The Nucleus

Rutherford showed that an atom’s positive charge and almost all its mass sit in a tiny nucleus. Inside it, protons and neutrons are held together by a force unlike anything in everyday life: enormously strong, but felt only at a distance of about a femtometre.

What a nucleus is made of#

A nucleus contains protons (charge +e, mass about 1840 electron masses) and neutrons (no charge, slightly heavier than a proton). Together they are called nucleons.

Masses as energies

Because mass and energy are equivalent (E = mc²), nuclear physicists often quote masses in energy units: the electron is 0.511 MeV/c², the proton 938.27 MeV/c² and the neutron 939.57 MeV/c². The energy mc² of a particle at rest is its rest-mass energy. Another common unit is the unified atomic mass unit, one twelfth of the mass of a neutral carbon-12 atom:

Protons and neutrons obey the Pauli exclusion principle: no two protons, and no two neutrons, can share a quantum state (a proton and a neutron can).

Naming nuclei

A nucleus has atomic number Z (protons), neutron number N, and mass number A = Z + N. It is written ^A_ZX with X the chemical symbol, e.g. ²³⁵₉₂U has 92 protons and 143 neutrons. Chemistry depends only on Z, because Z fixes the number of electrons.

Which nuclei are stable?#

Over a thousand nuclides are known, but most are unstable: they emit a particle and change. Plotting N against Z, the stable ones lie in a narrow band. Light stable nuclei have N ≈ Z; heavier ones need more and more neutrons, up to N/Z ≈ 1.6. Nuclides to one side of the band have too many neutrons, to the other side too many protons; both are radioactive. The heaviest stable nuclide is ²⁰⁹Bi.

How big is a nucleus?#

A nucleus has no sharp edge (it is described by a wave function), but experiments give an average radius

Nucleons also move in quantum states with orbital angular momentum, and each has spin (m_s = ±½). Their total is the nuclear spin.

The nuclear force#

Protons repel each other electrically, yet nuclei hold together. At separations of about 1 fm a new force acts between nucleons. Its main features:

• Short range: strongest at about 1 fm, negligible beyond about 10 fm.
• Very strong: within its range, roughly 50–60 times stronger than the electric force.
• Charge-independent: the same between p–p, n–n and n–p (the electric force acts on protons as well).
• Not central: it depends on more than distance, e.g. it is stronger when the two spins are parallel.

Because of its short range, each nucleon only feels its nearest neighbours, which is why nuclear density is the same everywhere. In light nuclei, nuclear attraction dominates and pairing favours N ≈ Z. Coulomb repulsion, however, is long-range: every proton repels every other one. In a big nucleus it grows important, so extra neutrons (which add attraction but no repulsion) are needed. Too many neutrons are unfavourable too, and beyond bismuth no N/Z ratio gives a stable nucleus.

Weigh a helium nucleus and you find it is lighter than the two protons and two neutrons it is made of. The missing mass is the energy that binds them, and how that energy varies across the elements explains why both splitting and fusing nuclei can release power.

Mass defect and binding energy#

To pull a hydrogen atom apart you must supply 13.6 eV; bringing a proton and electron together releases the same 13.6 eV. Nuclei are similar, on a scale a million times larger. The energy needed to separate a nucleus completely into free protons and neutrons is its binding energy B. Since energy is released when the nucleus forms, the nucleus has less rest-mass energy than its parts:

Binding energy per nucleon#

Dividing B by A tells you how tightly, on average, each nucleon is held. For the deuteron (one proton, one neutron) it is only 2.22/2 = 1.11 MeV. It rises quickly with A, reaches a broad maximum of about 8.7–8.8 MeV for A ≈ 50–80 (iron and nickel), then falls slowly to about 7.5 MeV for uranium. Middle-weight nuclei are the most tightly bound.

Explaining the shape

• Volume term (+a₁A): the nuclear force is short-ranged, so a nucleon inside interacts only with its immediate neighbours (about 12, packed like spheres). The binding energy from all those bonds is proportional to A.
• Surface term (−a₂A^2/3): nucleons at the surface have fewer neighbours, so binding is lost in proportion to the surface area R² ∝ A^2/3. This matters most for small nuclei, so the curve starts low.
• Coulomb term (−a₃Z(Z − 1)/A^1/3): every pair of protons repels, with energy ∝ 1/R. This grows fastest for heavy nuclei, making the curve droop at large A.

Mass excess#

Releasing nuclear energy#

Whenever nucleons regroup into nuclei with a higher total binding energy, the rest-mass energy drops and the difference is released (the Q-value). Both ends of the curve slope up towards the middle:

• Fusion: joining light nuclei into a heavier one moves up the steep left side.
• Fission: splitting a heavy nucleus into two middle-weight ones moves up the gentle right side.

A nucleus off the valley of stability will not stay as it is. It throws out an alpha particle, turns a neutron into a proton (or the reverse), or sheds extra energy as a gamma photon, moving step by step towards stability.

