Atoms, nuclei and radiation

1. Overview

The nuclear atom is defined by a small, dense, positively charged nucleus containing the vast majority of the atom's mass, surrounded by a cloud of orbiting electrons. This model, which superseded the "Plum Pudding" model, was established through the $\alpha$-particle scattering experiment, demonstrating that matter is mostly empty space. Radioactive decay is the process by which unstable nuclei emit $\alpha$, $\beta$, or $\gamma$ radiation to reach a more stable state. These processes are governed by strict conservation laws regarding charge and nucleon number. A critical distinction in nuclear physics is the discrete energy of $\alpha$-particles versus the continuous energy spectrum of $\beta$-particles, the latter of which necessitated the discovery of the neutrino.

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Key Definitions

• Nucleon Number ($A$): The total number of protons and neutrons (collectively called nucleons) in the nucleus of an atom.
• Proton Number ($Z$): The number of protons in the nucleus of an atom, which determines the chemical identity of the element.
• Isotope: Atoms of the same element (same proton number) that contain different numbers of neutrons (different nucleon number).
• Nuclide: A specific species of nucleus characterized by its specific combination of proton number and nucleon number.
• Antiparticle: A fundamental particle that has the same rest mass as its corresponding particle but opposite charge (and other opposite quantum numbers).
• Unified Atomic Mass Unit ($u$): A standard unit of mass defined as exactly $\frac{1}{12}$ of the mass of an isolated neutral atom of carbon-12 ($^{12}_{6}\text{C}$).
• Positron: The antiparticle of the electron, carrying a charge of $+1e$ and the same mass as an electron.
• Neutrino ($\nu$): A fundamental particle with zero charge and negligible rest mass that is emitted during $\beta^+$ decay.
• Antineutrino ($\bar{\nu}$): The antiparticle of the neutrino, emitted during $\beta^-$ decay.

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Content

3.1 The $\alpha$-particle Scattering Experiment (Geiger-Marsden)

In 1909, Ernest Rutherford directed Hans Geiger and Ernest Marsden to fire high-energy $\alpha$-particles at a very thin gold foil (only a few hundred atoms thick) in a vacuum.

Experimental Setup Requirements:

• Vacuum: Necessary because $\alpha$-particles have a very short range in air (a few cm) and would be stopped by collisions with air molecules.
• Thin Gold Foil: Gold is highly malleable, allowing for a foil thin enough that $\alpha$-particles generally only undergo a single collision.
• Movable Detector: A zinc sulfide screen that emits a flash of light (scintillation) when struck by an $\alpha$-particle, viewed through a microscope.

Observations and Inferences (The "Evidence-Conclusion" Link):

1. Observation: The vast majority (approx. 99.9%) of $\alpha$-particles passed straight through the foil with zero or negligible deflection.
   • Inference: The atom consists mostly of empty space.
2. Observation: A small number of $\alpha$-particles were deflected through small angles.
   • Inference: There is a concentration of positive charge within the atom that exerts an electrostatic repulsive force on the positive $\alpha$-particles.
3. Observation: A very small fraction (about 1 in 8000) were deflected through very large angles ($> 90^{\circ}$), with some reflecting almost directly back ($180^{\circ}$).
   • Inference: The nucleus is extremely small (dense) and contains most of the mass of the atom. The "backscattering" occurs when an $\alpha$-particle makes a head-on approach to the nucleus.

3.2 The Nuclear Atom Model

The atom is structured as follows:

• The Nucleus: A central core containing protons (positive) and neutrons (neutral). It is approximately $10^{-15}\text{ m}$ (1 femtometer) in diameter.
• Electrons: Negatively charged particles that orbit the nucleus at relatively large distances. The atomic radius is approximately $10^{-10}\text{ m}$ (1 angstrom).
• Comparison: The atom is roughly $100,000$ times larger than its nucleus. If the nucleus were the size of a pea in the center of a football stadium, the electrons would be orbiting in the highest stands.

3.3 Atomic Notation and Nuclides

A nuclide is represented using the notation: $${}^{A}_{Z}\text{X}$$

• $\text{X}$: Chemical symbol.
• $A$: Nucleon number (Mass number).
• $Z$: Proton number (Atomic number).
• $N$: Neutron number, calculated as $N = A - Z$.

Isotopes: Isotopes of an element have the same $Z$ but different $A$. Because they have the same number of protons, they have the same number of electrons in a neutral state, meaning they are chemically identical. However, they have different physical properties (e.g., density, stability/radioactivity).

3.4 Conservation Laws in Nuclear Processes

In any nuclear decay or reaction, the following quantities are strictly conserved:

1. Nucleon Number ($A$): The sum of $A$ on the left side of the equation must equal the sum of $A$ on the right.
2. Charge ($Z$): The total charge (proton number) must be conserved. Note that an electron has a charge of $-1$ and a positron has a charge of $+1$ in these equations.
3. Mass-Energy: While mass alone is not conserved (some mass is converted to kinetic energy of the products), the total mass-energy remains constant.

