| Briefly describe the sequence of reactions leading to the formation of the light nuclides 2H, 3He, 4He and 7Li in the early universe. | [4] |
| The early universe contains both protons and free neutrons, which can combine to form deuterium nuclei via p + n → d + γ. Initially this reaction is reversible (the deuterium can be photodissociated if struck by a gamma ray), but below 109 K the typical gamma ray energies become too small to do this, and the deuterium will accumulate. | [1] |
| Helium-3 is made directly when deuterium is struck by a proton, or when two deuterium nuclei collide (you'd expect this to make helium-4, but usually the energies are too high to make a stable nucleus; instead a neutron or proton is emitted, making either helium-3 or tritium). Similar reactions also make tritium, hydrogen-3: any of this which is not used up in later nucleosynthesis will decay to helium-3 by β-decay. | [1] |
| Helium-4 is made by a wide variety of reactions (it's so stable that it's very easy to make). Almost anything (neutron, deuterium nucleus, tritium nucleus, helium-3 nucleus) hitting a helium-3 nucleus will make helium-4; so will almost anything hitting a tritium nucleus. Occasionally two deuterium nuclei will collide to make helium-4, though usually they'll make helium-3 or tritium instead. | [1] |
| Lithium-7 is made directly by 3H + 4He, or indirectly by 3He + 4He (this makes 7Be, which β-decays to 7Li). | [1] |
|
Explain why: (i) this process cannot produce nuclides heavier than 7Li (even though fusion processes in stars can and do produce such nuclides); |
[2] |
|
There is no stable nuclide with mass 5, and none with mass 8;
thus it is extremely difficult to find two-body reactions which will go beyond
helium-4, and those which do require collisions of rare products such as lithium-7,
and therefore occur at undetectably low rates. In stars, the high densities and long timescales allow carbon-12 to be produced via the extremely unstable nucleus beryllium-8 (if this is struck by a helium nucleus during its fleeting existence, stable carbon-12 can be produced). This does not happen in the early universe, because the density is not as high as a stellar interior, and the time available before the universe cools to below nuclear-reaction temperatures is only a few minutes. |
[2] |
| (ii) you expect the final yield of 4He to be somewhat less than 30% by mass; | [2] |
| Given mn - mp = 2.3×10-30 kg, the exponential exp(-E/kT) = exp(-2.3×10-30 × 9×1016 /1.38×10-23 × 1010) = 0.22; thus, 2 neutrons to 9 protons. This would give a helium:hydrogen ratio of 1:7 (each helium nucleus uses up two neutrons and two protons, with 7 protons left over) by number, or 4:7 by mass, so the helium fraction would be 4/11 = 36%. But in fact free neutrons are unstable, so over the course of nucleosynthesis the neutron:proton ratio steadily decreases, hence we wind up with a smaller number. | [2] |
| (iii) the exact yield depends on the neutron lifetime, the number of neutrino species and H0 as well as on the baryon density. | [2] |
|
The neutron lifetime: because the faster neutrons decay, the fewer there will be
during nucleosynthesis, and the less helium-4 we will end up with. The number of neutrino species: because neutrinos are relativistic and therefore contribute to the radiation energy density of the early universe. A greater radiation density will increase the expansion rate (since for a flat geometry H2 ∝ ρ), and thus decrease the time it takes for the universe to cool down. This actually increases the yield of He-4, because fewer neutrons will have decayed before the point at which deuterium becomes stable. H0, for the same reason as above: faster expansion shortens the timescale. (You may notice that plots of nucleosynthesis yield are usually drawn against Ωbh2, where h = H0/100 km/s/Mpc—this is because this combination is what actually comes out of the calculations, and so the uncertainty on this is less than the uncertainty on Ωb alone.) The baryon density: because nucleosynthesis relies on collisions, and the larger the density the more likely collisions will be (the frequency is actually proportional to the square of the density). |
[2] |
| [Note that the mass of the proton is 1.6726×10-27 kg, and the neutron mass is 1.6749×10-27 kg.] |
(2006 Q7(a) and (b).)