Beta and Gamma Decay

A neutron becomes a proton

A nucleus above the stability belt has too many neutrons. It cannot throw a neutron out, because the neutron is bound. Instead one neutron inside the nucleus turns into a proton. An electron and a particle called an antineutrino are created at that moment, and both leave the nucleus: \[ n \;\to\; p + e^- + \bar\nu \] This is beta-minus decay. The electron was never stored inside the nucleus; it is made in the decay, as a photon is made when an atom emits light.

The proton stays in the nucleus, so \(Z\) rises by one. The total number of nucleons is the same, so \(A\) does not change. Carbon-14 decays this way to nitrogen-14: \[ {}^{14}_{6}\text{C} \;\to\; {}^{14}_{7}\text{N} + e^- + \bar\nu \] The equation balances: \(14 = 14 + 0\) on the top, and \(6 = 7 - 1\) on the bottom, since the electron has charge \(-1\) and mass number 0.

A free neutron outside any nucleus decays in the same way, with a half-life of about 10 minutes. A neutron is heavier than a proton and an electron together, so the decay releases energy. Inside a stable nucleus the same change would cost energy, and it does not happen.

The energy of a beta-minus decay

With atomic masses, the Q value of beta-minus decay is simply the difference of the two atomic masses. The parent atom has \(Z\) electrons. The daughter atom needs \(Z + 1\), and the extra one is the mass of the emitted electron. So the electron's mass is already counted inside the daughter's atomic mass.

For carbon-14 the atomic masses are \(14.003242\,\text{u}\) and \(14.003074\,\text{u}\) for nitrogen-14. The difference is \(0.000168\,\text{u}\), and \(Q = 0.000168 \times 931.5 = 0.156\,\text{MeV}\). This small energy is shared among the electron, the antineutrino and the recoiling nitrogen nucleus.

A proton becomes a neutron

A nucleus below the belt has too many protons. One proton inside it turns into a neutron, and a positron, the positive twin of the electron, is created with a neutrino: \[ p \;\to\; n + e^+ + \nu \] This is beta-plus decay. Now \(Z\) falls by one while \(A\) stays the same. Carbon-11 becomes boron-11: \({}^{11}_{6}\text{C} \to {}^{11}_{5}\text{B} + e^+ + \nu\).

A free proton never decays like this, because a neutron is heavier than a proton. Inside a nucleus the change can happen when the daughter nucleus is bound much more tightly than the parent. The extra binding pays for the heavier neutron and the positron.

The atomic masses now leave two electron masses over. The daughter atom has one electron fewer than the parent, and a positron of the same mass has been created as well. So \(Q = (M_X - M_Y - 2m_e)c^2\), and \(2m_ec^2 = 1.022\,\text{MeV}\). For carbon-11 the atomic masses differ by \(0.002128\,\text{u}\), which is \(1.982\,\text{MeV}\), so \(Q = 1.982 - 1.022 = 0.96\,\text{MeV}\).

Electron capture

A nucleus with too many protons has a second way out. It can take in one of its own inner electrons, usually from the K shell, closest to the nucleus. The electron and a proton become a neutron, and only a neutrino leaves: \[ p + e^- \;\to\; n + \nu \] This is electron capture. Beryllium-7 decays this way to lithium-7. As in beta-plus decay, \(Z\) falls by one and \(A\) stays the same. An outer electron then fills the empty place in the K shell. The atom gives out an X-ray, as the chapter on X-rays describes.

Fig 2.1 shows each decay as a move on the chart. Every beta decay keeps \(A\) fixed, so it moves along a line of constant \(A\), one square across and one square back. So after the first alpha decay, the uranium series in the figure stays on nuclei of mass number 234.

Electrons of every energy

Every carbon-14 nucleus releases the same \(0.156\,\text{MeV}\). If an electron and the daughter were the only products, each electron would carry the same energy, as each alpha particle does. Measurements in the 1910s and 1920s showed something else. The electrons of one beta emitter come out with every energy from nearly zero up to a highest value, and the highest value equals \(Q\).

The answer is a third particle. With three products, momentum no longer fixes the sharing. The electron and the antineutrino can fly off at any angle to each other, and the energy is shared differently in every decay. An electron with nearly all of \(Q\) leaves the antineutrino almost nothing, and an electron with almost nothing leaves it nearly all. Fig 2.2 builds the spread one decay at a time.

The daughter nucleus takes part in the momentum balance, but it is thousands of times heavier than an electron. By the result of lesson 1, its share of the energy is tiny. So in practice \(K_e + K_{\bar\nu} = Q\).

Energy levels of the nucleus

A nucleus, like an atom, has energy levels. Recall from the chapter on the Bohr atom that an atom falling to a lower level gives out a photon of the energy difference. A nucleus does the same. The difference is the size. The levels of an atom are a few eV apart, and those of a nucleus about a million times farther.

In gamma decay an excited nucleus, written with a star, falls to a lower level and gives out a gamma photon: \[ {}^{A}_{Z}\text{X}^* \;\to\; {}^{A}_{Z}\text{X} + \gamma \] A photon has no charge and no nucleons, so \(A\) and \(Z\) stay the same. Only the energy of the nucleus falls.

An alpha or beta decay often leaves the daughter in an excited state, and a gamma photon follows at once. Cobalt-60 is the standard example. It decays by beta-minus to nickel-60, but the nickel nucleus is born on a level \(2.506\,\text{MeV}\) above its ground state. It then falls in two steps, giving out two gamma photons.

The energies add up, as Fig 2.3 shows. The beta particles share at most \(2.823 - 2.506 = 0.317\,\text{MeV}\). The two photons carry \(2.506 - 1.333 = 1.173\,\text{MeV}\) and \(1.333\,\text{MeV}\). A photon of \(1.33\,\text{MeV}\) has a wavelength of \(1240/(1.33\times10^6)\,\text{nm}\), about \(0.93\,\text{pm}\), far shorter than any X-ray from an atom.

From uranium to lead

The daughter of a decay is often radioactive itself. Uranium-238 decays to thorium-234, which decays to protactinium-234, and so on. The chain runs through fourteen decays before it reaches lead-206, which is stable. Such a chain is a decay series. The order of the steps needs a table, but the number of each kind of decay needs only two subtractions.

Only alpha decay changes the mass number, and it lowers \(A\) by exactly 4. So the number of alpha decays is the fall in \(A\) divided by 4. Each alpha decay also lowers \(Z\) by 2, and each beta-minus decay raises it by 1. The final \(Z\) then fixes the number of beta-minus decays.

The rule has a neat consequence. Since \(A\) changes only in steps of 4, every member of one series has the same remainder when \(A\) is divided by 4. For the uranium-238 series the remainder is 2, since \(238 = 4 \times 59 + 2\). The thorium-232 series has remainder 0, and the uranium-235 series has remainder 3. So the natural series never cross into one another.