3.5 Properties of $\alpha$, $\beta$, and $\gamma$ Radiations

Property	Alpha ($\alpha$)	Beta-minus ($\beta^-$)	Beta-plus ($\beta^+$)	Gamma ($\gamma$)
Composition	2 protons, 2 neutrons	Electron	Positron	Photon (EM Wave)
Symbol	${}^{4}{2}\text{He}$ or ${}^{4}{2}\alpha$	${}^{0}{-1}e$ or ${}^{0}{-1}\beta$	${}^{0}{+1}e$ or ${}^{0}{+1}\beta$	$\gamma$
Charge	$+2e$	$-1e$	$+1e$	$0$
Mass	$4\text{ u}$	$1/1840\text{ u}$	$1/1840\text{ u}$	$0$
Speed	Slow ($\approx 0.05c$)	Fast ($\approx 0.9c$)	Fast ($\approx 0.9c$)	Speed of light ($c$)
Ionizing Power	Very High	Weak	Weak	Very Weak
Penetrating Power	Low (Stopped by paper/skin)	Medium (Stopped by few mm Al)	Medium (Stopped by few mm Al)	High (Reduced by thick lead/concrete)
Deflection in B-field	Small deflection	Large deflection	Large deflection (opposite to $\beta^-$)	No deflection

3.6 Antiparticles and Neutrinos

• Antiparticles: Every particle has an antiparticle. For the electron ($e^-$), the antiparticle is the positron ($e^+$). They have the same mass but opposite charge.
• The Neutrino Hypothesis: During $\beta$-decay, it was observed that the emitted electrons did not have a single fixed energy, which seemed to violate the conservation of energy. Wolfgang Pauli proposed the neutrino ($\nu$) to carry away the "missing" energy and momentum.
  • $\beta^-$ decay: Produces an electron antineutrino ($\bar{\nu}_e$).
  • $\beta^+$ decay: Produces an electron neutrino ($\nu_e$).

3.7 Radioactive Decay Equations

Alpha ($\alpha$) Decay: A heavy nucleus becomes more stable by ejecting a helium nucleus. $${}^{A}{Z}\text{X} \rightarrow {}^{A-4}{Z-2}\text{Y} + {}^{4}_{2}\alpha$$

Beta-minus ($\beta^-$) Decay: A neutron in the nucleus decays into a proton, an electron, and an antineutrino. $${}^{A}{Z}\text{X} \rightarrow {}^{A}{Z+1}\text{Y} + {}^{0}_{-1}e + \bar{\nu}_e$$ (Note: $Z$ increases by 1 because a new proton is formed).

Beta-plus ($\beta^+$) Decay: A proton in the nucleus decays into a neutron, a positron, and a neutrino. $${}^{A}{Z}\text{X} \rightarrow {}^{A}{Z-1}\text{Y} + {}^{0}_{+1}e + \nu_e$$ (Note: $Z$ decreases by 1 because a proton is lost).

3.8 Energy Spectra: Discrete vs. Continuous

• Discrete Energy ($\alpha$): $\alpha$-particles are emitted with specific, constant kinetic energies (e.g., 4.2 MeV). This is because the decay is a two-body problem (the daughter nucleus and the $\alpha$-particle). To conserve momentum and energy, the energy must be partitioned in a fixed ratio.
• Continuous Energy ($\beta$): $\beta$-particles are emitted with a range of energies from zero up to a maximum ($E_{max}$). This is because the decay is a three-body problem (daughter nucleus, $\beta$-particle, and neutrino). The total energy released is shared among the three particles in varying proportions. The neutrino carries whatever energy the $\beta$-particle does not take to reach $E_{max}$.

3.9 Unified Atomic Mass Unit ($u$)

Because the kilogram is too large for atomic scales, we use $u$.

• $1\text{ u} = 1.66 \times 10^{-27}\text{ kg}$
• Mass of a proton $\approx 1.007\text{ u}$
• Mass of a neutron $\approx 1.009\text{ u}$
• Mass of an electron $\approx 0.0005\text{ u}$

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4. Worked Examples

Worked Example 1 — Specific Charge Calculation

Question: Calculate the specific charge of an $\alpha$-particle. Give your answer in standard form to three significant figures.

Step 1: Identify the properties of an $\alpha$-particle. An $\alpha$-particle is a helium nucleus (${}^{4}_{2}\text{He}^{2+}$).

