Nuclear decay mechanisms and equations

There is now one extra proton and one fewer neutron. The total number of nucleons has not changed, so the mass number stays the same. The proton number increases by one:

The electron’s lower number is $-1$ because its relative charge is negative. Its upper number is zero because it is not a nucleon—not because it has no mass. Charge balances: the extra positive charge of the daughter nucleus is balanced by the negative charge carried away by the electron.

Beta-plus decay: a proton becomes a neutron

In beta-plus decay, a proton inside the nucleus changes into a neutron, and a positron is produced and emitted. A positron has the same mass as an electron, but positive rather than negative charge.

The nucleus now has one fewer proton and one extra neutron. Again, one nucleon has changed type rather than left the nucleus, so the mass number is unchanged. The proton number decreases by one:

The daughter nucleus has one less unit of positive charge, and the emitted positron carries that positive charge away.

Alpha and neutron emission: nucleons leave the nucleus

An alpha particle is a helium nucleus containing two protons and two neutrons. When it leaves, the daughter nucleus has four fewer nucleons and two fewer protons:

The element changes because the proton number decreases by two.

In neutron emission, a neutron leaves the nucleus. For the emission of one neutron:

The mass number decreases by one, but the proton number stays the same. The daughter is therefore a different isotope of the original element. If more than one neutron is emitted, the mass number decreases by the number emitted.

Gamma emission: losing energy without losing nucleons

After radioactive decay, the daughter nucleus often still has excess energy. It can undergo nuclear rearrangement, releasing this energy as gamma radiation. Gamma radiation is electromagnetic radiation, not a proton or neutron leaving the nucleus.

Neither the mass number nor the proton number changes:

The identical nuclear symbols on each side show that the nucleus contains the same numbers of protons and neutrons. What has changed is its energy: the nucleus is now in a lower-energy state. Gamma emission may therefore follow another type of decay without causing a further change of element.

Balancing a nuclear equation

A nuclear equation must balance in two separate ways:

• The sum of the upper numbers—the nucleon numbers—is equal on both sides.
• The sum of the lower numbers—representing electrical charge—is equal on both sides.

For nuclei, the lower number is the proton number. For emitted particles, it represents their relative charge: an electron has $-1$, a positron has $+1$, and a neutron has $0$.

Consider this beta-plus practice equation, where $X$ and $Y$ stand for elements:

Balance the top numbers first: $22=A+0$, so $A=22$. Then balance the bottom numbers: $11=Z+1$, so $Z=10$. The completed daughter symbol is ${}^{22}_{10}Y$. It has ten protons and twelve neutrons, compared with eleven protons and eleven neutrons in the parent—exactly the proton-to-neutron change expected in beta-plus decay.

The same method identifies a missing emission. As an abstract balancing exercise, consider

The missing particle must have upper number $24-23=1$ and lower number $11-11=0$. It is therefore a neutron, ${}^{1}_{0}n$. This exercise identifies the particle required to balance the numbers; balancing alone does not establish that a particular nucleus actually undergoes that decay. When an equation contains several emitted particles, add all their upper numbers and all their lower numbers before finding the missing values.