HookPauli invented an invisible particle rather than break physics
In December 1930 Wolfgang Pauli sent a letter to a radioactivity meeting in Tübingen that opened 'Dear radioactive ladies and gentlemen'. Beta decay had a scandal on its hands: the emitted electrons carried a continuous smear of energies, from almost nothing up to a maximum — yet a nucleus splitting into just two pieces must, by conservation of energy and momentum, give the electron one fixed energy every time. Some physicists, Niels Bohr among them, were prepared to let conservation of energy fail inside the nucleus. Pauli proposed what he called a desperate remedy instead: a third particle leaves the nucleus too — neutral, nearly massless, invisible to every detector of the day — quietly carrying off the missing energy. He fretted that he had postulated something no experiment could ever see. Fermi named it the neutrino, the little neutral one. Twenty-six years later, in 1956, Cowan and Reines parked detectors beside the Savannah River reactor, caught the antineutrinos streaming out, and telegraphed Pauli: it was real.
That episode is the whole of section 3.2 in miniature. Physicists trust conservation laws — energy, charge, baryon number, lepton number, and (in strong interactions) strangeness — so deeply that inventing an undetectable particle beat abandoning one. This section hands you the cast list: nucleons, leptons, quarks, antiparticles and photons; the four interactions and their exchange particles; and the bookkeeping rules that decide which reactions nature permits. Then it turns the same machinery on light and matter themselves — the photoelectric effect forces light to behave as particles, and electron diffraction forces electrons to behave as waves. Nearly every mark is careful counting, done fast.
ModelThe nucleus by numbers: A, Z and specific charge
An atom is a nucleus of protons and neutrons (collectively nucleons) orbited by electrons. The notation \({}^{A}_{Z}X\) compresses everything: \(Z\), the proton number, fixes which element you have and the charge \(+Ze\) of the nucleus; \(A\), the nucleon number, counts protons plus neutrons, so the neutron count is \(A-Z\). A neutral atom carries \(Z\) electrons; an ion has gained or lost a few. Isotopes share \(Z\) but differ in \(A\) — identical chemistry, different nuclear behaviour, which is why carbon-12 sits stably in your body while carbon-14 decays and dates archaeology.
The first calculation AQA drills is specific charge: total charge divided by total mass, in \(\text{C kg}^{-1}\). It is a fingerprint. The electron's, at \(1.76\times10^{11}\ \text{C kg}^{-1}\), is the largest of any particle — a huge charge-to-mass ratio was the first clue, back in Thomson's day, that something far lighter than an atom existed. The proton manages \(9.58\times10^{7}\ \text{C kg}^{-1}\); neutral atoms score zero. The exam habit: build charge from the count of protons minus electrons times \(e = 1.60\times10^{-19}\) C, build mass from \(A\) times the nucleon mass \(1.67\times10^{-27}\) kg, and ignore the electrons' mass — at roughly 1/1800 of a nucleon each, they change nothing at two significant figures.
Find the specific charge of an \({}^{27}_{13}\text{Al}^{3+}\) ion. Charge: three electrons missing, so \(Q = +3e = 3 \times 1.60\times10^{-19} = 4.80\times10^{-19}\) C. Mass: \(27 \times 1.67\times10^{-27} = 4.51\times10^{-26}\) kg (the ten remaining electrons contribute about 0.02% — ignore them). Specific charge \(= 4.80\times10^{-19} \div 4.51\times10^{-26} = 1.1\times10^{7}\ \text{C kg}^{-1}\). Sense-check: smaller than the proton's \(9.58\times10^{7}\) because 27 nucleons share only three units of charge, and dwarfed by the electron's \(1.76\times10^{11}\). If your answer beats the electron's, you have divided the wrong way round — the most common error the mark scheme anticipates.
ModelThe strong force, and the three ways nuclei fall apart
Squeeze protons into a femtometre-sized nucleus and Coulomb's law demands they fly apart. They do not, because a second force outguns electrostatic repulsion at short range: the strong nuclear force. Its profile is the exam's favourite sketch — strongly repulsive below about 0.5 fm (so nuclei cannot collapse), attractive from roughly 0.5 fm out to about 3 fm, and effectively zero beyond that. It acts on protons and neutrons alike, charge is irrelevant to it, and its short range is why very large nuclei struggle: distant protons still feel every other proton's repulsion, but only their nearest neighbours' strong attraction.
