Nuclei

Rutherford's alpha-scattering experiment showed that almost the entire mass and all the positive charge of an atom is concentrated in a tiny central core called the nucleus. The nucleus is made of two kinds of particles, together called nucleons: positively charged protons and electrically neutral neutrons (discovered by Chadwick in 1932). A nucleus is written as ^A_ZX, where Z is the atomic number (number of protons, which fixes the element and equals the number of electrons in the neutral atom) and A is the mass number (total number of nucleons). The number of neutrons is N = A − Z.

Nuclei of the same element (same Z) but different A are called isotopes — for example ¹₁H, ²₁H (deuterium) and ³₁H (tritium) are isotopes of hydrogen. They have identical chemical behaviour since chemistry depends only on Z, but different masses and nuclear properties. Nuclei with the same mass number A but different Z are called isobars (e.g. ¹⁴₆C and ¹⁴₇N). Nuclei with the same neutron number N are called isotones.

Nuclear and atomic masses are measured in atomic mass units (u), defined as 1/12th the mass of one atom of the carbon isotope ¹²₆C. By Einstein's mass-energy equivalence, 1 u of mass corresponds to 931.5 MeV of energy, a conversion used throughout this chapter.

Scattering experiments show that the nuclear radius depends on the mass number as R = R₀A^1/3, with R₀ ≈ 1.2 fm (1 fm = 10⁻¹⁵ m). Since the nuclear volume (4/3)πR³ is then directly proportional to A, and the nuclear mass is also directly proportional to A, the ratio mass/volume — the nuclear density — comes out the same for every nucleus, regardless of size. This is strong evidence that nucleons are packed together at a fixed, saturated density inside every nucleus, like incompressible spheres.

Worked example: Find the radius and check the density of the aluminium nucleus ²⁷₁₃Al (A = 27).
R = 1.2 × 27^1/3 fm = 1.2 × 3 fm = 3.6 fm.
Mass of the nucleus ≈ 27 × 1.66 × 10⁻²⁷ kg = 4.48 × 10⁻²⁶ kg.
Volume = (4/3)πR³ = (4/3)π(3.6 × 10⁻¹⁵)³ m³ = 1.954 × 10⁻⁴³ m³.
Density ρ = mass/volume = 4.48 × 10⁻²⁶ / 1.954 × 10⁻⁴³ ≈ 2.29 × 10¹⁷ kg/m³ — essentially the same enormous value obtained for any other nucleus, confirming that nuclear density is independent of A.

Einstein's special theory of relativity gives the equivalence of mass and energy, E = mc². Careful mass measurements show that the mass M of any nucleus ^A_ZX is always less than the sum of the masses of Z free protons and (A − Z) free neutrons that make it up. This difference is called the mass defect, Δm = [Zm_p + (A−Z)m_n] − M. (In practice, tabulated atomic masses already include the electrons, so using the mass of the hydrogen atom m_H in place of the bare proton mass automatically accounts for the Z atomic electrons on both sides of the equation.)

The energy equivalent of this missing mass is called the binding energy, E_b = Δm c². It is the energy that would have to be supplied to completely separate the nucleus into its individual, free nucleons; equivalently, it is the energy released when the nucleons come together to form the nucleus. A more useful quantity for comparing the stability of different nuclei is the binding energy per nucleon, E_b/A — the average energy needed to pull out one nucleon. The larger E_b/A is, the more tightly bound (stable) the nucleus is.

Worked example: Find the binding energy and binding energy per nucleon of ⁵⁶₂₆Fe, given M(⁵⁶₂₆Fe) = 55.934939 u, m_H = 1.007825 u, m_n = 1.008665 u.

Z = 26, N = A − Z = 30.
Sum of free-nucleon masses = 26(1.007825) + 30(1.008665) = 26.20345 + 30.25995 = 56.46340 u.
Δm = 56.46340 − 55.934939 = 0.528461 u.
E_b = 0.528461 × 931.5 MeV = 492.26 MeV.
E_b/A = 492.26 / 56 = 8.79 MeV per nucleon.

This value, close to 8.8 MeV/nucleon, is near the maximum binding energy per nucleon found in nature, which is why iron and its neighbours are among the most stable nuclei — a fact that will explain why both fission of heavy nuclei and fusion of light nuclei release energy (covered later in this chapter).

