Why does a nucleus not fly apart?
A nucleus packs its protons into a sphere a few femtometres across. Every proton repels every other. Two protons \(2\,\text{fm}\) apart push on each other with a force of \(ke^2/r^2 = 58\,\text{N}\), the weight of a 6 kg bag. That force acts on a particle of mass \(1.67 \times 10^{-27}\,\text{kg}\). Gravity between them is about \(10^{36}\) times weaker. Yet uranium holds 92 protons together for billions of years.
So a third force must act inside the nucleus, an attraction stronger than the electric repulsion. It is called the strong nuclear force. We cannot see it in daily life, and that is itself a clue: it must fade away over very short distances.
What the nuclear force is like
Experiments with colliding nucleons, and the properties of nuclei, show the force has five features.
- It is attractive and very strong at separations of about 1 fm, far stronger there than the electric repulsion between two protons.
- It has a short range. It falls off very fast beyond about 2 fm and is almost nothing beyond a few femtometres. The electric force falls only as \(1/r^2\).
- It is charge-independent. It pulls a proton on a proton, a neutron on a neutron and a proton on a neutron almost equally.
- It saturates. A nucleon is bound only to its nearest neighbours, not to every nucleon in the nucleus.
- It turns repulsive below about 0.7 fm, so nucleons do not crush into one another.
Read Fig 2.1 as you would any energy diagram. The force is minus the slope of the curve, as in the chapter on potential energy. Where the curve falls towards the well, the force pulls the nucleons together. Where the curve is flat, beyond about \(3\,\text{fm}\), there is almost no force at all. Nucleons in a nucleus sit about \(1\) to \(2\,\text{fm}\) apart, in the well.
Saturation explains the constant density of the last lesson. Each nucleon pulls only on the few neighbours within its reach. So a nucleon added to a nucleus simply sits beside the others, at the same spacing. Adding nucleons adds volume in proportion, like marbles in a bag.
Think of the nucleons as a drop of glue. Each bit of glue sticks only to the bits touching it. A small drop and a large drop are equally dense. A bit on the surface is held less firmly than a bit inside, because it has fewer neighbours. The electric repulsion between protons acts like a weak spring stretched across the whole drop.
Which of these are properties of the strong nuclear force?
Two protons are held \(10\,\text{fm}\) apart. What is the main force between them?
A nucleus is lighter than its parts
Now weigh a deuteron, the nucleus of deuterium. It is one proton and one neutron. You might expect its mass to be the sum of the two. It is not. The deuterium atom has a mass of \(2.014102\,\text{u}\). A hydrogen atom and a free neutron together have \(1.007825 + 1.008665 = 2.016490\,\text{u}\). The bound pair is lighter by \(0.002388\,\text{u}\).
The mass defect of a nucleus is the mass of its separate nucleons minus the mass of the nucleus. Worked with atomic masses, it is \(\Delta m = Z\,m_{\text{H}} + N\,m_n - M\). Here \(m_{\text{H}}\) is the mass of a hydrogen atom, \(m_n\) of a neutron, and \(M\) of the whole atom.
Why use the hydrogen atom and not the bare proton? Tables list the masses of whole atoms, electrons included. The atom of mass \(M\) holds \(Z\) electrons. Counting \(Z\) hydrogen atoms brings in the same \(Z\) electrons, so the electrons cancel in the subtraction. The same answer comes from nuclear masses: proton \(1.007276\,\text{u}\) plus neutron \(1.008665\,\text{u}\), minus the bare deuteron, \(2.013553\,\text{u}\).
The mass has not been lost carelessly. When the proton and the neutron came together, the nuclear force pulled them into the well of Fig 2.1. Energy was given out as a gamma ray. By \(E = mc^2\), from the chapter on Matter Waves and Special Relativity, energy has mass. The energy that left took its mass with it.
For helium-4 the drop is dramatic. Two hydrogen atoms and two neutrons have \(4.032980\,\text{u}\), but a helium-4 atom has only \(4.002603\,\text{u}\). The defect is \(0.030377\,\text{u}\), nearly all of the mass above 4 u. Helium-4 is far more tightly bound than the deuteron.
Why is a nucleus lighter than its separated nucleons?
The binding energy \(B\) of a nucleus is the energy needed to separate it into its free protons and neutrons, at rest far apart. It equals the energy given out when those nucleons come together, and \(B = \Delta m\,c^2\).
Converting a mass in u to an energy needs \(c^2\) once, so it is worth doing once. With the precise values \(1\,\text{u} = 1.66054 \times 10^{-27}\,\text{kg}\), \(c = 2.99792 \times 10^{8}\,\text{m/s}\) and \(1\,\text{eV} = 1.60218 \times 10^{-19}\,\text{J}\), the energy of \(1\,\text{u}\) is \(931.5\,\text{MeV}\). Rounded constants shift this by about 2 MeV. So always convert with the number \(931.5\) directly.
