Nuclear Emissions and Decay Equations

Unstable nuclei release radiation

An atomic nucleus contains positively charged protons and uncharged neutrons. Some combinations of these particles are stable; others are unstable. For example, carbon-12 is stable, whereas carbon-14, which has two more neutrons, is radioactive.

An unstable nucleus can release energy by emitting particles or electromagnetic waves. This process is radioactive decay, and it changes the nucleus towards a more stable state. The remaining nucleus may still be unstable and decay again.

Radioactivity is the property of unstable nuclei that spontaneously emit radiation. Radiation is what travels away from them. Nuclear radiation can take four forms here: alpha particles, beta particles, neutrons and gamma rays. The first three are particles; gamma rays are electromagnetic radiation.

Reading a nuclear equation

To describe a decay, we need to keep track of the particles in the nucleus. A nucleus is written as:

$${}^{A}_{Z}\mathrm{X}$$

$\mathrm{X}$ is the element symbol. The upper number, $A$, is the mass number: the total number of protons and neutrons. The lower number, $Z$, is the : the number of protons. The number of neutrons is therefore .

For example, ${}^{14}_{6}\mathrm{C}$ is carbon-14. Its nucleus contains six protons and eight neutrons. Its nuclear charge is $+6e$, where $e$ is the size of one proton's charge.

A decay equation shows the parent nucleus on the left and the daughter nucleus plus the emitted radiation on the right. Two totals must balance separately:

• The sum of the upper numbers must be equal on both sides.
• The sum of the lower numbers must be equal on both sides.

For nuclei, the lower number counts protons. For an emitted particle, it represents its charge in units of $e$. This allows the equation to keep track of electric charge as well as nucleon number.

Alpha emission removes two protons and two neutrons

An alpha particle contains two protons and two neutrons. It is a helium nucleus, written ${}^{4}_{2}\mathrm{He}$ or , with charge .

When a nucleus emits an alpha particle, its mass number decreases by four and its atomic number decreases by two. Its mass and positive charge both decrease. Because its proton number changes, it becomes a different element.

For example, polonium-212 decays into lead-208:

$${}^{212}_{84}\mathrm{Po}\rightarrow{}^{208}_{82}\mathrm{Pb}+{}^{4}_{2}\mathrm{He}$$

The upper numbers balance because $212=208+4$. The lower numbers balance because $84=82+2$. The daughter nucleus has lost exactly the two protons and two neutrons carried away by the alpha particle.

Beta emission changes a neutron into a proton

The beta emission considered here is beta-minus decay. Inside the nucleus, a neutron changes into a proton and an electron. The proton stays in the nucleus; the electron is emitted at high speed as a beta particle.

The total number of protons and neutrons stays the same, so the mass number is unchanged. However, the nucleus now has one more proton. Its atomic number increases by one, its positive charge increases by $e$, and it becomes a different element.

The emitted electron is written ${}^{0}_{-1}\mathrm{e}$ or ${}^{0}_{-1}\beta$. Its upper number is zero because it is not a nucleon; its mass is very small compared with that of a proton or neutron.

Carbon-14 undergoes beta-minus decay to form nitrogen-14:

$${}^{14}_{6}\mathrm{C}\rightarrow{}^{14}_{7}\mathrm{N}+{}^{0}_{-1}\mathrm{e}$$

The upper numbers balance as $14=14+0$. The charge numbers balance as $6=7+(-1)$. Although negative charge leaves the nucleus, charge is conserved: the daughter nucleus is correspondingly more positive.

Gamma emission releases energy without removing nucleons

A gamma ray is a high-energy electromagnetic wave. It carries energy but has no electric charge and does not contain protons or neutrons. In nuclear equations it is written ${}^{0}_{0}\gamma$.

Gamma emission often follows another type of decay, leaving the nucleus with less energy. Neither its proton number nor its neutron number changes, so both atomic number and mass number remain unchanged. It remains the same isotope.

An excited uranium-238 nucleus, for example, can release energy as a gamma ray:

$${}^{238}_{92}\mathrm{U}^{*}\rightarrow{}^{238}_{92}\mathrm{U}+{}^{0}_{0}\gamma$$

The star indicates an excited, higher-energy nucleus. Both sets of numbers balance, even though energy has been carried away.

Neutron emission changes the isotope, not the element

An emitted neutron is written ${}^{1}_{0}\mathrm{n}$. It contributes one to the mass number but nothing to the charge number.

When a nucleus emits one neutron, its mass number decreases by one. Its atomic number and nuclear charge remain unchanged because it has not lost a proton. The daughter is a different isotope of the same element.

For example, beryllium-13 can emit a neutron to form beryllium-12:

$${}^{13}_{4}\mathrm{Be}\rightarrow{}^{12}_{4}\mathrm{Be}+{}^{1}_{0}\mathrm{n}$$

Here, $13=12+1$ and $4=4+0$.

Building and checking an equation

Start by writing the parent nucleus and the emitted radiation in nuclear notation. Find the missing daughter numbers by making the upper and lower totals balance independently. Then use the daughter's atomic number to identify its element.

For instance, if a nucleus with mass number 210 and atomic number 84 emits an alpha particle, the daughter must have mass number $210-4=206$ and atomic number . Atomic number 82 identifies lead, so the daughter is . Finally, check that adding the daughter's numbers to the alpha particle's numbers reconstructs the parent.

Emissions and nuclear changes

Unstable nuclei emit radiation as they change towards a more stable state.