22.5 Nuclear Reactions | Nuclear Physics | Physics 12

A nuclear reaction is a process in which two nuclei, or a nucleus and a subatomic particle, collide to produce one or more new nuclides. Nuclear reactions must obey four conservation laws:

Conservation of charge number ( $Z$ ): total proton number is conserved.
Conservation of mass number ( $A$ ): total nucleon number is conserved.
Conservation of mass-energy: total rest-mass energy plus kinetic energy is conserved.
Conservation of momentum: total linear momentum is conserved.

A general nuclear reaction is written as:

$a + X \to Y + b$

where $a$ is the projectile, $X$ is the target nucleus, $Y$ is the product nucleus, and $b$ is the ejected particle.

Transmutation

Transmutation is the conversion of one element or isotope into another through a nuclear reaction. It is typically achieved by bombarding a target nucleus with particles such as protons, alpha particles, or neutrons.

Example — Rutherford's first artificial transmutation (1919):

$_{7}^{14} N +_{2}^{4} He \to_{8}^{17} O +_{1}^{1} H$

Why are neutrons effective projectiles?

Neutrons carry no electric charge, so they experience no Coulomb (electrostatic) repulsion from the positively charged nucleus. This allows neutrons to penetrate the nucleus even at very low kinetic energies, making them highly effective for inducing nuclear reactions.

Q-Value of a Nuclear Reaction

The Q-value is the energy released (or absorbed) in a nuclear reaction. It equals the difference in rest-mass energy between reactants and products:

$Q = (m_{reactants} - m_{products}) c^{2}$

Using atomic mass units, since $1 u = 931.5 MeV / c^{2}$ :

$Q = Δ m \times 931.5 MeV$

Condition	Type	Meaning
$Q > 0$	Exothermic (exoergic)	Energy is released; products are more stable
$Q < 0$	Endothermic (endoergic)	Energy must be supplied; products are less stable

Binding Energy and Nuclear Reactions

The binding energy per nucleon (B.E./A) curve peaks near iron-56 ( $_{26}^{56} Fe$ ) at approximately $8.8 MeV/nucleon$ .

Heavy nuclei (large $A$ ) have a lower B.E./A than iron.
Light nuclei (small $A$ ) also have a lower B.E./A than iron.

Energy is released in a nuclear reaction whenever the B.E./A of the products is greater than the B.E./A of the reactants — i.e., the products are more tightly bound (more stable).

Nuclear Fission

Nuclear fission is the process in which a heavy nucleus (such as $_{92}^{235} U$ ) splits into two smaller nuclei of roughly equal mass, releasing energy and typically 2–3 neutrons.

Mechanism

When $_{92}^{235} U$ absorbs a slow (thermal) neutron, it forms the highly unstable $_{92}^{236} U^{*}$ , which then splits:

$_{0}^{1} n +_{92}^{235} U \to_{92}^{236} U^{*} \to_{56}^{141} Ba +_{36}^{92} Kr + 3_{0}^{1} n + energy$

Average energy released per fission: $\approx 200 MeV$
Slow (thermal) neutrons are preferred because $_{92}^{235} U$ has a much higher fission cross-section for slow neutrons.

Chain Reaction

The neutrons released in each fission can trigger further fissions, creating a chain reaction. For a self-sustaining chain reaction, the mass of fissile material must equal or exceed the critical mass.

Why does fission release energy?

The fission fragments ( $_{56}^{141} Ba$ , $_{36}^{92} Kr$ ) have a higher B.E./A than $_{92}^{235} U$ . The products are more tightly bound, so the excess binding energy is released as kinetic energy of the fragments and radiation.

Nuclear Fusion

Nuclear fusion is the process in which two or more light nuclei combine to form a heavier, more stable nucleus, releasing a large amount of energy.

Deuterium–Tritium (D–T) Fusion Reaction

$_{1}^{2} H +_{1}^{3} H \to_{2}^{4} He +_{0}^{1} n + 17.6 MeV$

Conditions Required for Fusion

Because both nuclei are positively charged, they must overcome the Coulomb barrier. This requires:

Extremely high temperatures: $\sim 1 0^{7} K$ or higher (thermonuclear conditions).
High pressure/density: to ensure sufficient collision frequency.

Fusion is the energy source of the Sun and other stars.

Why does fusion release energy?

Light nuclei (e.g., $_{1}^{2} H$ , $_{1}^{3} H$ ) have a lower B.E./A than the product $_{2}^{4} He$ . Combining them moves the product toward the peak of the B.E./A curve, releasing the difference in binding energy.

Fusion vs. Fission: Energy per Nucleon

Fusion releases more energy per nucleon than fission (~3.5 MeV/nucleon for D-T fusion vs. ~0.85 MeV/nucleon for U-235 fission), making it a potentially more efficient energy source.

Calculating Energy Released in Nuclear Reactions

Steps:

Write the balanced nuclear equation.
Find the mass defect: $Δ m = m_{reactants} - m_{products}$ (in u).
Convert to energy: $Q = Δ m \times 931.5 MeV$ .

