Home / Topics / Atomic & Nuclear Physics
Atomic & Nuclear Physics
Atomic models, spectra, X-rays, radioactivity, fission and fusion. 63 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Atomic and nuclear physics explains the structure of atoms and the nuclei at their centres. This area follows the development of atomic models, from Thomson and Rutherford to Bohr, explains line spectra and X-rays, and covers radioactivity, half-life, binding energy, nuclear fission and fusion. These ideas underlie spectroscopy, medical imaging, nuclear power and the energy of stars.
How the topics connect
Atomic and nuclear physics start with the structure of atoms and move to the nucleus. Spectra, energy levels and X-rays show how electrons are arranged, while radioactivity, binding energy, fission and fusion describe the nucleus. Together they explain the origin of the elements and the power of nuclear technology.
Real-world uses
Use radioactive tracers in PET scans and X-rays in diagnosis, and understand the dose and shielding needed to protect patients and staff.
Estimate the age of archaeological and geological samples from the remaining fraction of carbon-14, using the half-life.
Explain how controlled fission in a reactor releases energy from the binding energy difference between heavy nuclei and their fragments.
Identify the chemical composition of stars and galaxies from the lines in their spectra, each one a fingerprint of an element.
Hospitals use radioactive tracers, PET and MRI scans and radiotherapy, all based on nuclear and atomic physics.
Nuclear power plants use fission, while carbon-14 dating uses radioactive decay to date ancient remains.
Thomson's plum pudding
Early atom model with electrons embedded in positive charge.
J.J. Thomson found the electron in 1897 and proposed atoms as a positive sphere with electrons embedded like plums. It explained neutrality but not spectra or scattering.
Cathode rays were deflected by E and B fields to give e/m.
Replaced by Rutherford's model.
Rutherford's scattering experiment
Alpha particles revealed a tiny dense nucleus.
Geiger and Marsden fired alpha particles at gold foil; most passed straight through but about 1 in 8000 bounced back, showing a tiny, dense, positive nucleus (~10⁻¹⁵ m) in an atom 10⁻¹⁰ m across.
Atoms are mostly empty space.
Backscattering was impossible for a plum pudding atom.
Rutherford model limits
Classical orbiting electrons would spiral into the nucleus.
A classical electron orbiting a nucleus would radiate and spiral in within 10⁻¹¹ s, and would give a continuous spectrum. Atoms are stable and emit lines, which classical physics can't explain.
Led directly to Bohr's quantisation.
Fails because it ignores quantum effects.
Bohr model
Electrons occupy quantised orbits.
bohrElectrons move in stable circular orbits with quantised angular momentum and emit or absorb a photon only when jumping between orbits: hf = E_i − E_f.
Accounts for the hydrogen spectrum accurately.
Fails for atoms with more than one electron.
Bohr's postulates
Quantised angular momentum and photon emission by transitions.
L = nħ(1) Electrons stay in stationary orbits without radiating, (2) angular momentum L = nħ, (3) transitions emit or absorb photons with E = hf.
n = 1 is the ground state.
The model treats orbits as classical paths, which quantum mechanics later refined.
Radius of Bohr orbit
Orbit radius grows as n².
r_n = 0.529 n² År_n = n²a₀ with a₀ = 0.529 Å, the Bohr radius. Orbits grow quickly with n.
n = 3: 4.76 Å.
Rydberg atoms with n = 100 are micrometres across.
Energy of hydrogen levels
Binding energy falls as 1/n².
E_n = −13.6/n² eVE_n = −13.6 eV/n²: negative because bound; ground state −13.6, first excited −3.4 eV. Level spacing shrinks as n increases and converges to 0 at ionisation.
Ionisation from n = 2 needs 3.4 eV.
Z² scaling for hydrogen-like ions.
Hydrogen spectrum
Lyman, Balmer and Paschen series.
1/λ = R(1/n₁²−1/n₂²)Photons emitted when electrons fall: 1/λ = R(1/n_f² − 1/n_i²). Series: Lyman (to n = 1, UV), Balmer (n = 2, visible), Paschen (n = 3, IR).
Balmer Hα (3→2): 656 nm red.
Each series ends at a series limit.
Rydberg constant
Constant in the spectral formula.
R = 1.097×10⁷ m⁻¹R = 1.097×10⁷ m⁻¹ appears in the spectral formula; theoretically R = mₑe⁴/(8ε₀²h³c).
Lyman limit: λ = 1/R = 91.2 nm.
Small correction for nuclear mass.
