22.3.2 Beta Decay | Nuclear Physics | Physics 12

Beta decay is a type of radioactive decay in which a nucleus emits a beta particle — either an electron ( $e^{-}$ ) or a positron ( $e^{+}$ ) — along with a neutrino or antineutrino. It is governed by the Weak Nuclear Force and occurs spontaneously and randomly in unstable nuclei.

Types of Beta Decay

Beta-Minus ( $β^{-}$ ) Decay

In $β^{-}$ decay, a neutron inside the nucleus transforms into a proton, emitting an electron and an antineutrino ( $\overset{ν}{ˉ}$ ):

$n \to p + e^{-} + \overset{ν}{ˉ}$

The general nuclear equation is:

$_{Z}^{A} X \to_{Z + 1}^{A} Y + e^{-} + \overset{ν}{ˉ}$

Changes in nuclear numbers:

Mass number $A$ : unchanged (total nucleons conserved)
Atomic number $Z$ : increases by 1 (one more proton)

Example: Carbon-14 undergoes $β^{-}$ decay: $_{6}^{14} C \to_{7}^{14} N + e^{-} + \overset{ν}{ˉ}$

Beta-Plus ( $β^{+}$ ) Decay

In $β^{+}$ decay, a proton inside the nucleus transforms into a neutron, emitting a positron ( $e^{+}$ ) and a neutrino ( $ν$ ):

$p \to n + e^{+} + ν$

The general nuclear equation is:

$_{Z}^{A} X \to_{Z - 1}^{A} Y + e^{+} + ν$

Changes in nuclear numbers:

Mass number $A$ : unchanged
Atomic number $Z$ : decreases by 1 (one fewer proton)

Example: Sodium-22 undergoes $β^{+}$ decay: $_{11}^{22} Na \to_{10}^{22} Ne + e^{+} + ν$

Beta Decay at the Quark Level

Beta decay is explained at the fundamental level by the Weak Nuclear Force acting on quarks:

Decay Type	Quark Change	Result
$β^{-}$	$d \to u$	Neutron ( $u dd$ ) → Proton ( $uu d$ )
$β^{+}$	$u \to d$	Proton ( $uu d$ ) → Neutron ( $u dd$ )

A down quark ( $d$ , charge $- \frac{1}{3} e$ ) changes into an up quark ( $u$ , charge $+ \frac{2}{3} e$ ) in $β^{-}$ decay, with the charge difference carried away by the emitted electron.

The Role of the Neutrino

Before the neutrino was discovered, beta decay appeared to violate conservation of energy and momentum — beta particles were observed with a continuous range of energies rather than a single discrete value (as in alpha decay).

In 1930, Wolfgang Pauli postulated the existence of a new particle — the neutrino — to account for the missing energy and momentum. The neutrino:

Has negligible (near-zero) mass
Is electrically neutral
Carries away the remaining energy and momentum
Conserves angular momentum (spin)

Continuous Energy Spectrum

The total decay energy (Q-value) is shared between the beta particle and the neutrino/antineutrino in varying proportions for each decay event. Therefore:

The beta particle can have any kinetic energy from zero up to a maximum value $E_{ma x}$
$E_{ma x}$ equals the total Q-value (when the neutrino carries negligible energy)

$E_{β} + E_{ν} = Q$

Conservation Laws in Beta Decay

Beta decay conserves:

Quantity	Conserved?	Notes
Mass number ( $A$ )	✓ Yes	Nucleon number unchanged
Charge	✓ Yes	Total charge before = total charge after
Energy	✓ Yes	Shared between $β$ particle and neutrino
Linear momentum	✓ Yes	Neutrino carries missing momentum
Angular momentum	✓ Yes	Neutrino has spin $\frac{1}{2}$
Lepton number	✓ Yes	$e^{-}$ and $\overset{ν}{ˉ}$ in $β^{-}$ ; $e^{+}$ and $ν$ in $β^{+}$

Spontaneous and Random Nature of Beta Decay

Like all radioactive decay, beta decay is:

Spontaneous: It occurs without any external trigger. No change in temperature, pressure, or chemical state can initiate or prevent it.
Random: It is impossible to predict which specific nucleus in a sample will decay next, or exactly when it will decay. Only statistical predictions about large numbers of nuclei are possible.

This spontaneous and random nature is a fundamental property of the weak nuclear force acting within the nucleus.