Unstable nuclei rebalance in three ways, and you must write each as a balanced equation — top row (A) and bottom row (Z) summing equally on both sides. Very heavy nuclei shed bulk by alpha emission: \({}^{238}_{92}\text{U} \to {}^{234}_{90}\text{Th} + {}^{4}_{2}\alpha\). Neutron-rich nuclei use beta-minus: a neutron becomes a proton, emitting a fast electron and an electron antineutrino — \({}^{90}_{38}\text{Sr} \to {}^{90}_{39}\text{Y} + \beta^{-} + \bar{\nu}_e\) (strontium-90 is a real fission product; its 29-year half-life is why it haunts reactor accidents). Proton-rich nuclei use beta-plus: a proton becomes a neutron, emitting a positron and a neutrino. The (anti)neutrino carries \(A=0\), \(Z=0\) — invisible to the balancing act, which is exactly why Pauli needed the continuous energy spectrum to infer it. Forgetting it is the most reliably dropped mark in the whole section.
ModelAntimatter, photons, and the exchange rate between mass and energy
Electromagnetic radiation travels as photons, packets of energy \(E = hf = hc/\lambda\) — the single most used equation in this section. Matter, meanwhile, comes in mirror pairs: every particle has an antiparticle with the same mass and rest energy but opposite charge (and opposite baryon number, lepton number and strangeness). Dirac's equations predicted the positron in 1928; Carl Anderson photographed one curving the wrong way through a cloud chamber in 1932. Rest energy is the mass-energy a particle has sitting still, quoted in MeV: 0.511 MeV for an electron or positron.
The two showpiece processes convert between the currencies. Annihilation: a particle meets its antiparticle and the pair becomes two photons — two, travelling in opposite directions, because a single photon could not conserve momentum. Each photon carries at least the rest energy of one particle. Hospitals run this reaction on purpose: in a PET scan, a patient receives glucose tagged with fluorine-18, a positron emitter; each positron annihilates with a tissue electron and the detector ring waits for pairs of 0.511 MeV photons arriving back-to-back, triangulating exactly where the glucose is being burned. Pair production is the reverse: a photon passing near a nucleus (which recoils, conserving momentum) materialises into a particle–antiparticle pair — possible only if the photon carries at least the pair's combined rest energy, with any surplus becoming kinetic energy.
Minimum photon frequency to create an electron–positron pair: the photon must supply both rest energies, \(E_{min} = 2 \times 0.511 = 1.022\) MeV. Convert: \(1.022\times10^{6} \times 1.60\times10^{-19} = 1.64\times10^{-13}\) J. Then \(f = E/h = 1.64\times10^{-13} \div 6.63\times10^{-34} = 2.47\times10^{20}\) Hz — deep in the gamma range, which is why pair production is a gamma-ray phenomenon and not something a desk lamp does. Run it backwards for PET: each annihilation photon carries 0.511 MeV, so \(f = (0.511\times10^{6} \times 1.60\times10^{-19}) \div 6.63\times10^{-34} = 1.23\times10^{20}\) Hz. The two conversions — MeV to joules, then joules to frequency — are the entire skill.
MechanismFour interactions, four messengers
Modern physics models every force as particles swapping exchange particles (gauge bosons) — momentum couriers between the interacting particles. Four interactions cover everything. Electromagnetic: acts on anything charged, infinite range, carried by virtual photons. Weak: acts on all particles including neutrinos, range about \(10^{-18}\) m because its carriers — the massive \(W^{+}\) and \(W^{-}\) bosons — are heavy; it is the only interaction that changes quark flavour, so every beta decay is its work. Strong: binds hadrons and nuclei, range about \(10^{-15}\) m; at A-level the nucleon–nucleon attraction is the Yukawa picture whose predicted exchange particle, the pion, turned up in cosmic rays in 1947. Gravity: negligible for particles; its hypothetical carrier, the graviton, has never been observed.
AQA examines this through Feynman diagrams, and marks them on conventions: straight lines with arrows for particles, a wavy line for the exchange particle, every line labelled, and charge conserved at each junction. Four weak-interaction diagrams are compulsory. Beta-minus: a neutron becomes a proton, the \(W^{-}\) materialising into \(e^{-} + \bar{\nu}_e\). Beta-plus: proton to neutron via \(W^{+}\), giving \(e^{+} + \nu_e\). Electron capture: a proton absorbs an inner atomic electron, \(p + e^{-} \to n + \nu_e\). Electron–proton collision: same particles in and out as capture, but the electron arrives as a projectile. Plus the electromagnetic staple: two electrons repelling by exchanging a virtual photon. Check every vertex before moving on — a \(W^{-}\) emitted where charge does not balance is the standard planted error in 'spot the mistake' questions.
