Recall that in any nuclear reaction the following must be conserved:
1. The baryon number – protons, neutrons and their anti-particles
2. The lepton number – light particles, electrons, positrons, neutrinos, and anti-
neutrinos
3. Charge
Note also that the anti-particles have the opposite baryon/lepton number to
their particles
The binding energy of the atomic nucleus
The general description of a nuclear reaction is
Where Ai = the baryon number, or nucleon number (nuclear mass)
and Zi = the nuclear charge
The nucleus of any element is uniquely defined by the two integers Ai and Zi
𝐼 𝐴𝑖, 𝑍𝑖 + 𝐽 𝐴𝑗, 𝑍𝑗 ↔ 𝐾 𝐴 𝑘, 𝑍 𝑘 + 𝐿(𝐴𝑙, 𝑍𝑙)
The binding energy of the atomic nucleus
• The total mass of a nucleus is known to be less than the mass of the
constituent nucleons
• Hence there is a decrease in mass if a companion nucleus is formed
from nucleons, and from the Einstein mass-energy relation E = mc2 the
mass deficit is released as energy
• This difference is known as the binding energy of the compound
nucleus
• Thus if a nucleus is composed of Z protons and N neutrons, it’s binding
energy is
𝑄 𝑍, 𝑁 ≡ 𝑍𝑚 𝑝 + 𝑁𝑚 𝑛 − 𝑚 𝑍, 𝑁 𝑐2
• For our purposes, a more significant quantity is the total binding energy
per nucleon
• We can then consider this number relative to the hydrogen nucleus
𝑄(𝑍, 𝑁)
𝐴
The binding energy per nucleon
The variation of binding energy per nucleon with baryon
number A
• General trend is an increase of Q with atomic mass up
to A= 56 (Fe), then slow monotonic decline
• There is steep rise from H through 2H, 3He, to 4He =>
fusion of H to He should release larger amount of
energy per unit mass than fusion of He to C
• Energy may be gained by fusion of light elements to
heavier, up to iron
• Or from fission of heavy nuclei into lighter ones down
to iron
Abundance - Sun
Asplund et al.
The binding energy per nucleon
Fusion Fission
Coulomb Barrier
ZaZb … number of protons
in each nuclei
RaRb … interaction radii
0 … permittivity of free
space (8.85 10-12 C2N-1 m-2)
e … charge of electron
D-T reaction: VC is 0.38 MeV
Gas temperature of
4.4 109 K
Hydrogen and helium burning
The most important series of fusion reactions are those converting H to He (Hburning).
As we shall see this dominates ~90% of lifetime of nearly all stars.
• Fusion of 4 protons to give one 4He is completely negligible
• Reaction proceeds through steps – involving close encounter of 2 particles
• We will consider the main ones: the PP - chain and the CNO cycle
The PP - chain has three main branches called the PPI, PPII and PPIII chains.
PPI Chain
1 p + p → d + e+ + e
2 d + p → 3He + 
3 3He + 3He → 4He + 2p
PPII Chain
3' 3He + 4He → 7Be + 
4' 7Be + e− → 7Li + e
5' 7Li + p → 4He + 4He
PPIII Chain
4'' 7Be + p → 8B + 
5'' 8B → 8Be + e+ + e
6'' 8Be → 24He
Relative importance of
PPI and PPII chains
(branching ratios)
depend on conditions
of H-burning (T, ,
abundances). The
transition from PPI to
PPII occurs at
temperatures in excess
of 1.3107 K.
Above 3107 K the PPIII
chain dominates over
the other two, but
another process takes
over in this case.
