Atomic and Nuclear Physics (SL) — IB Physics SL SL Study Guide

For: IB Physics SL candidates sitting IB Physics SL.

Covers: Rutherford-Bohr atomic structure, atomic spectra, mass defect and binding energy, radioactive decay and half-life, and nuclear fission and fusion reactions, fully aligned to the latest IB DP Physics SL syllabus.

You should already know: IGCSE Physics, basic algebra.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Physics SL style for educational use. They are not reproductions of past IBO papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official IBO mark schemes for grading conventions.

1. What Is Atomic and Nuclear Physics?

Atomic and nuclear physics is the branch of physics focused on the structure, properties, and interactions of atoms and their dense central nuclei. For IB Physics SL, this is core Topic 7, tested on both Paper 1 (multiple choice) and Paper 2 (structured response), often combined with energy production and waves topics for multi-part questions worth 5-10 marks. It bridges classical physics and introductory quantum mechanics, with many questions requiring both qualitative explanation and quantitative calculation.

2. Atomic structure — Rutherford-Bohr model

Before 1911, the accepted atomic model was J.J. Thomson's "plum pudding" model, which described the atom as a uniform sphere of positive charge with negatively charged electrons embedded within it. Ernest Rutherford's gold foil experiment disproved this model, leading to the first nuclear atomic model:

• Rutherford fired positively charged alpha particles at a thin sheet of gold foil, measuring their deflection angles
• 3 key observations and conclusions:

1. Most alpha particles passed through undeflected: atoms are mostly empty space (radius ~$10^{-10}$ m)
2. A small fraction were deflected at large angles: a small, dense, positively charged nucleus sits at the center of the atom
3. ~1 in 8000 particles bounced back directly: the nucleus contains nearly all the atom's mass, with a radius of ~$10^{-15}$ m

Rutherford's model had a critical flaw: classical physics predicted accelerating orbiting electrons would emit radiation, lose energy, and collapse into the nucleus in microseconds. Niels Bohr revised the model in 1913 with 3 postulates:

1. Electrons orbit the nucleus in fixed, discrete stationary states with specific energy values, no radiation emitted while in these orbits
2. Electrons only emit or absorb energy when moving between stationary states
3. Angular momentum of orbiting electrons is quantized: $$mvr=n\frac{h}{2\pi }$$ where $m$ = electron mass, $v$ = orbital speed, $r$ = orbital radius, $n$ = principal quantum number (integer 1,2,3...), $h$ = Planck's constant.

Worked Example: The Bohr radius (radius of the $n=1$ orbit for hydrogen) is $5.29\times 10^{-11}$ m. If orbital radius scales with $n^2$, calculate the radius of the $n=3$ orbit. Solution: $r_3=r_1\times n^2=5.29\times 10^{-11}\times 9=4.76\times 10^{-10}$ m. Exam tip: Examiners regularly ask to link Rutherford's observations directly to conclusions, so avoid listing observations without explaining what they reveal about atomic structure.

3. Atomic spectra

Atomic spectra are the unique patterns of light emitted or absorbed by atoms, providing direct evidence for Bohr's discrete energy levels. There are 3 core types of spectra:

1. Continuous spectrum: Produced by hot, dense objects (e.g. the Sun's core, incandescent light bulbs), contains all wavelengths of visible light with no gaps
2. Emission spectrum: Produced by hot, low-pressure gas, appears as bright discrete lines on a dark background. Lines correspond to electrons falling from higher to lower energy levels, emitting photons of specific energy
3. Absorption spectrum: Produced when cool, low-pressure gas sits between a continuous light source and an observer, appears as dark discrete lines on a bright continuous background. Lines correspond to electrons absorbing photons of specific energy to jump from lower to higher energy levels

The energy of the photon emitted or absorbed in a transition equals the difference between the two energy levels: $$\Delta E=E_{higher}-E_{lower}=hf=\frac{hc}{\lambda }$$ where $f$ = photon frequency, $\lambda$ = photon wavelength, $c$ = speed of light. For hydrogen, transitions to $n=1$ produce the ultraviolet Lyman series, transitions to $n=2$ produce the visible Balmer series, and transitions to $n=3$ produce the infrared Paschen series.

Worked Example: A hydrogen electron transitions from $n=3$ (energy = -1.51 eV) to $n=2$ (energy = -3.40 eV). Calculate the wavelength of the emitted photon in nanometers. Solution: First calculate the energy difference: $\Delta E=-1.51-(-3.40)=1.89$ eV. Use the shortcut $hc=1240$ eV·nm to avoid converting to joules: $\lambda =\frac{hc}{\Delta E}=\frac{1240}{1.89}=656$ nm, which corresponds to red light in the Balmer series.

4. Mass defect and binding energy

Protons and neutrons in the nucleus are collectively called nucleons. The unified atomic mass unit (u) is defined as 1/12 the mass of a neutral carbon-12 atom, with $1\text{u}=1.6605\times 10^{-27}\text{kg}=931.5\text{MeV}/c^2$ (this conversion factor is given in your exam data booklet).