ModelThe particle zoo, the quark code and the conservation checklist
Hadrons feel the strong interaction and are built from quarks. They split into baryons — three quarks, like the proton (uud) and neutron (udd) — and mesons — a quark–antiquark pair, like pions and kaons. The proton is the only stable baryon; every other baryon ends its decay chain as a proton. Leptons are fundamental, immune to the strong force: the electron, the muon (a heavier electron that decays), their neutrinos, and the four antiparticles. Muons rain down on you now, made by cosmic rays hitting the upper atmosphere.
The quark card is short. Up: charge \(+\tfrac{2}{3}e\). Down: \(-\tfrac{1}{3}e\). Strange: \(-\tfrac{1}{3}e\) and strangeness −1. Each carries baryon number \(+\tfrac{1}{3}\); antiquarks flip every sign. Strange particles like kaons earned the name by behaving oddly: created in pairs through the strong interaction (which conserves strangeness) yet decaying slowly through the weak interaction (which can change strangeness by ±1). Beta-minus decay in quark language is one flavour flip: \(d \to u + e^{-} + \bar{\nu}_e\), the \(W^{-}\) doing the carrying.
Whether any proposed reaction happens is a checklist, not an opinion. Conserve charge; conserve baryon number; conserve electron-lepton number and muon-lepton number separately; conserve strangeness in strong interactions (weak decays may break it by one). Run the list line by line, and say which law fails — 'not allowed' without a named law scores nothing.
Test \(\bar{\nu}_\mu + p \to n + \mu^{+}\) — the reaction by which muon antineutrinos are actually detected. Charge: \(0 + 1 \to 0 + 1\) ✓. Baryon number: \(0 + 1 \to 1 + 0\) ✓. Muon-lepton number: the antineutrino counts −1, the antimuon −1, so \(-1 \to -1\) ✓. Electron-lepton number: \(0 \to 0\) ✓. Strangeness: \(0 \to 0\) ✓. Allowed — and, with a neutrino involved and a quark changing flavour (u to d), it is a weak interaction. Now flip one ingredient: \(\nu_\mu + p \to n + \mu^{+}\). Muon-lepton number reads \(+1 \to -1\) ✗. Forbidden, regardless of energy. One sign kills the whole reaction — which is precisely how examiners construct these questions.
ModelThe photoelectric effect: light arrives in lumps
Charge a zinc plate negatively on a gold-leaf electroscope and shine ultraviolet on it: the leaf falls as electrons are ejected. Flood it with visible light instead — any brightness you like — and nothing happens. Wave theory cannot survive this. If light delivered energy continuously, any frequency should eventually eject electrons, brighter light should mean faster electrons, and dim light should mean a delay while energy accumulates. Experiment says no three times: there is a sharp threshold frequency \(f_0\) below which no electrons leave at any intensity; emission is instantaneous; and the maximum kinetic energy rises with frequency but ignores intensity, which only changes how many electrons leave per second.
Einstein's 1905 answer: light is absorbed in whole photons, \(E = hf\), one photon to one electron, all or nothing. The work function \(\phi\) is the minimum energy needed to escape the metal's surface, so the energy budget reads \(hf = \phi + E_{k(max)}\) — 'max' because electrons from deeper in the metal pay more than \(\phi\). The stopping potential \(V_s\) is the reverse potential difference that just halts the fastest electrons, giving \(eV_s = E_{k(max)}\) and a way to measure the whole equation. Millikan spent a decade trying to prove Einstein wrong and instead confirmed the equation to high precision; the 1921 Nobel Prize citation for Einstein names the photoelectric effect, not relativity.
Sodium has work function \(\phi = 2.28\) eV. Shine blue light of wavelength 450 nm on it. Photon energy: \(E = hc/\lambda = (6.63\times10^{-34} \times 3.00\times10^{8}) \div 4.50\times10^{-7} = 4.42\times10^{-19}\) J \(= 2.76\) eV. Then \(E_{k(max)} = 2.76 - 2.28 = 0.48\) eV \(= 7.7\times10^{-20}\) J, so the stopping potential is 0.48 V. Threshold frequency: \(f_0 = \phi/h = (2.28 \times 1.60\times10^{-19}) \div 6.63\times10^{-34} = 5.50\times10^{14}\) Hz, equivalent to about 545 nm — green. So green and blue light eject electrons from sodium; orange and red never will, however bright the source. Keep the working in eV until the final conversion and the arithmetic stays humane.
ModelElectrons in, photons out: the electronvolt, energy levels and line spectra
Atomic physics is priced in electronvolts: the energy one electron gains crossing a potential difference of 1 V, so \(1\ \text{eV} = 1.60\times10^{-19}\) J. Convert at the last moment, not the first.
