A nucleus that changes by itself
Most nuclei around us never change. The carbon in a piece of charcoal and the oxygen in the air have been the same nuclei for billions of years. A few kinds of nucleus are different. Left alone, such a nucleus changes into a nucleus of another element. As it does so, it gives out a fast particle or a burst of light. This is radioactivity.
Radioactivity is the spontaneous change of an unstable nucleus into another nucleus. It gives out an alpha particle, a beta particle (an electron or a positron) or a gamma photon.
The change happens inside the nucleus, far from the electrons that take part in chemistry. So heating a radioactive salt, squeezing it or dissolving it in acid makes no difference to how fast it decays. A uranium atom decays in the same way in a rock, in a solution or in a compound. This is the first clue that radioactivity belongs to the nucleus and not to the atom.
Two forces inside the nucleus
The chapter on the nucleus describes the force that holds protons and neutrons together, the strong nuclear force. It is far stronger than the electric force, but it reaches only a few femtometres. So each nucleon feels the pull of its nearest neighbours only. The electric repulsion between protons is weaker, but it reaches across the whole nucleus. Every proton pushes on every other proton.
A nucleus is stable while the strong force wins this contest with some energy to spare. In a light nucleus the two kinds of nucleon come in about equal numbers. In a heavy nucleus there are many more pairs of protons to repel one another. Extra neutrons help, because they add strong attraction without adding charge. So heavier stable nuclei carry more and more neutrons per proton.
The belt of stability
Plot every known nuclide as a point, with the proton number \(Z\) across and the neutron number \(N\) up. The stable nuclei fall in a narrow band, the stability belt. For light nuclei it runs along \(N = Z\): helium-4, carbon-12 and oxygen-16 all have equal numbers. Then it bends upward. Lead-208 has 126 neutrons and 82 protons, a ratio of about 1.54.
The belt ends at \(Z = 83\). Every nucleus with more protons is radioactive, whatever its number of neutrons, because no neutron count can hold so much charge together. Fig 1.1 shows the four ways a nucleus can be unstable, and each has its cure.
- Too big. A heavy nucleus beyond the end of the belt, such as uranium-238, sheds a bundle of two protons and two neutrons. This is alpha decay.
- Too many neutrons. A nucleus above the belt turns a neutron into a proton. This is beta-minus decay.
- Too many protons. A nucleus below the belt turns a proton into a neutron, by beta-plus decay or by electron capture.
- Too much energy. A nucleus with the right numbers but left in an excited state gives out a photon. This is gamma decay.
Three radiations
The three radiations were named alpha, beta and gamma before anyone knew what they were. They differ in charge and in how far they go through matter. An alpha particle is a helium nucleus with a charge of \(+2e\). A sheet of paper or a few centimetres of air stops it. A beta particle is an electron or a positron; it needs a few millimetres of aluminium. A gamma ray is a photon, with no charge, and it needs several centimetres of lead.
The order reverses for the damage done along the way. A slow, heavily charged alpha particle tears electrons off every atom it passes, so it spends its energy in a very short path. A gamma photon rarely meets an atom, so it goes far but ionises little. That is why an alpha source is harmless outside the body and dangerous if swallowed.
A nucleus lies well above the stability belt: it has too many neutrons for its protons. Which decay brings it towards the belt?
A radioactive salt decays at a certain rate. Which of these changes would leave that rate the same?
In 1896 Henri Becquerel in Paris was testing whether uranium salts, left in sunlight, gave out X-rays. A week of cloudy weather stopped him, and he put a salt and a wrapped photographic plate away in a drawer. When he developed the plate anyway, it was strongly darkened. The uranium had given out rays by itself, with no sunlight at all. Marie Curie took up the study. She found that the strength of the rays depended only on the amount of uranium, whatever its compound. She concluded that the rays came from inside the atom itself, and she named the effect radioactivity. With Pierre Curie she found two new radioactive elements in 1898, polonium and radium.
Shedding an alpha particle
In alpha decay a heavy nucleus gives out an alpha particle, a helium-4 nucleus of two protons and two neutrons. The parent nucleus X loses four nucleons, two of them protons, and becomes the daughter nucleus Y. Uranium-238 becomes thorium-234: \[ {}^{238}_{92}\text{U} \;\to\; {}^{234}_{90}\text{Th} + {}^{4}_{2}\text{He} \]
Every decay equation balances in two ways. The mass numbers on the top add up to the same total on both sides, \(238 = 234 + 4\). The charges on the bottom do too, \(92 = 90 + 2\). The first says that nucleons are neither made nor destroyed. The second is the conservation of charge. With these two sums, the daughter of any decay can be found without a table.
The energy of a decay
A decay happens by itself only if it releases energy. The products must have less rest mass than the parent, and the lost mass appears as kinetic energy. This energy is the Q value of the decay. From the chapter on relativity, a loss of mass \(\Delta m\) releases \(\Delta m\,c^2\), and \(1\,\text{u}\) of mass is \(931.5\,\text{MeV}\).
Tables list the masses of neutral atoms, electrons included, not of bare nuclei. For alpha decay this causes no trouble. The parent atom has \(Z\) electrons. The daughter atom has \(Z - 2\), and the helium atom has 2. The electron masses on the two sides are equal and cancel, so atomic masses may be used directly.
