Nuclear Decay Equations

Nuclear decay equations show how the nucleus changes when it emits radiation. The nucleon number and proton number must balance on both sides.

Alpha decay

An alpha particle ($\alpha$) is a helium-4 nucleus: $^4_2\text{He}$.

$^A_Z X \rightarrow ^{A-4}_{Z-2} Y + ^4_2 \text{He}$

The nucleon number decreases by 4; the proton number decreases by 2.

Example: Uranium-238 undergoes alpha decay:

$^{238}_{92}\text{U} \rightarrow ^{234}_{90}\text{Th} + ^4_2\text{He}$

Beta-minus decay

A beta-minus particle ($\beta^-$) is a high-speed electron: $^0_{-1}\text{e}$.

A neutron converts to a proton, emitting an electron:

$^A_Z X \rightarrow ^{A}_{Z+1} Y + ^0_{-1} \text{e}$

The nucleon number stays the same; the proton number increases by 1.

Example: Carbon-14 undergoes beta decay:

$^{14}_6\text{C} \rightarrow ^{14}_7\text{N} + ^0_{-1}\text{e}$

Gamma emission

Gamma rays ($\gamma$) have no mass and no charge: $^0_0\gamma$.

$^A_Z X^* \rightarrow ^A_Z X + ^0_0 \gamma$

Neither the nucleon number nor the proton number changes. The nucleus simply releases excess energy.

Balancing equations

• Total nucleon numbers on left = total nucleon numbers on right.
• Total proton numbers on left = total proton numbers on right.

Common errors and how to correct them

• Getting the direction of proton number change wrong in beta decay. Beta-minus emission increases $Z$ by 1 (a neutron becomes a proton).
• Forgetting that gamma emission does not change $A$ or $Z$.

How to apply this in an exam

Write the parent nucleus on the left. Write the daughter nucleus and the emitted particle on the right. Check that both $A$ and $Z$ balance.