# Fission reactors

Fission reaction, neutron transport, criticality, and reactor types. (Draft)

# Fission reaction

Fission reactors are based on a neutron-driven fission chain reaction producing heat. Most commonly:

$${}^{235}_{92}U + {}^1_0n \to [{}^{236}_{92}U] \to \text{fission products} + 2.44\ {}^1_0n + \gamma\text{ or }\beta + \overline{v} + Q$$

Where $$\overline{v}$$ are antineutrinos.

When a neutron hits a uranium nucleus (most of which happens in the thermal energy region), it is absorbed and they form a compound nucleus, which lasts for about $$10^{-14}$$ seconds and then splits. The thus produced fission products decay, either by emitting neutrons (almost instantaneously, after around $$10^{-17}$$ seconds) or by beta or gamma decay (after around $$10^{-13}$$ seconds). After about $$10^{-10}$$ to $$10^{-6}$$ seconds, we arrive at the right-hand side. The fission products carry kinetic energy in the order of nano meters, i.e. they stay in the fuel, which corresponds to the release of energy $$Q$$ (the Q-value of the reaction).

## Fuel

Fissionable material are isotopes that can undergo fission with either slow or fast neutrons.

Fissile material is the subset of isotopes that can undergo fission with thermal neutrons. For example: U-235, Pu-239 (where U-235 is the one with the largest cross-section for fission).

Fertile material are isotopes that can be converted into fissile material (through the absorption of neutrons and subsequent decay). For example U-238, Th-232.

## Moderator

The moderator slows down neutrons to the thermal energy range, where the probability of fission is much higher.

## Poisoning and slagging

Fuel burn-up and reactor poisoning/slagging

# Neutron transport

The neutron population $$n(\overrightarrow{r},E,\overrightarrow{\Omega},t)$$ is the number of neutrons at each position $$\overrightarrow{r}$$ (e.g. $$(x,y,z)$$ coordinates), for each energy level $$E$$ , each direction $$\overrightarrow{\Omega}$$ (solid angle $$(\Theta, \Phi)$$ , usually ignored when the number of neutrons within a reactor is considered) and each time $$t$$ .

Balance equation:

$$\cfrac{\delta n(\overrightarrow{r},E,\overrightarrow{\Omega},t)}{\delta t} = \text{gains} - \text{losses}$$

Where the gains are:

• fission
• $$(n, i\times n)$$ reactions
• external neutron sources (e.g. Cf to jumpstart reactor)
• scattering (neutrons from another energy range scatter and enter energy range $$E$$ )

And the losses are:

• any reaction (fission, capture, scattering, etc.) that changes the energy or angle of neutrons, combined in the total macroscopic cross-section $$\Sigma_\text{total}$$
• leakage (neutron leaves the control volume without reaction)

Each of the gain and loss terms (except for leakage) has the general form: $$\text{multiplier} \times \int_\text{stuff} \text{reaction rate}\ d\ \text{stuff}$$ Where the reaction rate is a cross-section times flux.

To make things easier, we neglect the angle of the neutrons and assume that the reactor is in a steady state and homogeneous (the latter doesn’t hold in or near fuel and control rods, though). Also, the energy can be discretized in groups (for LWR, this can be a two-group approximation: thermal and non-thermal energy ranges), taking averages of the quantities depending on the energy (like cross-sections). This allows for transforming the complex neutron transport equation into a much simpler (and analytically solvable) neutron diffusion equation.

## Delayed neutrons

Not all neutron production is instantaneous.

# Criticality

$$k = \dfrac{\text{gains}}{\text{losses}}$$
• If $$k=1$$ , the reactor is critical, i.e. in steady state.
• If $$k<1$$ , the reactor is subcritical.
• If $$k>1$$ , the reactor is supercritical.

If the temperature is raised, macroscopic cross-sections $$\Sigma = N \times \sigma$$ tend to go down, as atoms spread out and thus the number density $$N$$ go down. The relative change of a physical property associated with a given change in temperature is the temperature coefficient, and we definitely don’t want a positive temperature coefficient for $$k$$ in a reactor.

# Reactor types

## Light water reactors (LWRs)

Coolant: Water Moderator: water, pressurized

## Heavy water reactors

Fuel: natural or low-enriched uranium Coolant: D2O (deuterium oxide) Moderator: D2O (deuterium oxide), unpressurized

smaller absorption cross-section, so uranium doesn’t need to be enriched (so much).

deuterium oxide is expensive.

## Gas cooled reactors

Fuel: low-enriched uranium Coolant: CO2 Moderator: Graphite

Low power density, because CO2 is less dense and has a lower heat capacity than water.

## Liquid metal cooled reactors

Coolant: liquid metals

• lead-bismuth eutectic (LBE, eutectic = lowest possible melting point alloy)
• sodium-potassium allow (NaK)

Fast reactors, because liquid metals are not good moderators fast fission with U-238

(lead-bismuth reactor e.g. used in Russian Alfa submarines, which can even outrun torpedos (at the risk of a nuclear run-away))

## Molten salt reactors (Flüssigsalzreaktor)

Fuel: molten salt Coolant: molten salt Moderator: Graphite