Physics · Atomic and quantum physics · Grade 11

Thermonuclear fusion

How does the Sun turn hydrogen into helium, and can this be repeated on Earth?

The situation

The Sun converts about 600 million tonnes of hydrogen into helium every second and loses more than 4 million tonnes of its mass as radiation. Its central temperature is ≈ 15 million K. That is very hot, but to overcome the Coulomb repulsion of protons by classical means would need billions of degrees. Nevertheless the Sun has been shining for 4.6 billion years. How is this possible — and can the ITER tokamak being built in France light this “stellar fire” on Earth?

Topic

Thermonuclear fusion. In the heart of the Sun and in a tokamak: hydrogen to helium

Key ideas

Coulomb barrier and tunnelling
Lawson criterion
Tokamak

Formulas

U_{C} = \frac{e^{2} Z_{1} Z_{2}}{4 π ε_{0} R}, R = r_{0} (A_{1}^{1 / 3} + A_{2}^{1 / 3})

Coulomb barrier: the potential energy that must be overcome for nuclei to “touch”.

P \approx e^{- \sqrt{E_{G} / E}}, E_{G} = 2 μ c^{2} {(π α Z_{1} Z_{2})}^{2}

Probability of passing through by tunnelling (the Gamow factor); $E_{G}$ is the Gamow energy, $μ$ the reduced mass.

^{2} H +^{3} H \to^{4} He (3.5 MeV) + n (14.1 MeV)

D–T reaction, $Q = 17.6$ MeV. The momenta are equal, so the lighter neutron takes 4/5 of the energy.

n τ_{E} \geq \frac{12 k T}{⟨ σ v ⟩ E_{α}} \approx 1.5 \cdot 10^{20} {s/m}^{3}

Lawson criterion (D–T burning, most favourable $k T \approx 25$ keV): alpha heating compensates the losses.

4^{1} H \to^{4} He + 2 e^{+} + 2 ν_{e}, Q = 26.7 MeV

The net result of the pp chain in the Sun; $Q$ includes the annihilation of the two positrons with electrons (neutrinos carry away ≈ 2 % of it).

Theory

Thermonuclear reaction

When light nuclei fuse into a heavier nucleus, the binding energy increases and the difference is released: for D–T it is 3.5 MeV per nucleon — four times more than in uranium fission (≈ 0.85 MeV per nucleon). But the nuclei must come to within ≈ 3 fm, while Coulomb repulsion creates a barrier of 0.4–0.9 MeV. Thermal motion gives the particles such energy — hence the reaction is called thermonuclear, and the matter is in a fully ionised plasma state. Radioactive decay, fission and chain reactions — in the “Radioactivity, half-life” and “Nuclear reactions” experiments.

The tunnel effect and the Gamow peak

At the centre of the Sun $k T \approx 1.35$ keV — about 450 times smaller than the p–p barrier (≈ 600 keV). In classical physics fusion would not occur. In quantum mechanics a particle can pass through a barrier by tunnelling, with a probability $e^{- \sqrt{E_{G} / E}}$ that rises sharply with energy. In the Maxwell distribution, however, fast particles are exponentially few. The product of the two factors gives the Gamow peak: in the Sun $E_{0} \approx 6$ keV for p–p. In the p + p reaction a proton must also turn into a neutron during the tunnelling (weak interaction), so a proton in the solar core on average “waits its turn” for billions of years — which is why the Sun lives so long.

The Sun and its standard model

Mass ≈ 2·10³⁰ kg, radius ≈ 696 thousand km, radiated power $L = 3.83 \cdot 10^{26}$ W, surface temperature ≈ 5 800 K. The dark (Fraunhofer) lines in its spectrum show its composition: by mass ≈ 73 % hydrogen, ≈ 25 % helium, the rest heavier elements. The core (≈ 1/4 of the radius, 15.7 million K, ≈ 150 g/cm³) produces the energy; in the radiative zone photons carry the energy for tens of thousands, even hundreds of thousands of years; in the outer convection zone hot matter rises. The standard model is a set of equations of hydrostatic equilibrium and energy transport; it is confirmed by helioseismology and neutrino measurements.

Nucleosynthesis in stars

≈ 99 % of the Sun's energy comes from the pp chain. In heavier stars the CNO cycle dominates: C, N and O act as catalysts to turn 4 protons into helium; its rate depends extremely strongly on temperature (about $\propto T^{20}$ ), and above roughly 17–20 million K it overtakes the pp chain. When the hydrogen is used up, the star becomes a red giant: at ≈ 100 million K the triple-alpha process forms carbon. In massive stars fusion goes as far as iron–nickel, then the core collapses, a supernova explodes and a neutron star or black hole remains. Elements heavier than iron form by neutron capture: the slow (s-) process in giants, the rapid (r-) process in neutron-star mergers and supernovae. Some of the lithium, beryllium and boron forms when cosmic rays break up interstellar nuclei.

The solar neutrino problem

With each helium nucleus 2 electron neutrinos are born; they leave the Sun freely, and ≈ 6.5·10¹⁰ of them fall on each cm² of the Earth every second. R. Davis's chlorine detector (from the late 1960s) recorded only ≈ 1/3 of the expected number — the solar neutrino problem. In 2001–2002 the SNO detector counted neutrinos of all types and showed that the total flux matches the standard model: electron neutrinos change (oscillate) into muon and tau neutrinos on the way, so neutrinos have mass. These discoveries were awarded the Nobel Prize in 2002 and 2015.

Controlled fusion

The Lawson criterion: for ignition, the product of density and energy confinement time, $n τ_{E}$ , must be large enough. A tokamak (the idea of I. Tamm and A. Sakharov, 1950s) holds the plasma in a toroidal chamber by the field of the coils and the field of the current in the plasma. ITER in France (34 countries) plans to obtain 500 MW of fusion power ( $Q = 10$ ) with 50 MW of heating. JET obtained 16 MW ( $Q \approx 0.67$ ) in 1997. In a stellarator the field is made by twisted coils and no current is needed in the plasma (Wendelstein 7-X in Germany). In inertial fusion lasers compress a fuel capsule ≈ 1000 times. Model: a 0-dimensional power balance and a homogeneous plasma; impurities and helium “ash” are neglected — in real devices $Q$ is lower.

Did you know?

Every second the Sun converts about 4.3 million tonnes of mass into radiation ( $Δ m = L / c^{2}$ ). But it is so massive that in 4.6 billion years it has lost only about 0.03 % of its mass.
On 5 December 2022, at the NIF facility in the USA, fusion in a laboratory for the first time released more energy than the laser energy put in: 3.15 MJ of fusion energy was obtained from 2.05 MJ of laser energy.

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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