Physics · Atomic and quantum physics · Grade 11

Nuclear reactions

Why doesn't a nuclear power plant reactor explode, and how is it controlled?

The situation

A coal-fired power station of 1000 MW burns more than a hundred wagons of coal a day. The reactor of a nuclear power plant (NPP) of the same power “burns” only about 3 kg of uranium-235 a day. Work has begun on Uzbekistan's first NPP project too. Inside a reactor, trillions upon trillions of uranium nuclei undergo fission every second. Why does this chain reaction not explode like a bomb, and how does the operator hold it at exactly 100 % power?

Topic

Nuclear reactions. Nuclear power plant reactor

Key ideas

Chain reaction
Delayed neutrons
Fission energy

Formulas

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

One of the fission reactions: each fission releases 2–3 new neutrons (on average ≈ 2.4) and ≈ 200 MeV of energy.

k_{eff} = \frac{N_{i + 1}}{N_{i}}, ρ = \frac{k_{eff} - 1}{k_{eff}}

Multiplication factor and reactivity: $k_{eff} = 1$ — the critical state, the power does not change.

\frac{d n}{d t} = \frac{ρ - β}{Λ} n + λ C, \frac{d C}{d t} = \frac{β}{Λ} n - λ C

Point kinetics: $β = 0.0065$ is the delayed-neutron fraction, $λ \approx 0.08 s^{- 1}$ , $Λ \approx 10^{- 4}$ s.

T \approx \frac{β - ρ}{λ ρ} (ρ ≪ β), T_{prompt} = \frac{Λ}{ρ}

The reactor period is the time in which the power changes by a factor of $e$ . Without delayed neutrons it would be thousands of times shorter.

E = \frac{m}{M} N_{A} E_{1}

Energy released when a mass $m$ of U-235 fissions completely: $M = 235$ g/mol, $E_{1} \approx 200$ MeV $= 3.2 \cdot 10^{- 11}$ J — one fission.

Theory

Chain reaction

When a neutron is absorbed by a U-235 nucleus, the nucleus splits into two fragments and 2–3 fast neutrons fly out. The fast neutrons hit the hydrogen nuclei in the water and are slowed down (red → blue in the scene) — slow (thermal) neutrons cause fission of U-235 far more readily. Some of the neutrons of one generation are absorbed by U-238 or by the boron carbide in the control rods. If on average exactly one neutron in each generation causes a new fission, $k_{eff} = 1$ — the reactor is critical and the power does not change.

Nuclear reactions and conservation laws

A nuclear reaction $a + A \to b + B$ is written in short as $A (a, b) B$ . In it electric charge and the number of nucleons are conserved (in the fission equation: 92 = 56 + 36, 1 + 235 = 141 + 92 + 3), and so are energy, momentum and angular momentum. The energy released is $Q = (m_{a} + m_{A} - m_{b} - m_{B}) c^{2}$ : for $Q > 0$ energy is released, for $Q < 0$ the energy of the incoming particle must exceed the threshold: $E_{t h r} = | Q | (1 + m_{a} / m_{A})$ — this follows from conservation of momentum (reaction kinematics). The probability of a reaction is measured by the cross-section $σ$ (1 barn = 10⁻²⁸ m²); the fraction of particles that react in a thin target — the reaction yield — is $w = σ n$ , where $n$ is the number of nuclei per 1 m² of the target. At low energies the particle is absorbed by the nucleus to form an excited compound nucleus, which decays having “forgotten” how it was formed: U-235 + n → U-236* → fission. At high energies direct reactions (for example deuteron stripping (d,p)) take place instantly at the surface of the nucleus; they are described by the optical model — the nucleus partly refracts and partly absorbs the incoming wave. A charged particle can excite a nucleus with its electric field even at energies below the Coulomb barrier — Coulomb excitation. In the “Nuclear reactions” section, calculate historical reactions.