Alpha decay#

The nucleus emits an alpha particle, a helium-4 nucleus (2 protons, 2 neutrons):

Z changes, so the element changes: the parent becomes a different daughter. For example ²¹²Bi (83 protons) → ²⁰⁸Tl (81 protons) + α. Alpha decay is typical of nuclei with A > 210, which are too large for the nuclear force to hold against Coulomb repulsion.

Beta decay#

Beta decay changes N/Z by converting one kind of nucleon into the other, through the weak force.

β⁻: a neutron becomes a proton

A nucleus with too many neutrons converts one, creating an electron and an antineutrino (uncharged, almost massless, spin ½):

The heavy daughter barely moves, so Q is shared between the electron and the antineutrino in any proportion. The electron’s kinetic energy can be anything from zero up to Q.

β⁺: a proton becomes a neutron

A nucleus with too many protons converts one, creating a positron (the electron’s antiparticle: same mass, charge +e) and a neutrino. An electron and a positron that meet annihilate into energy; the neutrino and antineutrino are also antiparticles of each other.

A free proton cannot do this: the neutron is heavier, so Q would be negative. A free neutron, however, does decay (half-life about 10 minutes).

Electron capture

Instead of emitting a positron, the nucleus may swallow one of its own atomic electrons, usually from the K shell: p + e⁻ → n + ν, so ^A_ZX + e⁻ → ^A_Z−1Y + ν, with Q = [m(X) − m(Y)]c² (46.15). The hole left in the K shell is filled from above, so X-rays follow.

Decay series#

The daughter may itself be unstable and decay again, and so on until a stable nucleus is reached. Uranium-238 passes through fourteen decays before ending as lead-206.

Gamma decay#

Nucleons occupy discrete energy levels, typically MeV apart, far too far to excite by heating. But an α or β decay often leaves the daughter in an excited state. It then drops to its ground state by emitting one or more photons, just as an excited atom emits light, only with far more energy. This electromagnetic radiation is a gamma ray, with λ = hc/E. Neither Z nor N changes.

α, β and γ emission are together called radioactive decay, and the rays nuclear radiation.

Properties of the radiations#

α and (weakly) β particles make some fluorescent materials flash (scintillation), and all three ionize gases.

Nobody can say when a particular nucleus will decay. Yet a large sample shrinks with clockwork regularity, halving in the same time over and over. The decay law turns that pattern into a clock for dating rocks and bones.

Chance, not destiny#

Prepare many identical radioactive nuclei at the same moment, in identical conditions. They do not decay together: some go almost at once, some last a very long time. No law of physics predicts the moment for any one nucleus. This is a quantum effect; classical physics would say identical starting conditions give identical futures. Quantum mechanics gives only probabilities (an idea Einstein never fully accepted).

The decay law#

If N nuclei are active, the number decaying in the next short time dt is proportional to N (more nuclei, more decays) and to dt (each has more chance in a longer interval):

The SI unit of activity is the becquerel (1 Bq = 1 decay per second). The older curie is 1 Ci = 3.7 × 10¹⁰ decays per second, named after Marie and Pierre Curie, who in 1902 isolated 0.19 g of radium chloride from uranium ore and shared the 1903 Nobel Prize with Henri Becquerel. Activity per unit mass is the specific activity.

Half-life and average life

The half-life t_1/2 is the time for half the active nuclei, and therefore half the activity, to go:

Adding up how long every nucleus survives and dividing by N₀ gives the average life:

Radioactive clocks#

Dating rocks

Carbon dating

Two ways to decay, and decay chains

If a nucleus can decay by two independent routes with constants λ₁ and λ₂, the chances simply add: λ = λ₁ + λ₂, so 1/t = 1/t₁ + 1/t₂ for the half-lives. For example, half-lives of 3 h and 6 h for the two routes give an overall half-life of 2 h.

If the daughter is also radioactive, it is produced at the rate λ_pN_p and decays at λ_dN_d. Its number stays constant when λ_pN_p = λ_dN_d, i.e. when the two activities are equal.

Uses and dangers#

Gamma rays from sources such as cobalt-60 treat cancer; radioactive tracers follow substances through the body or through pipes; the ionization from a weak source lets static charge leak away from moving machinery in factories; carbon dating dates once-living material. Too much exposure damages living cells, so radiation is always used with shielding and limited doses.

Split one uranium nucleus and you release about 200 MeV, some fifty million times more than burning one carbon atom. Make each split trigger exactly one more, and you have a nuclear reactor.

Energy from the binding curve#

Coal, oil and gas are limited, and harnessing sunlight on a large scale is still a challenge. Nuclei offer another source. Middle-weight nuclei are more tightly bound per nucleon than heavy ones. If the nucleons of a heavy nucleus regroup into two middle-weight fragments, the total binding energy rises, the rest mass falls, and the difference is released, mostly as kinetic energy of the fragments. This is nuclear fission.

Why heavy nuclei don’t all split at once#

Before splitting, the nucleus must stretch into a dumbbell shape, and that intermediate state has more energy than the original nucleus. Classically, without an energy kick to get over this hump, the nucleus would never split.

