule: for example, a die that shows a “6” is considered to have decayed in that time interval.
  • 2. Simulate Decay Over Time:
  • Roll all dice simultaneously (or one by one, if preferred) for the first-time interval.
  • Count and record the number of dice that do not show a “6.” Let this number be [math]N_![/math]
  • Remove or mark the dice that have “decayed” (i.e., show a 6).
  • Roll the remaining dice for the next time interval, and record the number of dice that have not decayed after this interval, [math]N_2[/math]
  • Repeat the process for several time intervals.
  • 3. Data Analysis:
  • Plot the number of undecayed dice N versus the number of time intervals (rolls).
  • You should observe an exponential decrease, analogous to the radioactive decay law:
  • [math]N(t) = N_0 e^{-\lambda t}[/math]
  • or, equivalently, using the concept of half-lives:
  • [math]N = \frac{N_0}{2x}[/math]
  • Where x is the number of half-lives elapsed.
  • 4. Discussion:
  • – The dice experiment visually and statistically demonstrates that decay is a random process with a constant probability per unit time.
  • – It shows how an ensemble of decaying nuclei exhibits exponential decay behavior.
  • – The experiment can be used to illustrate the concept of a half-life: after a certain number of intervals, roughly half of the dice (nuclei) will have “decayed.”
  • ⇒ Conclusion:
  • The dice analogy provides an accessible, hands-on method to simulate the statistical nature of radioactive decay and helps in understanding the exponential decay law that governs nuclear processes.

  • II)  Investigation of the Variation of Intensity of Gamma Radiation with Distance

  • ⇒  Purpose:
  • To experimentally verify how the intensity (or count rate) of gamma radiation decreases with distance, following the inverse square law.
  • ⇒  Theory:
  • Gamma rays are high-energy photons that emanate from a radioactive source.
  • According to the inverse square law, the intensity I of gamma radiation from a point source decreases with the square of the distance rrr from the source:
  • [math]I ∝ \frac{1}{r^2}[/math]
  • In practice, the measured count rate using a radiation detector (like a Geiger-Müller counter) should follow this relationship.
  • ⇒  Apparatus and Materials:
  • A gamma radiation source (commonly a low-activity source such as Cs-137 or Co-60, subject to safety regulations).
  • A Geiger-Müller (GM) counter or a scintillation detector to measure the gamma count rate.
  • A meter stick or measuring tape.
  • A stable mount for the radiation source.
  • A controlled environment (lab room) with minimal additional radiation sources.
  • Data recording tools (data logger or manual recording sheet).
  • Figure 6 Investigation of the Variation of Intensity of Gamma Radiation with Distance
  • ⇒  Procedure:
  • 1. Initial Setup:
  • Place the gamma radiation source securely on a stable platform.
  • Position the GM counter at a known initial distance from the source (for example, 10 cm). Ensure the detector is oriented so that its sensitive face is directed toward the source.
  • 2. Measure Baseline Radiation:
  • Record the count rate (number of gamma events per unit time) at the initial distance.
  • Ensure the measurement is taken over a fixed period (e.g., 1 minute) for consistency.
  • 3. Vary the Distance:
  • Move the GM counter to several different distances from the source (e.g., 15 cm, 20 cm, 25 cm, etc.).
  • At each distance, record the count rate over the same fixed time interval.
  • 4. Data Analysis:
  • Plot the measured count rate (intensity) I versus distance r on a graph
  • On a log-log scale, a straight line with a slope of −2 would indicate the inverse square relationship.
  • Alternatively, plot I against [math]\frac{1}{r^2}[/math] to check for linearity.
  • 5. Allowance for Background Radiation:
  • Before or during the experiment, measure the background radiation level with the source shielded or removed.
  • Subtract the background count rate from all measured count rates to obtain the net gamma radiation intensity.
  • ⇒  Discussion:
  • The experimental data should demonstrate that the intensity of gamma radiation falls off with the square of the distance.
  • Any deviations might be due to experimental uncertainties, geometric factors (if the source is not a perfect point source), or scattering effects.
  • This experiment reinforces the fundamental principle of the inverse square law in radiative processes.
  • ⇒   Conclusion:
  • The investigation confirms that gamma radiation intensity decreases with distance according to the inverse square law.
  • The experiment also illustrates the importance of background subtraction for accurate measurement of radiation intensity.
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