Scattering and tunnelling
Scattering and tunnelling

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Scattering and tunnelling

5.3 Stellar astrophysics

If tunnelling out of nuclei is possible then so is tunnelling in! As a consequence it is possible to trigger nuclear reactions with protons of much lower energy than would be needed to climb over the full height of the Coulomb barrier. This was the principle used by J.D. Cockcroft and E.T.S. Walton in 1932 when they caused lithium-7 nuclei to split into pairs of alpha particles by bombarding them with high-energy protons. Their achievement won them the 1951 Nobel prize for physics. The same principle is also at work in stars, such as the Sun, where it facilitates the nuclear reactions that allow the stars to shine. Indeed, were it not for the existence of quantum tunnelling, it's probably fair to say that the Sun would not shine and that life on Earth would never have arisen.

The nuclear reactions that allow stars to shine are predominantly fusion reactions in which low mass nuclei combine to form a nucleus with a lower mass than the total mass of the nuclei that fused together to form it. It is the difference between the total nuclear masses at the beginning and the end of the fusion process that (via E = mc2) is ultimately responsible for the energy emitted by a star. The energy released by each individual fusion reaction is quite small, but in the hot dense cores of stars there are so many fusing nuclei that they collectively account for the prodigious energy output that is typical of stars (3.8 × 1026 W in the case of the Sun).

In order to fuse, two nuclei have to overcome the repulsive Coulomb barrier that tends to keep them apart. The energy they need to do this is provided by the kinetic energy associated with their thermal motion. This is why the nuclear reactions are mainly confined to a star's hot central core. In the case of the Sun, the core temperature is of the order of 107 K. Multiplying this by the Boltzmann constant indicates that the typical thermal kinetic energy of a proton in the solar core is about 1.4 × 10−16 J ≈ 1 keV. However, the height of the Coulomb barrier between two protons is more than a thousand times greater than this. Fortunately, as you have just seen, the protons do not have to climb over this barrier because they can tunnel through it. Even so, and despite the hectic conditions of a stellar interior, where collisions are frequent and above-average energies not uncommon, the reliance on tunnelling makes fusion a relatively slow process.

Again taking the Sun as an example, its energy comes mainly from a process called the proton–proton chain that converts hydrogen to helium. The first step in this chain involves the fusion of two protons and is extremely slow, taking about 109 years for an average proton in the core of the Sun. This is one of the reasons why stars are so long-lived. The Sun is about 4.6 × 109 years old, yet it has only consumed about half of the hydrogen in its core. So, we have quantum tunnelling to thank, not only for the existence of sunlight, but also for its persistence over billions of years.

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