Protons repel each other electrically, yet nuclei stay together — so a much stronger, short-range attractive force called the strong nuclear force must act between nucleons. Its key properties are: (i) it is the strongest force in nature at nuclear distances, roughly 100 times stronger than the electromagnetic force; (ii) it has an extremely short range, acting strongly only over about 1–3 fm and becoming negligible beyond about 3–4 fm; (iii) it is essentially charge-independent — the force between two protons, two neutrons, or a proton and a neutron is (after correcting for the Coulomb repulsion between protons) approximately the same; and (iv) it shows saturation — each nucleon interacts strongly only with its nearest neighbours, not with every other nucleon in the nucleus, which is exactly why nuclear volume grows in direct proportion to A rather than as A².

If the binding energy per nucleon (E_b/A) is plotted against mass number A for all known nuclei, a characteristic curve results. It rises steeply for the lightest nuclei, reaches a broad maximum of about 8.7–8.8 MeV/nucleon for nuclei with A in the range of roughly 40 to 120 (peaking near A ≈ 56, iron), and then falls slowly for heavier nuclei, dropping to about 7.6 MeV/nucleon for uranium (A = 238). The slow decline for heavy nuclei happens because the short-range nuclear force saturates while the long-range Coulomb repulsion between the ever-increasing number of protons keeps growing, gradually destabilising very heavy nuclei.

This shape of the curve immediately explains why energy can be released in two opposite ways. Very light nuclei (e.g. deuteron, E_b/A ≈ 1.11 MeV/nucleon, from the earlier deuteron result of 2.224 MeV total) sit low on the curve; if they combine (fusion) into a nucleus nearer the peak, the nucleons become more tightly bound and energy is released. Very heavy nuclei (e.g. ⁵⁶Fe has E_b/A = 8.79 MeV/nucleon compared with only about 7.6 MeV/nucleon for uranium) sit on the downward slope; if such a nucleus splits (fission) into two mid-sized fragments closer to the peak, the fragments are more tightly bound than the original nucleus, and the difference in binding energy is released as energy. Both processes, in other words, are simply the nucleus moving towards the more stable, higher-Eb/A region of the curve.

Certain unstable nuclides spontaneously transform into other nuclides, emitting radiation in the process — a phenomenon discovered by Henri Becquerel and called radioactivity. It is a purely nuclear, statistical process: it is impossible to predict when any one nucleus will decay, but for a large number of identical nuclei the average decay rate follows a precise law. If N is the number of undecayed radioactive nuclei present at time t, the number decaying per unit time is proportional to N itself:

dN/dt = −λN, where λ is a constant called the decay constant, characteristic of the nuclide and independent of external conditions such as temperature or pressure. Integrating this equation gives the exponential decay law N = N₀e^−λt, where N₀ is the number of nuclei present at t = 0.

Two related, more intuitive quantities are commonly used. The half-life T½ is the time in which exactly half the nuclei present decay; setting N = N₀/2 in the decay law gives T½ = ln2/λ = 0.693/λ. The mean life τ is the average lifetime of a nucleus before it decays, τ = 1/λ = T½/0.693 ≈ 1.44 T½. The activity of a sample, A = λN = A₀e^−λt, is the number of disintegrations occurring per second; its SI unit is the becquerel (1 Bq = 1 decay/s), and an older but still common unit is the curie (1 Ci = 3.7 × 10¹⁰ Bq).

Worked example: A radioactive sample has a half-life of 20 days. (a) Find its decay constant. (b) What fraction of the original sample remains after 60 days?

(a) λ = 0.693/T½ = 0.693/20 = 0.03465 per day.
(b) 60 days = 3 half-lives, so the remaining fraction is N/N₀ = (1/2)³ = 1/8 = 0.125 (12.5%). This can be checked directly: N/N₀ = e^−λt = e^{−0.03465×60} = e^−2.079 ≈ 0.125.

In alpha (α) decay, a heavy nucleus emits a helium nucleus (2 protons + 2 neutrons): ^A_ZX → ^A−4_Z−2Y + ⁴₂He. Both mass number and charge are conserved. Because the alpha particle is much lighter than the parent nucleus, it carries away most of the released kinetic energy, and alpha particles from a given nuclide are emitted with essentially one discrete energy (or a few discrete energies).