Quantum mechanics allows barrier penetration (tunnelling): if the final state has lower energy, the system can occasionally pass through the hump without the energy to climb it. Energy conservation seems violated, but only for a time Δt allowed by the uncertainty relation ΔE·Δt ≈ h/2π. The higher and wider the barrier, the less likely this is. For ²³⁸U the half-life for spontaneous fission is about 10¹⁶ years.

Fission of uranium-235#

Natural uranium is 99.3 % ²³⁸U and only 0.7 % ²³⁵U. A slow (thermal, about 0.04 eV) neutron is readily absorbed by ²³⁵U, forming ²³⁶U, which is very likely to split. Many pairs of fragments are possible, for example:

• On average about 2.5 neutrons are released per fission.
• Unequal fragments are most likely, with mass numbers around 95 and 140; equal halves are rare.
• Fragments inherit uranium’s high N/Z, too high for their size, so they emit neutrons and then β⁻-decay, with γ rays. Some neutrons come out only after a fragment’s β decay: these delayed neutrons are vital for controlling reactors.
• About 200 MeV is released per fission, mostly as the fragments’ kinetic energy; neutrons carry about 5 MeV, and the later decays release another 15–20 MeV. The fragments stop within the fuel, heating it.

Chain reactions#

Each fission’s neutrons can cause more fissions, which release more neutrons: a chain reaction. If, on average, more than one neutron from each fission causes another, the rate grows in geometric progression and the energy is released almost at once: an explosion. Neutrons are lost by escaping through the surface (more so for small pieces) and by absorption without fission.

Fission neutrons are fast (about 2 MeV), and ²³⁵U absorbs fast neutrons poorly. While slowing down they pass through 1–100 eV, where ²³⁸U absorbs them very strongly (forming ²³⁹U, which does not fission). Those neutrons are lost to the chain.

The nuclear reactor#

Over time the ²³⁵U is used up and the rods must be replaced.

Breeder reactors

²³⁹Pu also fissions easily but doesn’t occur naturally. It can be made from ²³⁸U: ²³⁸U + n → ²³⁹U → (β⁻) ²³⁹Np → (β⁻) ²³⁹Pu. If spent rods of ²³⁸U surround the core and at least one neutron per fission is captured in them, the reactor makes as much fuel as it burns, or more. Such a breeder reactor stretches limited uranium supplies.

The Sun shines by squeezing hydrogen into helium. Doing the same on Earth would give almost limitless, cleaner energy, if only we could hold a gas hotter than the Sun’s core in place.

Fusing light nuclei#

Bring two light nuclei within about 1 fm and the nuclear force can merge them into one heavier nucleus. Because B/A rises steeply for light nuclei, this releases energy: nuclear fusion. The difficulty is getting them that close. Atoms are about 10⁵ fm across, their electrons repel, and even bare nuclei repel each other strongly.

The answer is to heat the gas until the electrons are stripped off (a plasma) and the nuclei move so fast that some collide hard enough to fuse. Such reactions are called thermonuclear.

The Sun’s core is only about 1.5 × 10⁷ K, a hundred times cooler, yet it runs on fusion. Two reasons: in a hot gas some particles have far more than the average energy, and nuclei with too little energy can still tunnel through the Coulomb barrier occasionally.

Fusion in stars#

The proton–proton cycle

In the Sun and cooler stars (up to about 10⁷ K):

The Sun has been shining for about 4.5 × 10⁹ years, and its hydrogen should last roughly another 5 × 10⁹.

The carbon cycle

In hotter stars (about 10⁸ K), carbon-12 acts as a catalyst: it captures protons step by step (forming ¹³N, ¹³C, ¹⁴N, ¹⁵O, ¹⁵N, with two β⁺ decays) and finally gives back ¹²C plus a ⁴He. The net result is again four protons turned into one helium nucleus.

Heavier nuclei need hotter cores because Coulomb repulsion grows with Z. Hot enough stars fuse helium (3 ⁴He → ¹²C + γ) and so on up to the iron region, where B/A peaks. Elements heavier than iron are built by neutron capture followed by β decay.

Fusion on Earth#

Stars are held together by gravity. On Earth no solid wall can contain a plasma at fusion temperatures, so confinement is the main problem. The most practical reactions use deuterium and tritium:

The Lawson criterion

J. D. Lawson showed that to get out more energy than you put in, the plasma density n and confinement time τ must satisfy

Magnetic confinement: the tokamak

A tokamak holds the plasma in a doughnut-shaped (toroidal) chamber with carefully shaped magnetic fields: whenever a charged particle heads for the wall, the force qv × B turns it back. The plasma is heated with high-frequency currents, and fusion proceeds in pulses. Large machines such as JET in Europe have approached Q ≈ 1, and ITER is being built to go further; smaller ones such as Aditya at the Institute for Plasma Research, Ahmedabad, study plasma behaviour.

Inertial confinement

Intense laser pulses hit a tiny deuterium–tritium pellet from all sides. They vaporize it into plasma and compress it to 10³–10⁴ times its normal density, heating it so fast that fusion occurs before it can fly apart. The α particles produced stay inside and heat the plasma further.