Nuclear Chain Reaction

A nuclear chain reaction is a self-sustaining series of nuclear fission events, where neutrons released by one fission cause additional fission events in other nuclei. Chain reactions are fundamental to both nuclear reactors and nuclear weapons.

Basic Concept

When a fissile nucleus such as ${}^{235}_{92}U$ absorbs a neutron, it becomes unstable and splits into two lighter nuclei, releasing:

Energy (approximately 200 MeV per fission),
2 to 3 free neutrons,
Gamma radiation.

These free neutrons can then induce further fission events, creating a chain reaction.

Multiplication Factor $k$

The growth or decay of the neutron population depends on the multiplication factor $k$ , defined as:

$k = \frac{\text{number of neutrons in one generation}}{\text{number of neutrons in the previous generation}}$

If $k = 1$ , the chain reaction is critical (steady state),
If $k > 1$ , the reaction is supercritical (neutron population grows exponentially),
If $k < 1$ , the reaction is subcritical (reaction dies out).

Neutron Population Over Generations

The number of neutrons after $n$ generations is:

$N_n = N_0 k^n$

where $N_0$ is the initial neutron number.

Critical Mass

The critical mass is the minimum amount of fissile material needed so that $k \geq 1$ . Factors influencing critical mass include:

Shape and density of the material,
Presence of neutron reflectors,
Purity and type of fissile material.

Controlling Chain Reactions

In nuclear reactors, control rods made of neutron-absorbing materials (e.g., boron, cadmium) regulate $k$ to maintain a controlled, steady reaction. The reactor is kept critical ( $k = 1$ ) for stable power generation.

Delayed Neutrons

Not all neutrons are emitted immediately; some are delayed neutrons released seconds after fission from radioactive decay of certain fission products. These delayed neutrons are crucial for reactor control and safety because they slow the reaction's response time.

Mathematical Model of Chain Reaction Dynamics

The time evolution of neutron density $n(t)$ in a reactor can be approximated by:

$\frac{dn}{dt} = \frac{\rho - \beta}{\Lambda} n(t) + \sum_i \lambda_i C_i(t),$

where:

$\rho$ is the reactivity,
$\beta$ is the delayed neutron fraction,
$\Lambda$ is the neutron generation time,
$C_i$ are concentrations of delayed neutron precursors,
$\lambda_i$ are their decay constants.

This equation governs reactor kinetics and is essential for safety analysis.

Chain Reaction in Nuclear Weapons

In weapons, the chain reaction is prompt, meaning it proceeds without delay, reaching supercriticality rapidly and releasing massive energy in microseconds.

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Nuclear Chain Reaction

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