Download 10. Nuclear fusion in stars

10. Nuclear fusion in stars
10.1 Energy considerations
The energy source of the sun must be a very efficient one, for the energy radiated away over
the 4 billion years of its age amounts to
Etot ' (6 · 1029 g) c2 ' 3 · 10−4 M c2
(10.1)
The binding energies of protons and neutrons in the atomic nucleus are large enough to provide
the required energy efficiency in nuclear fusion reactions. The binding energy can be most easily
determined by measuring the rest mass of the atomic nuclei. It turns out that for example a
helium nucleus has the rest mass of only 3.97 protons, the difference to the expected 2 protons
and 2 neutrons being the binding energy of the Helium nucleus.
The binding energy per nucleon has a maximum for iron. We can therefore also gain energy by
breaking up a heavier atom, say uranium, in a nuclear fission process. We make use of that in
our nuclear reactors. However, the slow fusion processes operating in the core of the sun will
for the same reason rarely produce elements beyond iron during their normal life.
10.2 Slow fusion: the tunnel effect
In the quantum-mechanical theory forces between particles act by the transfer of virtual particles, that do not satisfy Heisenberg’s uncertainty principle and can thus not be observed.
Written in terms of the particle energy, the uncertainty principle states for the lifetime of particles δt · δE ≥ ~. For virtual particles we therefore have δt · δE < ~. Electromagnetic interactions
require the exchange of a photon, which can have arbitrarily low energy on account of its vanishing rest mass, thus allowing a in principle infinite range of the force. The atomic nucleus
on the other hand is held together by the strong force, which, as the name implies, is much
stronger than the other fundamental forces, but is carried by pi-mesons with a rest mass of 135
MeV. Its range is therefore limited to
δr ' c δt <
~c
m0 c
2
' 10−13 cm
(10.2)
If two protons wanted to come sufficiently close for the strong force to bind them together, they
would have to overcome the repulsive Coulomb force with their kinetic energy
k Tc ' V '
e2
' 10−6 erg
δr
⇒ Tc ' 1010 K
(10.3)
The core temperature of the sun is only of the order 107 K, thus insufficient to permit fusion in
classical physics. In quantum physics, however, the protons have a small, yet finite probability
to come sufficiently close to fuse. This tunnel effect allows the fusion reaction to proceed at a
low rate for the individual particle, each of which has a lifetime before fusing much in excess
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of 109 years. Only because some 1054 particles are involved does the sun radiate energy at the
large rate observed.
10.3 Thermonuclear reactions
The interstellar medium consists mostly of hydrogen with a significant contribution of helium,
approximately 10% by number or 30% by mass. The most important fusion reaction therefore
involves two protons or hydrogen nuclei. To understand the thermonuclear reactions, three
things must be understood:
• Protons and neutrons in an atomic nucleus are confined to specific energy levels, much like
the electrons in the atom shells. Protons and neutrons are distinguishable, however, and each
group of particles has its own series of allowed energies. The neutron levels are typically higher
than the proton levels of the same quantum number (higher mass), but often lower than the
proton level of the next higher quantum number. It is therefore energetically beneficial for the
nucleus to have roughly equal numbers of protons and neutrons. A neutron can turn into a
proton by emitting an electron. A proton can turn into a neutron either by emitting a positron
or by capturing an electron, e.g. from the innermost shell of the atom.
• Electrons and positrons have related particles, called electron neutrinos and antineutrinos,
all together forming a so-called lepton family. There is a conservation law stating that in
every reaction the total Lepton number must be conserved. Now electrons and neutrinos
contribute +1 to the lepton number and the antiparticles positron and antineutrino count 1. Nuclear reactions involving the creation or decay of leptons are called weak interactions,
for they happen much slower than strong interactions. The decay of a neutron therefore also
creates an antineutrino:
n −→ p+ + e− + ν¯e
(10.4)
• In each reaction energy and momentum must be conserved. Two particles of arbitrary energy
and finite rest mass can therefore not end up as one particle with finite rest mass.
This explains the fundamental pp-chain
p+ + p+ →21 H + e+ + νe
(10.5)
p +21 H →32 He + γ
(10.6)
He +32 He →42 He + p+ + p+
(10.7)
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Reaction 10.5 is the slowest, because is requires a weak interaction. The neutrino interacts
very little. It thus steals away energy, but carries direct information about the conditions in
the solar interior. The pp-chain accounts for 98 % of the solar luminosity, and we can thus
precisely predict the neutrino flux. The measurements of solar neutrinos have shown that they
must be able to transform themselves into other neutrinos related to the muon and the tau.
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Since the positrons will annihilate in collision with ambient electrons, we can write the overall
outcome of the pp-chain in the form
2 e− + 4 p+ →42 He + 2 νe + 26.7 [MeV]
(10.8)
The same basic reaction can also happen with a carbon nucleus as starting point and as catalyst
in the CNO-cycle (interactive hour). Carbon 12 has more binding energy than three helium
nuclei and thus is stable, but how do we get it? The reaction
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He +42 He →84 Be + γ
(10.8)
consumes energy, and thus should happen very rarely. Nevertheless, once in a while a beryllium
nucleus will interact before decaying to give
∗
Be +42 He →12
6 C +γ
8
4
(10.9)
By subsequent digestion of a helium nucleus we can produce oxygen, neon, magnesium, silicon,
etc. These nuclei carry larger and larger charges, thus their Coulomb barriers are higher, and
higher temperatures are required to spawn these reactions.
This temperature rule does not apply to the capture of neutrons, for they don’t carry an electric
charge. Free neutrons are produced in a number of nuclear reactions. If too many neutrons
have been absorbed by nucleus, it may become unstable against beta decay, i.e. reaction (10.4).
The sequence of elements formed in this way, by neutron captured followed by beta decay, are
called s-process elements (s=slow). In a runaway nuclear reaction in a Supernova (or a bomb)
a nucleus will absorb very many neutrons before it eventually undergoes multiple beta decay.
The sequence of elements thus formed is called r-process elements (r=rapid). The abundances
of elements and isotope ratios, in particular those involving unstable nuclei, can thus be used
to date meteorites or the solar system.
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