Neutrinos – Solar Fusion
9
The CNO Cycle
• Remember: [Z] < 2%, most abundant CNO
• CNO induce a chain of H-burning reactions in which
they act as catalysts
• The process is known as the CNO Cycle. There are
alternative names in the literature:
1. The CNO bi-cycle
2. The CNOF cycle
3. The CN and NO cycles
4. The CN and NO bi-cycles
• In this course we will just refer to it all as the CNO
cycle
The main CNO branch
1 12C + p → 13N + 
2 13N → 13C + e+ + e
3 13C + p → 14N + 
4 14N + p → 15O + 
5 15O → 15N + e+ + e
6 15N + p → 12C + 4He
In the steady state case, the
abundances of isotopes must take
values such that the isotopes which
react more slowly have higher
abundance. The slowest reaction is p
capture by 14N. Hence most of 12C is
converted to 14N.
Sun
CNO cycle for stars with M > 1.2 M dominant
Helium Burning: the triple -  reaction
• Simplest reaction should be the fusion of two helium nuclei
• 4He + 4He → 8Be
• There is no stable configuration with A = 8.
• Beryllium isotope 8Be has a lifetime of only 2.610-16 s
• But a third helium nucleus can be added to 8Be before decay, forming 12C by
the “triple-alpha” reaction
4He + 4He → 8Be
8Be + 4He → 12C + 
Carbon and oxygen burning
Carbon burning (fusion of two C nuclei) requires temperatures above 5 108 K, and
oxygen burning in excess of 109 K.
Interactions of C and O nuclei are negligible – as at the intermediate temperatures
required by the coulomb barrier the C nuclei are quickly destroyed by interacting with
themselves
The branching ratios for these reactions are temperature dependent probabilities.
12C + 12C → ~13 MeV
16O + 16O → ~16 MeV
These reactions produce p, n, He, which are immediately captured by heavy nuclei, thus
many isotopes created by secondary reactions.
Silicon burning
Two Si nuclei could fuse to create 56Fe – the end of the fusion
chain
But now very high Coulomb barrier, at T above O burning, but
below that required for Si burning, photodisintegration takes
place
Si disintegration occurs around 3109 K, and the light
particles emitted are recaptured by other Si nuclei.
Although the reactions tend to a state of equilibrium,
a leakage occurs towards the stable iron group nuclei
(Fe, Co, Ni), which resist photodisintegration up
to 7109 K.
Summary - nuclear burning processes
• Release of energy by consumption of nuclear fuel
• Rates of energy release vary enormously
• Nuclear processes can also absorb energy from radiation field
Nuclear
Fuel
Process Tthreshold
106K
Products Energy per nucleon (MeV)
H PP ~4 He 6.55
H CNO 15 He 6.25
He 3 100 C, O 0.61
C C+C 600 O, Ne, Na, Mg 0.54
O O+O 1000 Mg, S, P, Si ~0.3
Si Nuc eq. 3000 Co, Fe, Ni <0.18
IN(A+N, Z) → J(A+N, Z+1) + e− + 
If new element stable, it will resume neutron capture, otherwise undergo
series of -decays
J(A+N, Z+1) → K(A+N, Z+2) + e− + 
K(A+N, Z+2) → L(A+N, Z+3) + e− + 
The s-process and r-process
Interaction between nuclei and free neutrons (neutron capture) – the
neutrons are produced during C, O and Si burning.
Neutrons capture by heavy nuclei is not limited by the Coulomb barrier – so
could proceed at relatively low temperatures. The obstacle is the scarcity of
free neutrons. If enough neutrons available, chain of reactions possible:
I(A, Z) + n → I1(A+1, Z)
I1(A+1, Z) + n → I2(A+2, Z)
I2(A+2, Z) + n → I3(A+3, Z)
If a radioactive isotope is formed it will undergo  – decay, creating new
element.
The s-process and r-process
• Two types of reactions and two types of nuclei
1. Neutron captures and -decays
2. Stable and unstable nuclei
• Stable nuclei may undergo only neutron captures
• Unstable ones my undergo both
• Outcome depending on the timescales for the two
processes
• Neutron capture reactions may proceed more slowly
or more rapidly than the competing  – decays
• The different chains of reactions and products are
called the s – process and r – process
The s-process and r-process