The deuteron's defect of \(0.002388\,\text{u}\) is \(0.002388 \times 931.5 = 2.22\,\text{MeV}\). This is checked directly. A gamma-ray photon of \(2.22\,\text{MeV}\) or more can split a deuteron into a proton and a neutron; one of less energy cannot. Helium-4's \(0.030377\,\text{u}\) is \(28.3\,\text{MeV}\).
Masses in nuclear physics are often quoted as energies, in MeV/\(c^2\). The proton is \(938.3\,\text{MeV}/c^2\) and the neutron \(939.6\,\text{MeV}/c^2\). The deuteron is lighter than the pair by its binding energy: \(938.3 + 939.6 - 2.2 = 1875.7\,\text{MeV}/c^2\).
The atomic mass of iron-56, \({}^{56}_{26}\text{Fe}\), is \(55.934942\,\text{u}\). Find its mass defect and its binding energy. Take \(m_{\text{H}} = 1.007825\,\text{u}\), \(m_n = 1.008665\,\text{u}\) and \(1\,\text{u} = 931.5\,\text{MeV}/c^2\).
\(Z = 26\) and \(N = 56 - 26 = 30\).
\(\Delta m = 26\,m_{\text{H}} + 30\,m_n - M\), then \(B = \Delta m \times 931.5\).
\(26 \times 1.007825 = 26.203450\) and \(30 \times 1.008665 = 30.259950\), so the parts have \(56.463400\,\text{u}\). \(\Delta m = 56.463400 - 55.934942 = 0.528458\,\text{u}\), and \(B = 0.528458 \times 931.5 = 492\,\text{MeV}\).
Iron-56 has a mass defect of \(0.528\,\text{u}\) and a binding energy of \(492\,\text{MeV}\). Values used: \(1\,\text{u} = 931.5\,\text{MeV}/c^2\), \(m_{\text{H}} = 1.007825\,\text{u}\), \(m_n = 1.008665\,\text{u}\), \(M_{\text{Fe}} = 55.934942\,\text{u}\).
Compare these energies with chemistry. Pulling the electron out of a hydrogen atom takes \(13.6\,\text{eV}\), as the chapter on the Bohr atom found. Pulling a helium-4 nucleus apart takes \(28.3\,\text{MeV}\), two million times more. Nuclear energies are measured in MeV because the nuclear force is so strong over so short a range.
The atomic mass of carbon-12 is exactly \(12\,\text{u}\), by the definition of the unit. Find its binding energy in MeV. Take \(m_{\text{H}} = 1.007825\,\text{u}\), \(m_n = 1.008665\,\text{u}\) and \(1\,\text{u} = 931.5\,\text{MeV}/c^2\).
The binding energies of helium-4 and helium-3 are \(28.3\,\text{MeV}\) and \(7.7\,\text{MeV}\). How much energy, in MeV, is needed to pull one neutron out of helium-4, leaving helium-3?
The proton has a mass of \(938.3\,\text{MeV}/c^2\) and the neutron \(939.6\,\text{MeV}/c^2\). The binding energy of the deuteron is \(2.2\,\text{MeV}\). What is the mass of the deuteron, in \(\text{MeV}/c^2\)?
In 1919 Francis Aston built the mass spectrograph, which sorts atoms by mass as they fly through electric and magnetic fields. He found that the masses of atoms are nearly whole numbers. Nearly, but not quite: the helium atom was lighter than four hydrogen atoms by about 0.7 per cent. In 1920 Arthur Eddington read the tiny difference as energy. If the Sun turned hydrogen into helium, he argued, that missing mass could keep it shining for billions of years.
One helium-4 nucleus is bound by \(28.3\,\text{MeV} = 4.5 \times 10^{-12}\,\text{J}\). A mole of helium, \(6.02 \times 10^{23}\) nuclei and only 4 grams, is therefore bound by about \(2.7 \times 10^{12}\,\text{J}\). Burning coal gives about \(3 \times 10^{7}\,\text{J}\) per kilogram. To match the binding of 4 grams of helium, a thermal power station would have to burn about 90 tonnes of coal.
- Adding proton masses to an atomic mass.Tables list atomic masses. Use hydrogen atoms for the protons, so the electrons cancel, or use nuclear masses on both sides.
- Converting u to MeV with \(1.66 \times 10^{-27}\,\text{kg} \times (3 \times 10^8)^2\).Rounded constants give about 934 MeV. Use \(1\,\text{u} = 931.5\,\text{MeV}/c^2\) directly.
- Thinking the nucleus is heavier than its parts because it holds energy.Energy leaves when nucleons bind. The bound nucleus is lighter, by \(B/c^2\).
- Believing the nuclear force acts between all the nucleons of a nucleus.It reaches only about 2 fm, so each nucleon binds only its neighbours: the force saturates.