describe a simple model for the nuclear atom to include protons, neutrons and orbital electrons

Simple Nuclear Model of the Atom

1. Fundamental Particles

1.1 Nucleons and Electrons

1.2 Quark Flavours (relevant to the 9702 syllabus)

1.3 Quark composition of nucleons

1.4 β‑decay at quark level (required learning outcome)

• β⁻ decay: a down quark changes into an up quark, emitting an electron and an antineutrino

  \$d \;\rightarrow\; u + e^{-} + \bar{\nu}_{e}\$

  At the nucleon level: \$n \rightarrow p^{+}+e^{-}+\bar{\nu}_{e}\$.

• β⁺ decay (positron emission): an up quark changes into a down quark, emitting a positron and a neutrino

  \$u \;\rightarrow\; d + e^{+} + \nu_{e}\$

  At the nucleon level: \$p^{+} \rightarrow n + e^{+} + \nu_{e}\$.

2. Structure of the Atom

1. Nucleus

   • Contains \$Z\$ protons and \$N\$ neutrons (collectively called nucleons).
   • Radius ≈ \$1\times10^{-15}\,\text{m}\$ (1 fm); about 10 000 times smaller than the whole atom.
   • Mass number \$A = Z + N\$.
   • Holds > 99.9 % of the atomic mass.

2. Electron cloud

   • Electrons occupy *orbitals* that, for an introductory model, are represented by concentric shells \$K\$, \$L\$, \$M\$, … with principal quantum numbers \$n = 1,2,3,\dots\$.
   • Maximum capacity of a shell: \$2n^{2}\$ electrons (e.g. \$K\$ holds 2, \$L\$ holds 8, \$M\$ holds 18).
   • Atomic radius is typically \$0.1\$–\$0.3\,\$nm (1 Å = \$10^{-10}\$ m).

3. Nuclear Notation, Isotopes & Conservation Laws

The Cambridge notation for a nuclide is \$^{A}_{Z}\text{X}\$, where:

• \$\text{X}\$ = chemical symbol,
• \$Z\$ = atomic number (number of protons),
• \$A\$ = mass number (\$Z+N\$).

In all nuclear reactions the following are conserved:

• Charge (Z) – the total number of protons before and after the reaction is the same.
• Nucleon number (A) – the total number of protons + neutrons is unchanged.

4. Types of Radioactive Decay

4.1 Alpha (α) decay

• Emission of a \$^{4}_{2}\alpha\$ particle (2 p + 2 n).

  \$^{A}{Z}\text{X}\;\rightarrow\;^{A-4}{Z-2}\text{Y}+^{4}_{2}\alpha\$

• Low penetrating power (stopped by a sheet of paper); high ionising power.

4.2 Beta (β) decay

• β⁻ decay – neutron → proton + electron + antineutrino.

  \$^{A}{Z}\text{X}\;\rightarrow\;^{A}{Z+1}\text{Y}+e^{-}+\bar{\nu}_{e}\$

• β⁺ decay (positron emission) – proton → neutron + positron + neutrino.

  \$^{A}{Z}\text{X}\;\rightarrow\;^{A}{Z-1}\text{Y}+e^{+}+\nu_{e}\$

• Medium penetrating power; stopped by a few millimetres of aluminium.

4.3 Gamma (γ) decay

• Emission of a high‑energy photon from an excited nucleus.

  \$^{A}{Z}\text{X}^{*}\;\rightarrow\;^{A}{Z}\text{X}+ \gamma\$

• No change in \$Z\$ or \$A\$.
• Very high penetrating power; dense materials (lead, several centimetres) are required for shielding.

5. Radioactive Decay Law & Half‑Life

The number of undecayed nuclei after time \$t\$ is

\$\$

N(t)=N_{0}\,e^{-\lambda t}

\$\$

• \$N_{0}\$ – initial number of nuclei.
• \$\lambda\$ – decay constant (s⁻¹).
• Activity \$A\$ (disintegrations s⁻¹) is \$A=\lambda N\$.