The mass defect ($\Delta m$) is the difference between the total mass of individual, separated nucleons and the mass of the bound nucleus: $$\Delta m=Z{m}_p+(A-Z)m_n-m_{nucleus}$$ where $Z$ = atomic number (number of protons), $A$ = mass number (total number of nucleons), $m_p$ = mass of a proton, $m_n$ = mass of a neutron. The mass defect is always positive, as bound nucleons have less total mass than free nucleons.

Binding energy (BE) is the energy required to split a nucleus into its individual nucleons, equal to the mass defect converted to energy via Einstein's mass-energy equivalence: $$BE=\Delta m{c}^2$$ The binding energy per nucleon ($BE/A$) is a measure of nuclear stability: higher values mean more stable nuclei. The peak of the binding energy per nucleon curve is at iron-56, the most stable naturally occurring nucleus.

Worked Example: Calculate the binding energy per nucleon for helium-4, given: $m_p=1.00728$ u, $m_n=1.00866$ u, mass of helium-4 nucleus = 4.00153 u. Solution: Helium-4 has 2 protons and 2 neutrons. Total mass of free nucleons = $(2\times 1.00728)+(2\times 1.00866)=4.03188$ u. Mass defect $\Delta m=4.03188-4.00153=0.03035$ u. Total binding energy = $0.03035\times 931.5=28.27$ MeV. Binding energy per nucleon = $28.27/4=7.07$ MeV per nucleon.

5. Radioactive decay and half-life

Radioactive decay is a spontaneous, random process: it is impossible to predict when an individual unstable nucleus will decay, and external factors like temperature, pressure, or chemical state do not affect the decay rate of a sample. There are 3 core decay types tested in SL:

The decay law describes the exponential decrease of undecayed nuclei in a sample over time: $$N=N_0{e}^{-\lambda t}$$ where $N$ = number of undecayed nuclei at time $t$, $N_0$ = initial number of undecayed nuclei, $\lambda$ = decay constant (probability of decay per nucleus per unit time). The half-life ($T_{1/2}$) is the time taken for half the initial number of nuclei in a sample to decay, related to the decay constant by: $$T_{1/2}=\frac{\ln 2}{\lambda }$$ This relationship also applies to activity (number of decays per second, measured in becquerels, Bq) and total mass of the radioactive sample, as both are proportional to the number of undecayed nuclei.

Worked Example: Iodine-131, used in medical thyroid treatments, has a half-life of 8.0 days. If a patient is given a sample with initial activity of 800 Bq, what is the activity of the iodine in their body after 24 days? Solution: Number of half-lives elapsed = $24/8=3$. Activity after $n$ half-lives = $A_0\times (1/2{)}^n=800\times (1/2{)}^3=800/8=100$ Bq.

6. Nuclear reactions — fission, fusion

Both fission and fusion are nuclear reactions that convert mass to energy, with the total energy released equal to $\Delta E=\Delta m{c}^2$, where $\Delta m$ is the difference between the total mass of reactants and total mass of products.

Nuclear fission is the process where a heavy, unstable nucleus splits into two lighter nuclei of roughly equal mass, releasing energy and extra neutrons. Fission reactions power all commercial nuclear power plants and atomic bombs. A common fission reaction for uranium-235 is: $${}_0^{1}n{+}_{92}^{235}U{\to }_{56}^{141}Ba{+}_{36}^{92}Kr+3_0^{1}n+\text{energy}$$ The extra neutrons emitted can trigger further fission reactions, creating a self-sustaining chain reaction. Controlled chain reactions are used in power plants, while uncontrolled chain reactions are used in nuclear weapons.

Nuclear fusion is the process where two light nuclei combine to form a heavier, more stable nucleus, releasing far more energy per unit mass than fission. Fusion powers all stars, including our Sun, and requires extremely high temperatures (~100 million °C) and pressure to overcome the electrostatic repulsion between positively charged nuclei. A common fusion reaction is the deuterium-tritium reaction being tested for experimental fusion power plants: $${}_1^{2}H{+}_1^{3}H{\to }_2^{4}He{+}_0^{1}n+\text{energy}$$ Fusion produces no long-lived radioactive waste and uses abundant fuel sources, but has not yet been scaled for commercial power generation.