In beta (β) decay, a nucleus changes a neutron into a proton or vice versa without changing its mass number. In β⁻ decay, a neutron converts to a proton, emitting an electron and an antineutrino: n → p + e⁻ + ν̄, so ^A_ZX → ^A_Z+1Y + e⁻ + ν̄. In β⁺ decay, a proton converts to a neutron, emitting a positron and a neutrino: p → n + e⁺ + ν, so ^A_ZX → ^A_Z−1Y + e⁺ + ν. Unlike alpha decay, the energy released in a beta decay is shared randomly between the beta particle and the (anti)neutrino, so the emitted electrons/positrons show a continuous range of energies up to a maximum — historically, this observation is what led Pauli to propose the existence of the neutrino, since without it energy and momentum would not appear to be conserved.

In gamma (γ) decay, a nucleus left in an excited energy state after an earlier alpha or beta decay drops to a lower energy state (often its ground state) by emitting a high-energy photon: ^A_ZX* → ^A_ZX + γ. Neither A nor Z changes in gamma decay — only the internal energy of the nucleus changes, exactly analogous to an atom emitting a photon when an electron drops to a lower orbital, but with energies about a million times larger because nuclear energy levels are far more widely spaced.

Worked example (Q-value of alpha decay): Find the energy released when ²³⁸₉₂U decays to ²³⁴₉₀Th by alpha emission, given M(²³⁸U) = 238.05079 u, M(²³⁴Th) = 234.04363 u, M(⁴He) = 4.002603 u.

Q = [M(²³⁸U) − M(²³⁴Th) − M(⁴He)] × 931.5 MeV
= [238.05079 − 234.04363 − 4.002603] × 931.5
= [238.05079 − 238.046233] × 931.5 = 0.004557 × 931.5 = 4.25 MeV (positive Q confirms the decay is energetically allowed; this energy appears mainly as kinetic energy of the emitted alpha particle and the small recoil of the thorium nucleus).

In nuclear fission, discovered by Hahn and Strassmann, a heavy nucleus such as ²³⁵₉₂U splits into two medium-sized fragments when it absorbs a slow (thermal) neutron, releasing two or three additional neutrons and a large amount of energy: n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3n + energy. As explained by the binding-energy curve, the fragments are more tightly bound than the parent nucleus, and roughly 200 MeV is released per fission event, overwhelmingly as the kinetic energy of the fragments. Because each fission releases more neutrons than it consumes, these neutrons can trigger further fissions in neighbouring nuclei, producing a self-sustaining chain reaction if enough fissile material (at least a critical mass) is present. In a nuclear reactor, the chain reaction is controlled: a moderator (such as heavy water or graphite) slows the fast neutrons down to thermal speeds where they are far more likely to cause further fission, and control rods made of a strong neutron absorber (such as boron or cadmium) are inserted or withdrawn to keep the reaction rate steady rather than runaway; a coolant carries away the heat generated, which is ultimately used to generate electricity.

Worked example (reactor power): A reactor delivers a steady 500 MW of power. If each fission of U-235 releases about 200 MeV, how many fissions occur every second?

Energy per fission = 200 MeV = 200 × 1.6 × 10⁻¹³ J = 3.2 × 10⁻¹¹ J.
Number of fissions per second = Power/Energy per fission = (500 × 10⁶) / (3.2 × 10⁻¹¹) ≈ 1.56 × 10¹⁹ fissions every second.

In nuclear fusion, two light nuclei combine to form a heavier, more tightly bound nucleus, again releasing energy according to the binding-energy curve. Because both nuclei carry positive charge, they must overcome strong mutual Coulomb (electrostatic) repulsion before the short-range nuclear force can bind them — this repulsive barrier can only be overcome if the nuclei collide with very large kinetic energies, which requires extremely high temperatures (of order 10&sup7; K or more); such fusion reactions are therefore called thermonuclear reactions. This is exactly what happens inside the Sun and other stars, where the net effect of the proton-proton chain is four hydrogen nuclei fusing into one helium nucleus: 4 ¹₁H → ⁴₂He + 2e⁺ + 2ν + energy, releasing about 26.7 MeV per helium nucleus formed. This continuous fusion is the source of the energy the Sun radiates, and it is why controlled fusion (requiring extreme confinement and temperature) is pursued as a potential future clean-energy source on Earth, unlike fission, which is already used commercially in nuclear power plants.