Electrons bound in an atom occupy discrete energy levels — negative values, because a bound electron sits below the zero of a free one. A free electron colliding with the atom can promote an atomic electron to a higher level, but only by delivering at least an exact gap: that is excitation. Deliver the full depth of the level and the atomic electron escapes entirely: ionisation. Excited atoms relax almost immediately, emitting a photon of exactly the energy difference, \(hf = E_1 - E_2\). Because every element has its own ladder of levels, its emitted wavelengths form a unique line spectrum — the fingerprint that let astronomers identify helium in the Sun's spectrum in 1868, twenty-seven years before anyone found it on Earth.
The fluorescent tube runs the whole story at once, and AQA asks for it by name: a high potential difference accelerates free electrons through low-pressure mercury vapour; collisions excite (and ionise) mercury atoms; de-exciting atoms emit ultraviolet photons; the white phosphor coating absorbs the UV and its own electrons de-excite in smaller steps, re-emitting the energy as visible light. Every arrow in that chain is a markable point.
Hydrogen's level \(n=3\) sits at −1.51 eV and \(n=2\) at −3.40 eV. An electron dropping between them emits \(\Delta E = (-1.51) - (-3.40) = 1.89\) eV \(= 1.89 \times 1.60\times10^{-19} = 3.02\times10^{-19}\) J. Wavelength: \(\lambda = hc/\Delta E = (6.63\times10^{-34} \times 3.00\times10^{8}) \div 3.02\times10^{-19} = 6.58\times10^{-7}\) m ≈ 658 nm — the red hydrogen-alpha line that colours emission nebulae in astronomy photographs. Two traps: both levels are negative and the photon carries the difference, and the eV-to-joule conversion must happen before \(h\) gets involved.
MechanismWave–particle duality: the electron learns to diffract
By this point the section has argued both sides. The photoelectric effect only makes sense if light is particulate; interference and diffraction (waiting in section 3.3) only make sense if light is a wave. Both are true, and which face nature shows depends on the experiment — that is wave–particle duality. De Broglie's 1923 conjecture completed the symmetry: if waves act as particles, particles should act as waves, with wavelength \(\lambda = h/mv\) — inversely proportional to momentum.
The confirming experiment is beautifully direct. Fire electrons through a thin polycrystalline graphite film onto a phosphor screen: concentric bright rings appear, exactly the diffraction pattern waves produce when they meet regularly spaced scattering centres — and carbon's atomic spacing, around \(10^{-10}\) m, matches the de Broglie wavelength of electrons accelerated through a few kilovolts. The clincher is the control knob: raise the accelerating voltage and the electrons speed up, \(mv\) grows, \(\lambda\) shrinks, and the rings visibly tighten — precisely what the equation orders. Particles do not diffract; waves do; electrons diffract. The section closes where the neutrino opened it: nature keeps its accounts balanced, but it is under no obligation to be intuitive.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Conservation questions are marked as bookkeeping, so present them as bookkeeping: a line per quantity — charge, baryon number, electron-lepton number, muon-lepton number, strangeness — with totals each side and a tick or cross. Name the law that fails; 'the reaction is not possible' without the culprit scores nothing. Remember the asymmetry the examiners exploit: strangeness must balance in strong interactions but may change by ±1 in weak ones, and lepton numbers are conserved per flavour, not in aggregate.
Decay equations earn their marks for balance and completeness: nucleon numbers summing, proton numbers summing, and the electron antineutrino present in every beta-minus (neutrino in every beta-plus). Feynman diagrams are marked on conventions — arrows on particle lines, the exchange boson wavy and labelled, charge conserved at each vertex. For the photoelectric effect, know the graph of maximum kinetic energy against frequency as an object: gradient h, x-intercept f₀, y-intercept −φ. AQA asks for one of the three almost every series, and the 6-mark levels question 'explain why wave theory cannot account for these observations' wants three contrasts — threshold frequency, instantaneous emission, intensity affecting number not energy — each stated as prediction versus observation.
On the AO balance, this section is unusually recall-heavy (the quark table, boson list and rest energies are AO1), but the money is in AO2 application: eV-to-joule conversions done at the right moment, 'show that' answers carried to one extra significant figure, and data-sheet rest energies quoted rather than half-remembered. Practise writing the fluorescent-tube chain in four sentences — accelerate, excite by collision, de-excite emitting UV, phosphor converts to visible — because as a levels-marked extended response it rewards sequence and correct vocabulary over length.