Absorption and emission spectra
Atoms emit and absorb light at specific wavelengths.
Hot gases emit light at discrete wavelengths (emission lines); cool gas in front of a continuous source absorbs the same wavelengths (dark lines). Each element has a unique pattern.
Fraunhofer lines in sunlight reveal the Sun's composition.
Helium was discovered in the Sun first.
Ionisation energy
Energy needed to remove an electron.
Energy needed to remove an electron: 13.6 eV for hydrogen, 24.6 eV for helium (first). It rises across periods and drops down groups.
Alkali metals have low IE (~4–5 eV).
Successive ionisations get harder.
Franck–Hertz experiment
Collisions confirm quantised energy levels.
Electrons accelerated through mercury vapour lose energy in steps of 4.9 eV, causing current drops at 4.9 V intervals and UV emission at 254 nm, showing quantised atomic levels.
1914 experiment.
Direct proof of discrete levels.
Characteristic X-rays
Produced by inner-shell transitions.
√f ∝ (Z−b)When an inner-shell electron is knocked out and an outer electron drops in, an X-ray of fixed energy is emitted, e.g. K_α. These lines are unique to the element.
Copper K_α = 8.05 keV (0.154 nm).
They sit on top of the continuous spectrum.
Continuous X-ray spectrum
Bremsstrahlung with a cut-off wavelength.
λ_min = hc/eVElectrons slowing in the target emit bremsstrahlung with a continuous spectrum and a sharp minimum wavelength λ_min = hc/(eV), independent of the target.
30 kV tube: λ_min = 0.041 nm.
Depends on tube voltage, not current.
Moseley's law
X-ray frequency relates to atomic number.
√f ∝ (Z − b) for characteristic X-ray lines; it showed that atomic number, not mass, orders the elements, and predicted missing elements.
Corrected the periodic table.
b ≈ 1 for K_α lines.
X-ray diffraction
Bragg's law reveals crystal structure.
2d sinθ = nλX-rays reflect from crystal planes and interfere constructively when 2d sinθ = nλ (Bragg's law). Used to find crystal and biological structures like DNA.
d = 0.2 nm, λ = 0.154 nm: first order at 22.6°.
θ is measured from the plane.
Electron configuration
Filling of shells explains the periodic table.
Electrons fill orbitals in order of increasing energy (1s, 2s, 2p, 3s, 3p, 4s, 3d...) respecting Pauli and Hund's rules. Valence electrons determine chemistry.
Oxygen: 1s²2s²2p⁴.
Exceptions exist (Cr, Cu).
Atomic mass unit
Unit of mass for atoms.
1 u = 931.5 MeV/c²1 u = 1/12 of the mass of carbon-12 = 1.6605×10⁻²⁷ kg = 931.5 MeV/c². A proton is about 1.0073 u.
Helium-4 has mass 4.0026 u.
Not exactly integer because of binding energy.
Nucleus composition
Protons and neutrons make up nuclei.
A nucleus has Z protons and N neutrons (A = Z + N nucleons). Notation ᴬ_Z X. Protons determine the element.
Carbon-14: Z = 6, N = 8.
Neutrons give stability and isotopes.
Isotopes, isobars, isotones
Same Z, same A, same N.
Isotopes: same Z, different N (¹H, ²H); isobars: same A (¹⁴C, ¹⁴N); isotones: same N.
Uranium-235 and -238 are isotopes.
Chemical properties follow Z.
Nuclear size
Radius scales as cube root of A.
R = R₀A^{1/3}R = R₀A^{1/3} with R₀ ≈ 1.2 fm. Volume ∝ A, so all nuclei have about the same density.
Uranium-238: 7.4 fm.
Size is a rough, fuzzy boundary.
Nuclear density
Roughly 10¹⁷ kg/m³, constant across nuclei.
About 2.3×10¹⁷ kg/m³ and nearly the same for all nuclei; a matchbox-sized lump of nuclear matter would weigh billions of tonnes.
Neutron star matter has this density.
Atoms are mostly empty, nuclei are not.
Mass defect
Nuclear mass is less than the sum of the parts.
Δm = Zm_p+Nm_n−MThe mass of a nucleus is less than the sum of its separate nucleons: Δm = Zm_p + Nm_n − M. The missing mass is the binding energy.
Helium-4: Δm = 0.0304 u.
Use nuclear masses, not atomic masses, or correct for electrons.
Binding energy
Energy released forming the nucleus.
E_b = Δm c²E_b = Δm c², the energy needed to split a nucleus into nucleons. Helium-4: 28.3 MeV.
