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Lecture 30/3 Nuclear processes Ulf Torkelsson 1 Nuclear reactions Consider the reaction I (Ai , Zi ) + I (Aj , Zj ) → K (Ak , Zk ) + L (Al , Zl ) (1) The amount of energy released in this reaction is Qijk = (Mi + Mj − Mk − Ml ) c2 . (2) By summing over all possible nuclear reactions that have reaction rates Rijk we can get the amount of energy released per unit mass as q= ρ X 1 Xi Xj Rijk Qijk . m2H 1 + δij Ii Aj (3) ijk Strictly speaking this is an upper limit to how much energy that can heat the star, because some of the energy might be carried away by neutrinos, that pass straight through the star. The energy that is perhaps released in a nuclear reaction is due to that the constituent nucleons are bound more tightly by the strong force in the new nucleus than in the old nuclei. For light nuclei the binding energy per nucleon is in general increasing with the size of the nucleus up to 56 Fe, in which they are the most strongly bound, and in heavier nuclei the binding energy per nucleon decreases with the size of the nucleus. Therefore there are two ways of deriving energy from atomic nuclei, by fusing light nuclei, and by fissioning heavy nuclei. The nuclei that are involved in a fusion reaction are repelling each other through the Coulomb force, since they are both positively charged. This Coulomb repulsion is so strong that the kinetic energy of the nuclei is insufficient in bringing them any closer together than a thousand times the radius of a nucleus, which is the distance scale at which the strong force works, however because of quantum tunneling there is a small probability that the nuclei will anyway react with each other. 2 Hydrogen burning The most common element in the universe is hydrogen, so it is natural to assume that this is the main fuel in the nuclear reactions. There are two ways in which hydrogen can be fused into helium. 2.1 p-p chain Two protons cannot form a bound system so the first step in hydrogen burning is the weak reaction p + p →2 D + e+ + ν, (4) in which one of the protons is converted to a neutron. The deuteron can then react with a new proton 2 D + p →3 He + γ. (5) After this there are two possible reactions 3 He +3 He →4 He + 2p, (6) 3 (7) which completes the p-p I chain, and He +4 He →7 Be + γ. 1 In the p-p II chain we then get 7 Be + e− 7 Li + p → 7 Li + ν → 24 He, (8) (9) and in the p-p III chain 7 8 Be + p → 8 8 B Be 8 → B+γ (10) + Be + e + ν (11) 4 (12) → 2 He. The energy that is released in this reaction is Qp−p = (4mp − m4 He ) c2 = 26.73 MeV. (13) The amount of energy that is lost through the neutrino emission varies between 0.26 MeV during the production of deuterium to 7.2 MeV in the boron decay, which occurs only in a small fraction of the reactions. For this reason the amount of energy that on average becomes available to the star is 26 MeV, and qp−p ∝ ρT 4 . (14) 2.2 The CNO-cycle Hydrogen can also be converted to helium via the CNO-cycle that uses carbon 12 C +1 H → 13 N → 13 C+e +ν (16) C +1 H → 14 N+γ (17) → 15 O+γ (18) O → 15 C + He, (20) O+γ (21) 13 13 14 1 N+ H 15 1 N+γ + + N+e +ν 4 (15) (19) 15 N+ H → 12 14 N +1 H → 15 O → 15 N+e +ν (22) → 16 O+γ (23) → 17 F+γ (24) F → 17 O + e+ + ν (25) 14 N +4 He (26) or nitrogren 15 15 16 1 N+ H 1 O+ H 17 17 O +1 H → + to catalyse the reaction. It is important here that as the rate of the proton reactions will increase with temperature, the rates of the β-decays are independent of time, thus at sufficiently high temperatures the reaction rate will be limited by the β-decays. The energy that is available to the star is slightly lower, 25 MeV, than in the p-p chain. That is because more energy is lost via the neutrinos in this case. The energy generation rate can be written as 3 qCNO ∝ ρT 16 . (27) He +4 He →8 Be, (28) Helium burning The reaction 4 2 in itself does not work for burning helium, because the life time of 8 Be is only 2.6 × 10−16 s. However this is still longer than the mean collision time of α-particles at 108 K. There is therefore a non-negligible probability that 8 Be +4 He →12 C (29) during this short time. As a matter of fact this probability is enhanced by there being a resonant energy level at 7.65 MeV in 12 C, which agrees closely with the energy that is released in this triple-α reaction, 7.275 MeV. Because this is essentially a three-particle interaction the energy generation rate is ρ3α ∝ ρ2 T 40 . (30) As the carbon abundance increases one can also have 12 4 C +4 He →16 O. (31) Carbon and oxygen burning Carbon burning starts at 5 × 108 K 12 C +12 C 24 Mg + γ (32) 12 C+ C → 23 Mg + n (33) 12 C +12 C → 23 Na + p (34) 12 C +12 C → 20 Ne + α (35) → 16 O + 2α (36) 12 C+ 12 → 12 C and oxygen burning starts at 109 K 16 O +16 O → 32 S+γ (37) O → 31 S+n (38) O +16 O → 31 P+p (39) O → 28 Si + α (40) O → 24 Mg + 2α. (41) 16 O+ 16 16 16 16 O+ 16 O+ 16 On average 13 MeV is released in the carbon burning and 16 MeV in oxygen burning. Note that the light nuclei that are released in these reactions will quickly react with other heavy nuclei in the environment. 5 Nuclear statistical equilibrium As the temperature increases further, photodissociation of the nuclei starts to become important. For instance at 1.5 × 109 K we have 20 Ne + γ →16 O + α, (42) and at 3 × 109 K even silicon can photodisintegrate. Thus we enter a regime in which a statistical equilibrium is established between the fusion reactions that build up heavier nuclei, and photodisintegration processes. However there is still a small drift towards the iron group nuclei, which do not photodisintegrate until the temperature is 7 × 109 K. 6 Neutron capture Elements heavier than iron cannot be formed through fusion reactions since such reactions would be endothermic. Instead heavier elements are formed through neutron capture I (A, Z) + n → I1 (A + 1, Z) 3 (43) I1 (A + 1, Z) + n → I2 (A + 2, Z) (44) I2 (A + 2, Z) + n → I3 (A + 3, Z) . (45) Eventually the nucleus will have gained so many neutrons that it becomes unstable IN (A + N, Z) → J (A + N, Z + 1) + e− + ν̄, (46) and the new element may then continue to go through new β-decays or catch new neutrons. Depending on the rate of the neutron capture compared to that of the β-decay, the nucleus may either decay as soon as it becomes unstable the first time, the slow (s)-process, or it might capture more neutrons before it decays, the rapid (r)-process. The r-process will in general produce more neutron rich nuclei. 7 Pair production When the photons are sufficiently energetic, two photons can produce an electron-positron pair. The simple estimate of the temperature based on that kT = 2hν = 2me c2 , yields a temperature of 1.2 × 1010 K. However, some of the photons are sufficiently energetic already at temperatures above 109 K. 8 Iron photodisintegration At high temperatures there are also photons that can drive the reaction 56 Fe → 13 4 He + 4n, (47) and at 7 × 109 K there are even photons that can disintegrate helium into protons and neutrons. 4