Half‑life:

\$\$

t_{1/2}=\frac{\ln 2}{\lambda}= \frac{0.693}{\lambda}

\$\$

Example (exam style): A sample contains \$2.0\times10^{6}\$ atoms of a radionuclide. After 10 days the activity is measured to be \$5.0\times10^{5}\$ disintegrations s⁻¹. If the initial activity was \$1.0\times10^{6}\$ disintegrations s⁻¹, calculate the half‑life.

1. Determine \$\lambda\$ from \$A=\lambda N\$: \$\lambda = A/N_{0}=1.0\times10^{6}/2.0\times10^{6}=5.0\times10^{-1}\,\text{s}^{-1}\$ (note: convert days to seconds if required).
2. Half‑life \$t_{1/2}=0.693/\lambda\approx1.39\,\$s (illustrative – the numbers can be adapted for a realistic half‑life).

6. Mass Defect and Binding Energy

The mass of a nucleus is slightly less than the sum of the masses of its separate nucleons. The difference is the mass defect \$\Delta m\$.

\$\$

\Delta m = Zm{p}+Nm{n}-m_{\text{nucleus}}

\$\$

Using Einstein’s relation \$E=mc^{2}\$, the corresponding binding energy \$E_{b}\$ is

\$\$

E_{b}= \Delta m\,c^{2}

\$\$

Typical binding energies are a few MeV per nucleon; the curve of \$E_{b}/A\$ explains why:

• Heavy nuclei release energy by fission (splitting into fragments with higher \$E_{b}/A\$).
• Light nuclei release energy by fusion (joining to form a nucleus with higher \$E_{b}/A\$).

7. Nuclear Reactions and Applications

• Fission – a heavy nucleus splits, emitting neutrons and ≈ 200 MeV per event.

  \$^{235}{92}\text{U}+^{1}{0}n\;\rightarrow\;^{141}{56}\text{Ba}+^{92}{36}\text{Kr}+3^{1}_{0}n+\text{energy}\$

• Fusion – light nuclei combine, releasing energy (e.g. deuterium‑tritium reaction).

  \$^{2}{1}\text{H}+^{3}{1}\text{H}\rightarrow^{4}{2}\text{He}+^{1}{0}n+\;17.6\;\text{MeV}\$

• Medical applications

  • Positron Emission Tomography (PET) – uses β⁺ emitters such as \$^{18}_{9}\text{F}\$.
  • Radiotherapy – high‑energy γ‑rays from \$^{60}_{27}\text{Co}\$ or electron beams.
  • Diagnostic X‑rays – produced by bremsstrahlung when high‑energy electrons are decelerated in a target.

• Industrial & safety – Geiger–Müller counters, cloud chambers, shielding calculations and neutron activation analysis all rely on the concepts above.

8. Practical Skills (AO3)

• Always wear lab coat, gloves, and eye protection when handling radioactive material.
• Use a calibrated Geiger–Müller tube; record background counts and subtract them from sample counts.
• To determine a half‑life experimentally, plot \$\ln(N)\$ versus time – the slope equals \$-\lambda\$.

9. Key Points to Remember

• The nucleus contains almost all the atomic mass but occupies only ~10⁻⁵ of the atomic volume.
• Protons define the element (\$Z\$); neutrons affect stability and give rise to isotopes.
• Electrons balance the nuclear charge in a neutral atom and occupy discrete shells following the \$2n^{2}\$ rule.
• α, β⁻, β⁺ and γ radiations differ in composition, charge, penetrating power and ionising ability.
• In every nuclear reaction, both charge (\$Z\$) and nucleon number (\$A\$) are conserved.
• Radioactive decay follows an exponential law; the half‑life is a characteristic constant for each radionuclide.
• Mass defect and binding energy explain why energy is released in fission (heavy nuclei) and fusion (light nuclei).
• Understanding nuclear processes underpins applications in medicine, industry, energy generation and radiation safety.

10. Suggested Classroom Diagram