1 u of mass defect = 931.5 MeV.
Higher binding energy per nucleon means more stable.
Binding energy per nucleon
Peaks near iron; explains fission and fusion.
The curve rises quickly to about 8.8 MeV at iron-56 and slowly falls for heavy elements. Fusion of light nuclei and fission of heavy nuclei both move toward the peak and release energy.
Iron-56 is the most stable.
Helium-4 is unusually stable.
Nuclear force
Strong, short-range and charge independent.
The strong force between nucleons is attractive at ~1 fm, very short range (about 2–3 fm), charge independent and repulsive at very short distances. It overcomes the electric repulsion between protons.
Stronger than electromagnetism by about 100 times at that range.
Residual of the quark–gluon force.
Radioactivity discovery
Becquerel and the Curies discovered natural radiation.
Becquerel discovered spontaneous radiation from uranium in 1896; the Curies isolated polonium and radium. Rutherford classified alpha, beta and gamma rays.
Marie Curie won two Nobel prizes.
The radiation comes from nuclei, not electrons.
Alpha decay
Emission of a helium nucleus.
A nucleus emits a helium-4 nucleus: ᴬ_Z X → ᴬ⁻⁴_{Z−2}Y + ⁴₂He. Alphas are heavily ionising but stopped by paper.
Uranium-238 → thorium-234.
Energy ~4–9 MeV in discrete lines.
Beta decay
Neutron changes to a proton, emitting an electron and antineutrino.
β⁻: a neutron becomes a proton emitting an electron and antineutrino; β⁺: a proton becomes a neutron with positron and neutrino. Energy is shared continuously, so neutrinos were proposed by Pauli.
Carbon-14 → nitrogen-14.
Electron isn't in the nucleus beforehand.
Gamma decay
Emission of high-energy photons from excited nuclei.
An excited nucleus drops to a lower state by emitting a photon of MeV energy; Z and A don't change. Very penetrating.
Cobalt-60 emits 1.17 and 1.33 MeV.
Follows alpha or beta decay.
Radioactive decay law
Number of nuclei decays exponentially.
N = N₀e^{−λt}N = N₀e^{−λt} and the activity A = λN. Decay is random for individual nuclei but exact statistically.
After 3 half-lives 12.5% remain.
Rate doesn't depend on temperature or chemistry.
Half-life
Time for half the nuclei to decay.
T½ = ln2/λT½ = ln 2/λ = 0.693/λ: time for half the nuclei to decay, regardless of how many there are. Ranges from 10⁻²¹ s to 10²⁴ years.
Iodine-131: 8 days; carbon-14: 5730 years.
After two half-lives, ¼ remains, not zero.
Mean life
Average lifetime of a nucleus.
τ = 1/λτ = 1/λ = 1.44 T½ is the average lifetime of a nucleus; the number falls to 1/e after τ.
T½ = 5730 yr gives τ = 8267 yr.
Mean life is longer than half-life.
Activity
Decays per second in becquerels.
A = λNA = −dN/dt = λN in becquerels (1 Bq = 1 decay/s); the curie is 3.7×10¹⁰ Bq. Activity decays with the same half-life.
1 g of radium: about 1 Ci.
Not equal to the number of nuclei.
Radioactive series
Chains of successive decays.
Heavy nuclei decay through chains of alpha and beta steps to stable lead (uranium, thorium and actinium series). Daughter products can be in secular equilibrium with the parent.
Radon gas comes from the uranium-238 series.
Four series by mass number 4n, 4n+1, 4n+2, 4n+3.
Carbon dating
Age estimated from C-14 decay.
decayLiving things take in carbon-14 (T½ 5730 years) at a steady ratio; after death it decays. Measuring the remaining fraction gives age: t = (T½/ln 2)ln(N₀/N).
25% remaining: 11,460 years.
Valid up to about 50,000 years.
Nuclear reactions
Transformation of nuclei by bombardment.
Nuclei change when struck by particles, conserving charge, mass number, energy and momentum: ¹⁴N + α → ¹⁷O + p (Rutherford, 1919).
Check A and Z on both sides.
Q-value tells the energy balance.
Q-value
Energy released in a reaction.
Q = (m_i−m_f)c²Q = (m_initial − m_final)c² is the energy released (Q > 0) or required (Q < 0) in a reaction.
D + T → He + n releases 17.6 MeV.
Use masses in u × 931.5 MeV.
Nuclear fission
Heavy nuclei split releasing energy.