Neutrons in matter

A neutron has no charge: it passes through electron shells almost “unnoticed” and loses energy only by colliding with nuclei. In an elastic collision a light nucleus takes a lot of energy: in hydrogen, in one collision the logarithm of the energy falls on average by $ξ = 1$ , so to slow down from 2 MeV to 0.025 eV takes ≈ 18 collisions with hydrogen, ≈ 25 with deuterium and ≈ 115 with carbon (graphite). The absorption cross-section for a slow neutron grows as $σ \propto 1 / v$ : the fission cross-section of U-235 is ≈ 585 barn for thermal neutrons and ≈ 1 barn for fast neutrons. Neutron reactions: radiative capture (n,γ) — this is how Pu-239 is formed from U-238; (n,α) — boron-10 in the rods; (n,p). Heavy nuclei also fission spontaneously (K. Petrzhak and G. Flerov, 1940): in U-238 this is very rare (half-life ≈ 8·10¹⁵ years), while in Cf-252 it accounts for ≈ 3 % of decays — it serves as the neutron source for starting up a reactor. Some nuclei emit a whole “cluster”: Ra-223 → Pb-209 + C-14. Superheavy elements are synthesised by colliding heavy ions — elements 114–118 were obtained in Dubna with Ca-48 ions. There are also reactions with a γ-quantum (photonuclear) or a fast electron (electronuclear); reactions involving mesons are in the “Elementary particles” experiment.

Delayed neutrons — the key to control

99.35 % of neutrons are emitted at the moment of fission (prompt), but $β \approx 0.65 %$ are emitted later, in the radioactive decay of some fragments, delayed on average by $1 / λ \approx 12.5$ s. The prompt-neutron generation time is $Λ \approx 10^{- 4}$ s: if only they existed, at $ρ = 0.001$ the power would grow by a factor of $e$ every 0.1 s. Thanks to delayed neutrons the period is ≈ 70 s — both the operator and the rods have time to act. When a rod is moved, the power first makes a small “jump” ( $n_{1} / n_{0} = β / (β - ρ)$ ) and then changes slowly — see it on the graph. If $ρ \geq β$ , the reactor becomes prompt critical and goes out of control.

Feedback and protection

When the fuel heats up, the resonance absorption of U-238 increases (the Doppler effect) and the reactivity falls: $α_{T} \approx - 10^{- 5} K^{- 1}$ . This is why the power stabilises itself. SCRAM is the emergency protection: all the rods fall under their own weight in ≈ 2 s. The fission power falls sharply, then more slowly because of delayed neutrons. But the radioactive decay of the fragments immediately gives ≈ 6–7 % decay heat — cooling must not stop (Fukushima, 2011). Safety rests on several barriers in succession: the fuel pellet, the sealed cladding of the fuel element, the body of the primary circuit and the reinforced-concrete containment; for radioactive substances to escape, all of them would have to fail.

About the model

The whole core is treated as a single “point”; the delayed neutrons are in one group ( $β = 0.0065$ , $λ = 0.08 s^{- 1}$ ); the equations are solved exactly at each step (matrix exponential). The regulating group is worth 2.4 % and the protection groups 8 %; excess reactivity is assumed to be compensated by boric acid. Nominal thermal power is 3000 MW, turbine efficiency 33 %, and decay heat is an approximate 3-group model. The neutrons in the magnified window are an illustration only: they are tied to the real power, slowed down, and the delayed-neutron fraction is exaggerated so that it can be seen.

Did you know?

On 2 December 1942 in Chicago, a team led by Enrico Fermi carried out a controlled chain reaction in the first nuclear reactor, Chicago Pile-1.
Only ≈ 0.72 % of natural uranium is the isotope U-235. For NPP reactors the fuel is enriched to 3–5 %.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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The photoelectric effect — Grade 11
Atomic structure and spectra — Grade 11
Radioactivity, half-life — Grade 11
Orbitals of the hydrogen atom — Grade 11
Thermal radiation and Planck — Grade 11
Photon momentum and the Compton effect — Grade 11
De Broglie waves — Grade 11
Rutherford's experiment — Grade 11
The atomic nucleus and binding — Grade 11