Physics · Atomic and quantum physics · Grade 11

The atomic nucleus and binding

Why does the Sun gain energy by joining light nuclei, while a nuclear power plant gains it by splitting heavy uranium?

The situation

The Sun gets its energy by joining hydrogen nuclei, while a nuclear power station gets it by splitting uranium nuclei. One joins, the other splits — and both release energy. The secret lies in the nucleus's “mass account”: a nucleus is lighter than the protons and neutrons that make it up. In which nuclei is this missing mass — the mass defect — greatest, and why does fusion inside stars stop around iron?

Topic

Atomic nucleus and binding energy. Protons and neutrons: mass defect and the most stable nucleus — iron

Key ideas

Mass defect
Binding energy per nucleon
Nuclear forces

Formulas

A = Z + N, R = r_{0} A^{1 / 3}, r_{0} \approx 1.2 fm

Mass number (baryon number) and nuclear radius: the volume is proportional to $A$ — the density is almost constant.

Δ m = Z m_{p} + N m_{n} - M_{y a}, E_{b} = Δ m c^{2}

Mass defect and binding energy; $1 u \cdot c^{2} = 931.5$ MeV.

E_{b} = a_{V} A - a_{S} A^{2 / 3} - a_{C} \frac{Z (Z - 1)}{A^{1 / 3}} - a_{A} \frac{{(N - Z)}^{2}}{A} \pm δ

The Weizsäcker (liquid-drop model) formula: volume, surface, Coulomb, asymmetry and pairing terms.

V (r) = - g^{2} \frac{e^{- r / a}}{r}, a = \frac{ℏ}{m_{π} c} \approx 1.4 fm

Yukawa potential: the nuclear force is carried by the exchange of a massive particle — the pion.

E_{J} = \frac{ℏ^{2}}{2 I} J (J + 1)

Rotational levels of a deformed nucleus (collective model).

Theory

Composition and size of the nucleus

A nucleus consists of protons ( $Z$ is the atomic number) and neutrons ( $N$ ), together called nucleons; $A = Z + N$ is the mass number (baryon number). Nuclei with the same $Z$ are isotopes, with the same $A$ isobars, with the same $N$ isotones; pairs in which $Z$ and $N$ are swapped (³H–³He) are mirror nuclei; a long-lived nucleus in an excited state is an isomer. The radius is $R = r_{0} A^{1 / 3}$ : the volume is proportional to the number of nucleons, and the density is ≈ 0.14 nucleons/fm³ ≈ 2·10¹⁷ kg/m³. The charge is measured from Rutherford scattering and Moseley's law, the mass with a mass spectrometer, and the radius from the scattering of fast electrons (Hofstadter) and from the difference in Coulomb energy of mirror nuclei. A nucleon has spin ½; even–even nuclei have spin 0; nuclei with odd $A$ are fermions, those with even $A$ are bosons. The magnetic moment is measured in nuclear magnetons, and the electric quadrupole moment shows how far the shape of the nucleus differs from a sphere.

Mass defect and binding energy

To separate a nucleus into individual nucleons, the binding energy $E_{b} = Δ m c^{2}$ is needed. For ⁴He: $Δ m = 0.0304$ u, i.e. 28.3 MeV — 7.07 MeV per nucleon. The height of a column on the chart is the binding energy per nucleon $E_{b} / A$ . It rises steeply for light nuclei, reaches a peak of ≈ 8.8 MeV around ⁵⁶Fe–⁶²Ni, and falls slowly to 7.6 MeV at uranium. Joining light nuclei (fusion) and splitting heavy ones (fission) both bring the product closer to the peak, and the difference is released as energy. Model: for nuclei without measured values on the chart, the Weizsäcker formula is used.

The liquid-drop model (Weizsäcker formula)

A nucleus is like a drop of charged liquid. The volume term $a_{V} A$ : each nucleon is attracted only by its neighbours (the saturation of the forces). The surface term: nucleons at the surface have fewer neighbours. The Coulomb term: protons repel one another — which is why heavy nuclei have more neutrons, and the valley of stability bends towards $N > Z$ . The asymmetry term (the Pauli principle) favours $N = Z$ , and the pairing term strengthens even–even nuclei. Here $a_{V} = 15.8$ ; $a_{S} = 18.3$ ; $a_{C} = 0.714$ ; $a_{A} = 23.2$ ; $δ = 12 / \sqrt{A}$ MeV. The formula works to 1 % accuracy for heavy nuclei, but for ⁴He it gives 22 MeV (measured 28.3) — shell effects show up.

Nuclear forces, the deuteron and isospin

The nuclear force is short-range (≈ 1–2 fm), very strong, and repulsive at distances shorter than ≈ 0.8 fm (the core). It does not depend on charge: in the same state the pp, nn and np forces are equal (isospin invariance). So the proton and the neutron are two states of a single nucleon: isospin $T = ½$ , $T_{3} = + ½$ (p) and $- ½$ (n). The force depends on spin: an np pair binds only when the spins are parallel — the deuteron ( $E_{b} = 2.22$ MeV, size ≈ 2 fm; the nucleons spend most of the time outside the range of the force). The generalised Pauli principle: the wave function is antisymmetric under exchange of space·spin·isospin, so in an s-state the spins of two identical nucleons must be opposite — the diproton and the dineutron are not bound. The force is studied in low-energy np scattering. The stage uses the Malfliet–Tjon phenomenological potential.

Meson theory

Yukawa (1935) explained the nuclear force by the exchange of a massive particle: by the uncertainty principle a particle of mass $m$ lives “on borrowed time” for $Δ t \approx ℏ / (m c^{2})$ and travels a distance of ≈ $ℏ / (m c)$ . For a radius of 1.4 fm, $m c^{2} \approx 140$ MeV is needed. Such a particle — the pion — was discovered in cosmic rays in 1947 ( $m_{π^{\pm}} c^{2} = 139.6$ MeV). The exchange of a charged pion turns a proton into a neutron and vice versa — on the stage the colours of the nucleons swap.

Nuclear models and γ radiation

The shell model: nucleons fill levels in a common potential; spin–orbit interaction produces the magic numbers 2, 8, 20, 28, 50, 82, 126 (Mayer and Jensen, 1949). The doubly magic ¹⁶O, ⁴⁰Ca and ²⁰⁸Pb are especially stable; the spin of an odd nucleus is determined by the level of the last unpaired nucleon. The Fermi-gas model: nucleons are free fermions in a well about 45 MeV deep, with a Fermi energy of ≈ 38 MeV. The collective model: an elongated nucleus rotates, and its levels obey the $J (J + 1)$ law. An excited nucleus drops down by emitting a γ quantum (selection rule: in a rotational band $Δ J = 2$ , E2 transitions), and sometimes gives its energy to an atomic electron (internal conversion). Long-lived excited states are isomers: in medicine ⁹⁹ᵐTc (140.5 keV, $T_{1 / 2} \approx 6$ h). The emission and absorption of a γ quantum by a nucleus in a solid crystal without recoil is the Mössbauer effect (⁵⁷Fe, 14.4 keV). Radioactive decay is in the “Radioactivity, half-life” experiment.

Did you know?

The density of nuclear matter is about 2·10¹⁷ kg/m³: a teaspoon (5 cm³) of nuclear matter would have a mass of about a billion tonnes.
The greatest binding energy per nucleon is not in iron-56 but in nickel-62 (8.795 MeV). In stellar explosions, however, a lot of nickel-56 is produced, which decays into iron-56 via cobalt-56 — which is why there is so much iron in the universe.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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