A heavy nucleus such as U-235, hit by a neutron, splits into two medium nuclei plus 2–3 neutrons and about 200 MeV of energy.
1 kg of U-235 releases 8×10¹³ J, like 20,000 tonnes of TNT.
Most energy appears as kinetic energy of fragments.
Chain reaction
Neutrons from fission trigger further fission.
Neutrons from one fission cause further fissions; if each fission produces on average k > 1 following fissions, it grows exponentially. Critical mass is needed.
k = 1 for a steady reactor.
Moderated neutrons are more effective.
Nuclear reactors
Controlled chain reactions with moderators and control rods.
Fuel rods undergo controlled fission; a moderator slows neutrons, control rods absorb them, and coolant carries heat to turbines. About 33% efficient.
A 1 GW plant uses about 25 tonnes of fuel per year.
Needs safety systems.
Nuclear weapons
Uncontrolled chain reaction or fusion.
Fission bombs achieve supercritical mass quickly; thermonuclear weapons use fission to ignite fusion. Energy released in microseconds.
Hiroshima: 15 kt.
Arms control and non-proliferation treaties exist.
Nuclear fusion
Light nuclei combine releasing energy.
Light nuclei join to form heavier ones, releasing energy because binding energy per nucleon rises. It needs temperatures of 10⁷–10⁸ K to overcome Coulomb repulsion.
D + T → ⁴He + n + 17.6 MeV.
Tunnelling helps in stars.
Fusion in stars
Proton–proton chain powers the Sun.
In the Sun, the proton–proton chain converts 4 protons into helium-4, releasing 26.7 MeV, at about 15 million K. Heavier stars use the CNO cycle.
The Sun fuses 600 million tonnes of hydrogen each second.
Takes billions of years because the first step is a weak interaction.
Fusion reactors
Tokamaks confine plasma magnetically.
Tokamaks and stellarators confine plasma at 150 million K with magnetic fields; inertial confinement compresses fuel with lasers. Net energy gain was achieved at NIF in 2022.
ITER aims for Q = 10.
Engineering challenges remain.
Neutron discovery
Chadwick discovered the neutron in 1932.
Chadwick showed in 1932 that a neutral particle of about proton mass is ejected when alpha particles hit beryllium. It explained isotopes and enabled fission.
Neutrons penetrate easily as they are neutral.
Free neutrons decay with 10-minute half-life.
Neutrinos
Nearly massless neutral particles from beta decay.
Almost massless neutral leptons emitted in beta decay; they interact so weakly that trillions pass through you each second. Three flavours oscillate, implying mass.
Detected in giant tanks like Super-Kamiokande.
Solar neutrinos confirm fusion.
Radiation detection
Geiger counter, scintillators and cloud chambers.
Geiger–Müller tubes count ionisation pulses; scintillators convert radiation into flashes of light; cloud and bubble chambers show tracks; semiconductor detectors give energy.
Geiger counter clicks.
Different detectors suit different radiation types.
Interaction of radiation with matter
Alpha, beta and gamma penetrate differently.
Alpha: strongly ionising, range of cm in air. Beta: penetrates mm of aluminium. Gamma: attenuated exponentially I = I₀e^{−μx}, needing lead or concrete.
Half-value thickness of lead for 1 MeV gamma ≈ 1 cm.
Neutrons need hydrogen-rich shields.
Radiation units
Gray, sievert and becquerel.
Becquerel (activity), gray (absorbed dose, J/kg) and sievert (equivalent dose = dose × weighting factor). Natural background is about 2.4 mSv/year.
A chest X-ray ≈ 0.1 mSv.
Sv accounts for the type of radiation.
Biological effects of radiation
Ionising radiation damages cells.
Ionising radiation damages DNA, which can cause cell death or mutation. Risk depends on dose, type and tissue; the LNT model assumes some risk at any dose.
Acute dose of 4 Sv can be fatal.
Protect with time, distance and shielding.
Medical uses of radiation
Imaging, PET scans and radiotherapy.
X-rays and CT for imaging, PET (positron emitters like F-18), gamma cameras (Tc-99m), radiotherapy for cancer, and sterilisation.
Tc-99m T½ = 6 hours.
Dose balances benefit against risk.
Nuclear shell model
Magic numbers show extra stability.
Nucleons fill energy shells; closed shells at magic numbers 2, 8, 20, 28, 50, 82, 126 give extra stability.
Helium-4 and lead-208 are doubly magic.
Explains abundance and spin of nuclei.
