K323 / 2027
Radioactivity overview

Topic 3 of 7

Nuclear decay equations

A nuclear equation records the daughter nucleus and emitted radiation. Check both the total nucleon number and the total charge number.

In nuclide notation, A counts nucleons and Z counts protons. An alpha particle contains two protons and two neutrons; beta-minus is an electron; gamma carries away nuclear energy.

Balance the upper and lower numbers separately

  1. Upper numbers: the total number of nucleons is the same before and after the emission.
  2. Lower numbers: the total charge number is the same. For a nucleus, this is its proton number; for an emitted electron, it is -1.
  3. Identify the daughter: its new proton number determines the element. Use the supplied element information when needed.

The lower -1 on a beta-minus electron does not mean that it contains negative protons. Its upper 0 means zero nucleons, not zero electron mass.

Balance nucleon and charge numbers

For a nucleus, the lower number is its proton number. For the emitted electron, -1 is its charge number; its upper 0 means no nucleons, not zero mass. The nucleus boxes give counts, not a drawing of every nucleon.

Alpha: four nucleons leave together

Alpha: four nucleons leave togetherUranium-238, proton number 92, becomes thorium-234, proton number 90, and a helium-4 nucleus with proton number 2. The parent has 92 protons and 146 neutrons; the daughter has 90 protons and 144 neutrons. The emitted alpha particle contains the remaining two protons and two neutrons. Nucleon totals balance as 238 equals 234 plus 4; lower charge numbers balance as 92 equals 90 plus 2.23892U23490Th42He+Before: nucleusAfter: nucleus92 protons146 neutrons90 protons144 neutronsEmitted alpha particlep+nnp+2 protons + 2 neutronsCharge +2238 = 234 + 492 = 90 + 2

Both nucleon number and total charge balance. Two protons and two neutrons leave the parent nucleus together.

Beta-minus: a neutron changes into a proton

Beta-minus: a neutron changes into a protonCarbon-14, proton number 6, becomes nitrogen-14, proton number 7, with a newly emitted electron. The nucleus changes from 6 protons and 8 neutrons to 7 protons and 7 neutrons. The inset shows one neutron changing into a proton with an electron produced during the change; no electron is drawn waiting in the original nucleus. This school bookkeeping omits the antineutrino, which is also emitted. The upper totals balance as 14 equals 14 plus 0; the lower charge totals balance as 6 equals 7 plus negative 1.146C147N0-1e+Before: nucleusAfter: nucleus6 protons8 neutrons7 protons7 neutronsOne neutron changesnp++e-NeutronProtonElectronThe electron is produced during the change.14 = 14 + 06 = 7 + (-1)

The beta electron is produced during the nuclear change; it is not ejected from an existing electron shell or stored inside the nucleus. An antineutrino is also emitted; the displayed school-level bookkeeping omits it. Antineutrino recall is not needed here.

Gamma: the same nuclide at lower nuclear energy

Gamma: the same nuclide at lower nuclear energyAn excited nucleus X, with nucleon number A and proton number Z, emits a gamma photon and reaches a lower nuclear energy. A star on the parent symbol marks its higher-energy nuclear state. Both nuclei have Z protons and A minus Z neutrons. Neither count changes. The photon carries energy away; its upper and lower bookkeeping numbers are both zero. The starred and unstarred symbols describe the same nuclide, not different isotopes.AZX*AZX00γ+Before: nucleusAfter: nucleusZ protonsA - Z neutronsZ protonsA - Z neutronsHigher energyLower energy* = higher-energy nucleusPhoton carries energy awaySame A and Z; lower nuclear energyA = A + 0Z = Z + 0

The star marks extra nuclear energy. Removing it changes the energy state, while the proton and neutron counts remain the same.

Alpha removes two protons and two neutrons. Beta-minus changes one neutron into a proton while emitting an electron. Gamma lowers nuclear energy without changing either nucleon count. The upper and lower bookkeeping rows must both balance.
Alpha: A falls by 4; Z falls by 2
The daughter loses the four nucleons carried away in the helium nucleus.
Beta-minus: A is unchanged; Z rises by 1
One neutron becomes a proton, so the total nucleon count stays the same. The emitted negative charge balances the daughter's increased positive charge.
Gamma: A and Z are unchanged
The daughter has the same composition but lower nuclear energy. A star on the initial symbol marks a higher-energy nuclear state; it does not mean an extra proton or neutron.

Two emissions in order

Start with A = 214 and Z = 84

The nucleus emits an alpha particle, then a beta-minus particle.

  1. After alpha: A = 214 - 4 = 210; Z = 84 - 2 = 82.
  2. After beta-minus: A stays 210; Z = 82 + 1 = 83.
  3. Count the final neutrons: A - Z = 210 - 83 = 127.

The second step acts on the daughter from the first step. Do not apply both emissions separately to the original nucleus and discard one of the changes.

A nucleus losing energy need not lose nucleons: gamma emission is the example. Conversely, the beta electron's negative charge does not mean the daughter's proton number decreases.

Optional check A nucleus with A = 214 and Z = 84 emits an alpha particle, then a beta-minus particle. What are its final nucleon, proton and neutron numbers?
A nucleus with A = 214 and Z = 84 emits an alpha particle, then a beta-minus particle. What are its final nucleon, proton and neutron numbers?