Summary

Property	$β^{-}$ Decay	$β^{+}$ Decay
Particle emitted	Electron ( $e^{-}$ )	Positron ( $e^{+}$ )
Also emitted	Antineutrino ( $\overset{ν}{ˉ}$ )	Neutrino ( $ν$ )
Change in $Z$	$+ 1$	$- 1$
Change in $A$	$0$	$0$
Quark change	$d \to u$	$u \to d$
Force responsible	Weak Nuclear Force	Weak Nuclear Force

Types of Beta Decay

Beta-Minus ( $β^{-}$ ) Decay

In $β^{-}$ decay, a neutron inside the nucleus transforms into a proton, emitting an electron and an antineutrino ( $\overset{ν}{ˉ}$ ):

$n \to p + e^{-} + \overset{ν}{ˉ}$

The general nuclear equation is:

$_{Z}^{A} X \to_{Z + 1}^{A} Y + e^{-} + \overset{ν}{ˉ}$

Changes in nuclear numbers:

Mass number $A$ : unchanged (total nucleons conserved)
Atomic number $Z$ : increases by 1 (one more proton)

Example: Carbon-14 undergoes $β^{-}$ decay: $_{6}^{14} C \to_{7}^{14} N + e^{-} + \overset{ν}{ˉ}$

Beta-Plus ( $β^{+}$ ) Decay

In $β^{+}$ decay, a proton inside the nucleus transforms into a neutron, emitting a positron ( $e^{+}$ ) and a neutrino ( $ν$ ):

$p \to n + e^{+} + ν$

The general nuclear equation is:

$_{Z}^{A} X \to_{Z - 1}^{A} Y + e^{+} + ν$

Changes in nuclear numbers:

Mass number $A$ : unchanged
Atomic number $Z$ : decreases by 1 (one fewer proton)

Example: Sodium-22 undergoes $β^{+}$ decay: $_{11}^{22} Na \to_{10}^{22} Ne + e^{+} + ν$

Beta Decay at the Quark Level

Beta decay is explained at the fundamental level by the Weak Nuclear Force acting on quarks:

Decay Type	Quark Change	Result
$β^{-}$	$d \to u$	Neutron ( $u dd$ ) → Proton ( $uu d$ )
$β^{+}$	$u \to d$	Proton ( $uu d$ ) → Neutron ( $u dd$ )

A down quark ( $d$ , charge $- \frac{1}{3} e$ ) changes into an up quark ( $u$ , charge $+ \frac{2}{3} e$ ) in $β^{-}$ decay, with the charge difference carried away by the emitted electron.

The Role of the Neutrino

In 1930, Wolfgang Pauli postulated the existence of a new particle — the neutrino — to account for the missing energy and momentum. The neutrino:

Has negligible (near-zero) mass
Is electrically neutral
Carries away the remaining energy and momentum
Conserves angular momentum (spin)

Continuous Energy Spectrum

The total decay energy (Q-value) is shared between the beta particle and the neutrino/antineutrino in varying proportions for each decay event. Therefore:

The beta particle can have any kinetic energy from zero up to a maximum value $E_{ma x}$
$E_{ma x}$ equals the total Q-value (when the neutrino carries negligible energy)

$E_{β} + E_{ν} = Q$

Conservation Laws in Beta Decay

Beta decay conserves:

Quantity	Conserved?	Notes
Mass number ( $A$ )	✓ Yes	Nucleon number unchanged
Charge	✓ Yes	Total charge before = total charge after
Energy	✓ Yes	Shared between $β$ particle and neutrino
Linear momentum	✓ Yes	Neutrino carries missing momentum
Angular momentum	✓ Yes	Neutrino has spin $\frac{1}{2}$
Lepton number	✓ Yes	$e^{-}$ and $\overset{ν}{ˉ}$ in $β^{-}$ ; $e^{+}$ and $ν$ in $β^{+}$

Spontaneous and Random Nature of Beta Decay

Like all radioactive decay, beta decay is:

Spontaneous: It occurs without any external trigger. No change in temperature, pressure, or chemical state can initiate or prevent it.
Random: It is impossible to predict which specific nucleus in a sample will decay next, or exactly when it will decay. Only statistical predictions about large numbers of nuclei are possible.

This spontaneous and random nature is a fundamental property of the weak nuclear force acting within the nucleus.

Summary

Property	$β^{-}$ Decay	$β^{+}$ Decay
Particle emitted	Electron ( $e^{-}$ )	Positron ( $e^{+}$ )
Also emitted	Antineutrino ( $\overset{ν}{ˉ}$ )	Neutrino ( $ν$ )
Change in $Z$	$+ 1$	$- 1$
Change in $A$	$0$	$0$
Quark change	$d \to u$	$u \to d$
Force responsible	Weak Nuclear Force	Weak Nuclear Force