Liquid drop model
Semi-empirical mass formula describes nuclear binding.
Treats the nucleus as a charged drop; the semi-empirical mass formula includes volume, surface, Coulomb, asymmetry and pairing terms. It explains fission and the binding-energy curve.
Gives binding energies within a few MeV.
Fails for shell effects.
Nuclear spin and magnetic moment
Nuclei have spin and magnetic properties.
Nuclei have intrinsic angular momentum and magnetic moments (μ_N = eħ/2m_p, ~1/1836 of the Bohr magneton), giving hyperfine splitting and used in NMR.
Proton spin ½.
Even–even nuclei have spin 0.
Nuclear magnetic resonance
Nuclei absorb RF in a magnetic field.
Nuclei in a magnetic field absorb radio waves at the Larmor frequency f = γB/2π (42.6 MHz/T for protons). Basis of NMR spectroscopy and MRI.
1.5 T: 63.9 MHz.
Chemical shifts give molecular structure.
Mössbauer effect
Recoilless gamma resonance gives ultra precise spectroscopy.
Gamma emission and absorption in a solid lattice can occur without recoil, producing extremely narrow lines (Δf/f ~ 10⁻¹³). Used to test gravitational redshift.
Iron-57 14.4 keV.
Pound–Rebka experiment.
Nuclear waste
Radioactive waste requires long-term storage.
Spent fuel contains fission products and actinides, dangerous for up to 10⁵ years. Disposal options: geological repositories, reprocessing, transmutation.
Most volume is low-level waste.
Public acceptance is a key issue.
Hydrogen-like ions
Energy scales as Z².
Ions with one electron (He⁺, Li²⁺) have E_n = −13.6 Z²/n² eV and radii a₀n²/Z; spectral lines shift to shorter wavelength.
He⁺ ground state: −54.4 eV.
Applies only to single-electron ions.
Positronium
Electron–positron bound state.
An electron–positron bound state with half the hydrogen binding energy (6.8 eV); it annihilates into gamma rays in 0.1 ns (para) or 140 ns (ortho).
Used to study QED.
Reduced mass is mₑ/2.
Muonic atoms
Muon replaces an electron, shrinking the orbit.
A muon (207 times heavier) replaces the electron, orbiting 207 times closer, probing nuclear size; also used in muon-catalysed fusion.
Muonic hydrogen revealed a smaller proton radius.
Muons decay in 2.2 μs.
Atomic clocks
Transitions of caesium define the second.
Caesium-133 hyperfine transition at 9,192,631,770 Hz defines the second; optical clocks using strontium reach 10⁻¹⁸ accuracy, losing a second in billions of years. They underlie GPS.
Relativistic time dilation measured at 1 cm height.
Interrogate atoms in vacuum.
Laser cooling of atoms
Photon recoil slows atoms toward microkelvin temperatures.
Atoms absorb photons from lasers tuned just below resonance (Doppler effect) and re-emit isotropically, losing momentum, cooling to microkelvin and below. Enables BEC and atomic clocks.
Nobel Prize 1997.
Needs repumping lasers.
Frequently asked questions
Why do atoms emit only certain colours of light?
Electrons can occupy only specific energy levels. When an electron falls from a higher level to a lower one, it emits a photon whose energy equals the difference, giving a line spectrum with discrete colours.
What is half-life?
The half-life is the time it takes for half of the nuclei in a sample to decay. It is a fixed property of each radioactive isotope and does not depend on the sample size or temperature.
What is the difference between fission and fusion?
Fission splits a heavy nucleus into lighter ones, releasing energy. Fusion joins light nuclei into a heavier one, also releasing energy, and this is the process that powers the Sun.
Are all radioactive materials dangerous?
They can be harmful, depending on the type of radiation, the dose and the length of exposure. Shielding, distance and time reduce the risk.
Why does fusion release energy but iron does not?
Light nuclei gain binding energy per nucleon when they fuse, up to iron. Beyond iron the binding energy per nucleon falls, so fusion absorbs energy instead.
Other areas
Measurement & Math Tools · Kinematics · Newton's Laws & Forces · Work, Energy & Power · Momentum & Collisions · Rotational Motion · Gravitation & Orbits · Oscillations & SHM · Waves & Sound · Fluids & Elasticity · Heat & Thermodynamics · Kinetic Theory & Statistical Physics · Electrostatics · Current Electricity & Circuits · Magnetism & Induction · AC Circuits & EM Waves · Optics · Quantum Mechanics · Relativity · Particle Physics & Cosmology