• Charge $Q = +2e = 2 \times (1.60 \times 10^{-19}\text{ C}) = 3.20 \times 10^{-19}\text{ C}$
• Mass $m = 4\text{ u} = 4 \times (1.66 \times 10^{-27}\text{ kg}) = 6.64 \times 10^{-27}\text{ kg}$

Step 2: Use the specific charge formula. $$\text{Specific Charge} = \frac{Q}{m}$$ $$\text{Specific Charge} = \frac{3.20 \times 10^{-19}\text{ C}}{6.64 \times 10^{-27}\text{ kg}}$$

Step 3: Calculate the final value. $$\text{Specific Charge} = 4.8192... \times 10^7\text{ C kg}^{-1}$$ Answer: $4.82 \times 10^7\text{ C kg}^{-1}$

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Worked Example 2 — Balancing Nuclear Equations

Question: A nucleus of Thorium-234 (${}^{234}_{90}\text{Th}$) undergoes a sequence of decays: first a $\beta^-$ emission, followed by another $\beta^-$ emission, and finally an $\alpha$ emission. Determine the nucleon number and proton number of the resulting nuclide.

Step 1: First $\beta^-$ decay. $${}^{234}{90}\text{Th} \rightarrow {}^{234}{91}\text{Pa} + {}^{0}_{-1}e + \bar{\nu}_e$$

• $A = 234$, $Z = 91$ (Protactinium)

Step 2: Second $\beta^-$ decay. $${}^{234}{91}\text{Pa} \rightarrow {}^{234}{92}\text{U} + {}^{0}_{-1}e + \bar{\nu}_e$$

• $A = 234$, $Z = 92$ (Uranium)

Step 3: Final $\alpha$ decay. $${}^{234}{92}\text{U} \rightarrow {}^{230}{90}\text{Th} + {}^{4}_{2}\alpha$$

• $A = 234 - 4 = 230$
• $Z = 92 - 2 = 90$

Answer: The resulting nuclide is Thorium-230 (${}^{230}_{90}\text{Th}$).

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Key Equations

• Nucleon Number Calculation: $A = Z + N$ (Not on data sheet)
• Alpha Decay General Equation: ${}^{A}{Z}\text{X} \rightarrow {}^{A-4}{Z-2}\text{Y} + {}^{4}_{2}\alpha$ (Not on data sheet)
• Beta-minus Decay General Equation: ${}^{A}{Z}\text{X} \rightarrow {}^{A}{Z+1}\text{Y} + {}^{0}_{-1}e + \bar{\nu}_e$ (Not on data sheet)
• Beta-plus Decay General Equation: ${}^{A}{Z}\text{X} \rightarrow {}^{A}{Z-1}\text{Y} + {}^{0}_{+1}e + \nu_e$ (Not on data sheet)
• Unified Atomic Mass Unit: $1\text{ u} = 1.66 \times 10^{-27}\text{ kg}$ (On data sheet)
• Elementary Charge: $e = 1.60 \times 10^{-19}\text{ C}$ (On data sheet)

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Common Mistakes to Avoid

• ❌ Wrong: Forgetting the antineutrino ($\bar{\nu}_e$) in $\beta^-$ decay or the neutrino ($\nu_e$) in $\beta^+$ decay.
  • ✅ Right: Always include the (anti)neutrino. It is required for the conservation of energy and lepton number.
• ❌ Wrong: Thinking that $\beta^-$ decay increases the mass of the nucleus because $Z$ increases.
  • ✅ Right: The nucleon number $A$ remains constant in $\beta$ decay. A neutron is simply replaced by a proton.
• ❌ Wrong: Stating that $\alpha$-particles are "helium atoms."
  • ✅ Right: $\alpha$-particles are helium nuclei. They have no electrons and carry a $+2e$ charge.
• ❌ Wrong: Confusing the observation of "most particles passing through" with the conclusion of "small nucleus."
  • ✅ Right: "Most passing through" $\rightarrow$ "Atom is mostly empty space." "Large angle deflection" $\rightarrow$ "Small, dense, positive nucleus."
• ❌ Wrong: Using $1.67 \times 10^{-27}\text{ kg}$ for $u$.
  • ✅ Right: Use the exact value from the data sheet ($1.66 \times 10^{-27}\text{ kg}$). $1.67$ is the approximate mass of a proton/neutron, but $u$ is defined by Carbon-12.

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Exam Tips

1. Scattering Questions: If asked to "Describe the evidence for the nuclear model," you must provide a pair: Observation + Inference. You will lose marks if you only provide the inference (e.g., "the atom is empty space") without the observation ("most $\alpha$-particles pass through").
2. Specific Charge: This is a common "hidden" calculation. Remember it is Charge $\div$ Mass. For ions, the mass is the number of nucleons $\times u$, and the charge is the net number of protons/electrons $\times e$.
3. The Neutrino's Role: If a question asks why the neutrino was predicted, the standard 2-mark answer is: (1) To account for the continuous range of energies of $\beta$-particles and (2) to ensure the conservation of energy/momentum.
4. Notation Precision: When writing decay equations, ensure the symbols for the electron (${}^{0}{-1}e$) and positron (${}^{0}{+1}e$) have the correct mass and charge numbers. Even though the mass is not zero, in the context of $A$ (nucleon number), it is represented as $0$.
5. Conservation Check: Always perform a "top and bottom" sum check on your nuclear equations. The sum of the superscripts on the left must equal the sum of the superscripts on the right. Repeat for subscripts.