The atomic masses are: uranium-238, \(238.050788\,\text{u}\); thorium-234, \(234.043601\,\text{u}\); helium-4, \(4.002603\,\text{u}\). Find the Q value of the alpha decay of uranium-238.
The decay is \({}^{238}_{92}\text{U} \to {}^{234}_{90}\text{Th} + {}^{4}_{2}\text{He}\). Use atomic masses; the electrons cancel.
\(Q = (M_U - M_{Th} - M_{He}) \times 931.5\,\text{MeV/u}\).
\(M_{Th} + M_{He} = 238.046204\,\text{u}\), so \(\Delta m = 238.050788 - 238.046204 = 0.004584\,\text{u}\). Then \(Q = 0.004584 \times 931.5 = 4.27\,\text{MeV}\).
Each uranium-238 nucleus releases \(4.27\,\text{MeV}\) when it decays. Values used: the three atomic masses above, \(1\,\text{u} = 931.5\,\text{MeV}/c^2\).
The mass lost is small, 0.004584 u out of 238, less than two parts in a hundred thousand. Yet \(4.27\,\text{MeV}\) is about a million times the energy of a chemical reaction per atom. This is why a gram of radium stays warm by itself for centuries.
Inside the nucleus the alpha particle faces a hill of electric potential energy nearly \(30\,\text{MeV}\) high, far above its \(4\,\text{MeV}\). By classical physics it could never get out. In 1928 George Gamow showed that it leaks through the hill by quantum tunnelling, as a wave leaks through a thin barrier. The chance of leaking is tiny and depends very strongly on the energy. That is why alpha emitters have half-lives from a millionth of a second to billions of years.
Polonium-218 (\(Z = 84\)) decays by giving out an alpha particle. The daughter nucleus is
Radium-226 decays to radon-222 by alpha emission. The atomic masses are \(226.025410\,\text{u}\) for radium-226, \(222.017578\,\text{u}\) for radon-222 and \(4.002603\,\text{u}\) for helium-4. Find the Q value, in MeV. Take \(1\,\text{u} = 931.5\,\text{MeV}/c^2\).
Who gets the energy
The Q value is shared between the alpha particle and the daughter. Momentum decides the share, exactly as it does when a gun recoils. Before you read on, predict the answer.
A uranium-238 nucleus at rest splits into a thorium-234 nucleus and an alpha particle, releasing \(4.27\,\text{MeV}\). How is this energy shared?
The parent is at rest, so the total momentum is zero before the decay and after it. The alpha particle and the daughter fly apart with momenta of equal size \(p\), in opposite directions. The derivation below turns this into a share of energy.
\[ m_\alpha v_\alpha = m_Y v_Y = p \]
Why. the total momentum stays zero, so the two momenta are equal in size and opposite in direction.
\[ K = \dfrac{p^2}{2m} \]
Why. kinetic energy in terms of momentum, from \(K = \tfrac12 mv^2\) and \(p = mv\).
\[ \dfrac{K_\alpha}{K_Y} = \dfrac{m_Y}{m_\alpha} \]
Why. the same \(p\) appears in both energies, so they are in the inverse ratio of the masses.
\[ K_\alpha + K_Y = Q \]
Why. the whole released energy becomes the kinetic energy of the two products.
\[ K_\alpha = Q\,\dfrac{m_Y}{m_Y + m_\alpha}, \qquad K_Y = Q\,\dfrac{m_\alpha}{m_Y + m_\alpha} \]
Read it. solve the last two lines together.
For uranium-238, \(K_\alpha = 4.27 \times 234/238 = 4.20\,\text{MeV}\). The thorium nucleus gets \(4.27 \times 4/238 = 0.072\,\text{MeV}\), about \(72\,\text{keV}\). The figure above shows the two bars. The alpha particle leaves at about \(1.4 \times 10^7\,\text{m/s}\), and the daughter recoils 58.5 times more slowly.
There is a second message in this result. Every alpha particle from uranium-238 has the same energy, \(4.20\,\text{MeV}\), because two bodies with equal and opposite momenta can share \(Q\) in only one way. Alpha particles come out of a given decay with one sharp energy. The next lesson shows that beta particles do not, and why that mattered.
Radium-226 at rest decays by alpha emission with \(Q = 4.87\,\text{MeV}\). Find the kinetic energy of the alpha particle, in MeV.
A very slow neutron is captured by a lithium-6 nucleus at rest, and the reaction \(n + {}^{6}\text{Li} \to {}^{3}\text{H} + \alpha\) releases \(4.78\,\text{MeV}\). The tritium nucleus (mass number 3) and the alpha particle fly apart. Find the kinetic energy of the alpha particle, in MeV.
Why does every alpha particle from the decay of uranium-238 at rest come out with the same kinetic energy?
The parent is at rest, so the alpha particle and the thorium nucleus have equal and opposite momenta. With the same momentum, their kinetic energies are in the fixed ratio 234 to 4, and together they must add up to the fixed Q value. Only one pair of energies satisfies both conditions, so the alpha particle always gets 4.20 MeV.
- Thinking the heavier product gets more energy because it has more mass.The momenta are equal, and \(K = p^2/2m\). The light alpha particle takes almost all of \(Q\).
- Using the masses of nuclei in one place and of atoms in another.Use atomic masses for every term. In alpha decay the electrons then cancel exactly.
- Balancing only the mass numbers of a decay equation.Balance the charges as well. An alpha particle lowers \(Z\) by 2, so the element changes.