7. Common Pitfalls (and how to avoid them)

• Wrong move: Calculating mass defect as the mass of the nucleus minus the mass of free nucleons, resulting in a negative value. Why students do it: Mixing up the order of subtraction. Correct move: Always subtract the bound nucleus mass from the total mass of free nucleons: $\Delta m=(\text{mass of nucleons})-(\text{mass of nucleus})$, which will always be positive.
• Wrong move: Assuming beta decay changes the mass number of the nucleus. Why students do it: Confusing beta and alpha decay properties. Correct move: Only alpha decay changes mass number (decreases by 4). Beta decay converts a neutron to a proton, so mass number stays the same while atomic number increases by 1.
• Wrong move: Using kg to calculate mass defect instead of unified atomic mass units. Why students do it: Habit of using SI units for all calculations. Correct move: Use the $1\text{u}=931.5\text{MeV}/c^2$ conversion factor to calculate binding energy directly in MeV, saving time and reducing arithmetic errors.
• Wrong move: Stating that fusion is used in commercial nuclear power plants. Why students do it: Mixing up fission and fusion applications. Correct move: All operational commercial nuclear plants use fission; fusion is still in experimental stages.
• Wrong move: Using a linear model for decay instead of the exponential formula for non-integer half-life values. Why students do it: Overreliance on integer half-life practice questions. Correct move: For time values that are not multiples of $T_{1/2}$, use $N=N_0(1/2{)}^{t/T_{1/2}}$ or the exponential decay law with the decay constant.

8. Practice Questions (IB Physics SL Style)

Question 1

(a) Outline two conclusions Rutherford drew from his alpha particle scattering experiment, linking each to an observation. (2 marks) (b) A hydrogen electron transitions from the $n=4$ energy level (-0.85 eV) to the $n=1$ ground state (-13.6 eV). Calculate the frequency of the emitted photon, giving your answer in Hz. (3 marks)

Solution

(a) 1. Observation: Most alpha particles passed through the foil undeflected → Conclusion: The atom is mostly empty space. 2. Observation: A small number of alpha particles were deflected at angles greater than 90° → Conclusion: The atom has a small, dense, positively charged nucleus containing most of the atom's mass. (1 mark per linked conclusion) (b) Step 1: Calculate energy difference: $\Delta E=-0.85-(-13.6)=12.75$ eV = $12.75\times 1.60\times 10^{-19}=2.04\times 10^{-18}$ J. Step 2: Rearrange $\Delta E=hf$ to solve for frequency: $f=\Delta E/h=2.04\times 10^{-18}/6.63\times 10^{-34}=3.08\times 10^{15}$ Hz. (1 mark for energy difference, 1 mark for unit conversion, 1 mark for final answer)

Question 2

(a) Define binding energy per nucleon. (2 marks) (b) The mass of a lithium-7 nucleus is 7.01436 u. Given $m_p=1.00728$ u, $m_n=1.00866$ u, calculate the binding energy per nucleon for lithium-7 in MeV. (3 marks)

Solution

(a) Binding energy per nucleon is the average energy required to remove one nucleon from a bound nucleus, equal to the total binding energy of the nucleus divided by its mass number (total number of nucleons). (1 mark for energy per nucleon definition, 1 mark for link to total binding energy) (b) Step 1: Lithium-7 has 3 protons and 4 neutrons, total mass of free nucleons = $(3\times 1.00728)+(4\times 1.00866)=7.05648$ u. Step 2: Mass defect $\Delta m=7.05648-7.01436=0.04212$ u. Step 3: Total binding energy = $0.04212\times 931.5=39.23$ MeV. Binding energy per nucleon = $39.23/7=5.60$ MeV per nucleon. (1 mark for mass defect, 1 mark for total binding energy, 1 mark for final per-nucleon value)

Question 3

(a) The half-life of cobalt-60, used in radiation therapy, is 5.27 years. Calculate its decay constant in units of s⁻¹. (2 marks) (b) A cobalt-60 sample has an initial activity of 1600 Bq. What is its activity after 15 years? (2 marks)

Solution

(a) Step 1: Convert half-life to seconds: $5.27$ years = $5.27\times 365\times 24\times 3600=1.66\times 10^8$ s. Step 2: $\lambda =\ln 2/T_{1/2}=0.693/1.66\times 10^8=4.17\times 10^{-9}$ s⁻¹. (1 mark for unit conversion, 1 mark for final decay constant) (b) Step 1: Number of half-lives elapsed = $15/5.27=2.846$. Step 2: Activity = $1600\times (1/2{)}^{2.846}=1600\times 0.139=222$ Bq. (Accept values between 210 and 230 Bq for full marks)

9. Quick Reference Cheatsheet

10. What's Next

This topic connects directly to multiple other areas of the IB Physics SL syllabus: you will apply binding energy and mass-energy equivalence concepts to Topic 8 (Energy Production) when studying nuclear power generation and experimental fusion technology, and use photon energy calculations from atomic spectra in Topic 4 (Waves) when analyzing electromagnetic radiation. Understanding radioactive decay is also a required skill for practical assessment investigations involving radiation measurement, and appears frequently in Paper 3 data analysis questions that test your ability to interpret exponential decay graphs.

If you struggle with any of the concepts in this guide, from explaining Rutherford's experiment conclusions to solving non-integer half-life problems, you can ask Ollie, our AI tutor, for personalized explanations, extra practice questions, or step-by-step walkthroughs tailored to your learning gaps. You can also find more topic-specific study guides and full timed mock exams on the homepage to build confidence ahead of your IB Physics SL exams.