Worked Example — D-T Fusion:

Given:

$m (_{1}^{2} H) = 2.014102 u$
$m (_{1}^{3} H) = 3.016049 u$
$m (_{2}^{4} He) = 4.002602 u$
$m (_{0}^{1} n) = 1.008665 u$

$Δ m = (2.014102 + 3.016049) - (4.002602 + 1.008665) = 0.018884 u$

$Q = 0.018884 \times 931.5 = 17.59 MeV \approx 17.6 MeV$

Summary Table

Feature	Fission	Fusion
Nuclei involved	Heavy (e.g., U-235)	Light (e.g., H-2, H-3)
Trigger	Slow neutron	Extreme temperature
Energy released	~200 MeV/reaction	~17.6 MeV/reaction
Energy per nucleon	~0.85 MeV	~3.5 MeV
Products	Medium-mass fragments + neutrons	He-4 + neutron
B.E./A change	Increases toward Fe-56 peak	Increases toward Fe-56 peak

A nuclear reaction is a process in which two nuclei, or a nucleus and a subatomic particle, collide to produce one or more new nuclides. Nuclear reactions must obey four conservation laws:

Conservation of charge number ( $Z$ ): total proton number is conserved.
Conservation of mass number ( $A$ ): total nucleon number is conserved.
Conservation of mass-energy: total rest-mass energy plus kinetic energy is conserved.
Conservation of momentum: total linear momentum is conserved.

A general nuclear reaction is written as:

$a + X \to Y + b$

where $a$ is the projectile, $X$ is the target nucleus, $Y$ is the product nucleus, and $b$ is the ejected particle.

Transmutation

Example — Rutherford's first artificial transmutation (1919):

$_{7}^{14} N +_{2}^{4} He \to_{8}^{17} O +_{1}^{1} H$

Why are neutrons effective projectiles?

Q-Value of a Nuclear Reaction

The Q-value is the energy released (or absorbed) in a nuclear reaction. It equals the difference in rest-mass energy between reactants and products:

$Q = (m_{reactants} - m_{products}) c^{2}$

Using atomic mass units, since $1 u = 931.5 MeV / c^{2}$ :

$Q = Δ m \times 931.5 MeV$

Condition	Type	Meaning
$Q > 0$	Exothermic (exoergic)	Energy is released; products are more stable
$Q < 0$	Endothermic (endoergic)	Energy must be supplied; products are less stable

Binding Energy and Nuclear Reactions

The binding energy per nucleon (B.E./A) curve peaks near iron-56 ( $_{26}^{56} Fe$ ) at approximately $8.8 MeV/nucleon$ .

Heavy nuclei (large $A$ ) have a lower B.E./A than iron.
Light nuclei (small $A$ ) also have a lower B.E./A than iron.

Energy is released in a nuclear reaction whenever the B.E./A of the products is greater than the B.E./A of the reactants — i.e., the products are more tightly bound (more stable).

Nuclear Fission

Nuclear fission is the process in which a heavy nucleus (such as $_{92}^{235} U$ ) splits into two smaller nuclei of roughly equal mass, releasing energy and typically 2–3 neutrons.

Mechanism

When $_{92}^{235} U$ absorbs a slow (thermal) neutron, it forms the highly unstable $_{92}^{236} U^{*}$ , which then splits:

$_{0}^{1} n +_{92}^{235} U \to_{92}^{236} U^{*} \to_{56}^{141} Ba +_{36}^{92} Kr + 3_{0}^{1} n + energy$

Average energy released per fission: $\approx 200 MeV$
Slow (thermal) neutrons are preferred because $_{92}^{235} U$ has a much higher fission cross-section for slow neutrons.

Extremely high temperatures: $\sim 1 0^{7} K$ or higher (thermonuclear conditions).
High pressure/density: to ensure sufficient collision frequency.

Fusion is the energy source of the Sun and other stars.

Write the balanced nuclear equation.
Find the mass defect: $Δ m = m_{reactants} - m_{products}$ (in u).
Convert to energy: $Q = Δ m \times 931.5 MeV$ .

Worked Example — D-T Fusion:

Given:

$m (_{1}^{2} H) = 2.014102 u$
$m (_{1}^{3} H) = 3.016049 u$
$m (_{2}^{4} He) = 4.002602 u$
$m (_{0}^{1} n) = 1.008665 u$

$Δ m = (2.014102 + 3.016049) - (4.002602 + 1.008665) = 0.018884 u$

$Q = 0.018884 \times 931.5 = 17.59 MeV \approx 17.6 MeV$

Summary Table

Feature	Fission	Fusion
Nuclei involved	Heavy (e.g., U-235)	Light (e.g., H-2, H-3)
Trigger	Slow neutron	Extreme temperature
Energy released	~200 MeV/reaction	~17.6 MeV/reaction
Energy per nucleon	~0.85 MeV	~3.5 MeV
Products	Medium-mass fragments + neutrons	He-4 + neutron
B.E./A change	Increases toward Fe-56 peak	Increases toward Fe-56 peak