1. Guest Lecturer: Dr W J Chaplin
Somak Raychaudhury
Lecture 11
Guest Lecturer: Dr W J Chaplin
Nuclear energy generation in stars
Revision: Central temperature and pressure in a star
Can fusion occur at the centre of stars: overcoming Coulomb repulsion
Gamow’s solution
The p-p chain

2. Revision: Density, temperature, pressure within a star
The mass density profile of the Sun looks like this
A typical model would be
Likewise temperature and pressure would peak at the core of the star

3. Revision
We derived the equation of hydrostatic equilibrium, balancing gravity and thermal pressure
Central pressure of a star like the Sun
Central temperature of the Sun

4. Atomic nuclei
Usually we express the mass of nuclei in atomic mass units u, defined to be 1/12 the mass of 12C (the 12 is the mass number A C has 6 protons, 6 neutrons and 6 electrons)
1u= x10-27 kg = MeV/c2
The mass of 1 proton + 1 electron is u
Note 6 p + 6 n + 6 e- =

5. Binding energy
The mass of an atom (protons+neutrons+electrons) is not equal to the mass of the individual particles.
There is also a binding energy associated with the nucleons themselves.
The mass of the Carbon-12 atom:
6 p + 6 n + 6 e- =
The mass difference is 0.099u, equivalent to MeV!
This is the binding energy of the C-12 atom

6. Einstein’s relation: E = mc2
Energy is released in fusion reaction if the sum of masses of initial nuclei is larger that the mass of the final nucleus
hydrogen
mp + mp
hydrogen
Positron (antielectron)
Deuterium
MD + me < 2 mp
M = 2 mp- MD - me
Energy released E = M c2
neutrino
Einstein’s relation: E = mc2
Deuterium has larger binding energy than protons (more tightly bound)

7. Energy and nuclear reactions
Proton rest energy
When one proton and one neutron fuse to form a Deuteron nucleus, the final mass is less than the sum of the mass of the four particles. The deficit is the “binding energy”, amounting to 2.2 MeV
Here 0.1% of the mass of the particles is being converted to energy

8. |Ub| There are no heavy elements in the stars
Figure 7.16: The red line in this graph shows the binding energy (the energy that holds an atomic nucleus together) for all the different atoms plotted by atomic mass (the number of protons and neutrons in their nucleus). Both fission and fusion nuclear reactions move downward in the diagram (arrows) toward more tightly bound nuclei. Iron has the most tightly bound nucleus, so no nuclear reactions can begin with iron and release energy.
There are no heavy elements
in the stars

9. Nuclear time-scale
What if you could convert the entire mass of the Sun into energy?
At the current luminosity of the Sun, this would be spent in
If 0.1% of the mass is converted to energy, the Sun could
still last for 1010 yr if powered by nuclear fusion energy. The Sun is currently 4.5 x 109 yr old.

10. Nuclear energy: fusion
Nuclear energy is sufficient to sustain the Sun’s luminosity. But can it actually occur naturally in the Sun?

11. Coulomb repulsion
The repulsive force between like-charged particles results in a potential barrier that gets stronger as the particles get closer:
The strong nuclear force becomes dominant on very small scales, m
What temperature is required to overcome the Coulomb barrier?

12. Protons should be hot!

13. Statistical mechanics
If the gas is in thermal equilibrium with temperature T, the atoms have a range of velocities described by the Maxwell-Boltzmann distribution function.
The number density of gas particles with speed between v and v+dv is:
The most probable velocity:
The average kinetic energy:

14. Overcoming the Coulomb barrier
Fusion is possible if the average particle kinetic energy (3/2 kT) is equal to or greater than the Coulomb potential energy:
For two protons of z1=z2=1 separated by a typical distance of r=10-15 m
This is much larger than the central temperature of the Sun

15. Maxwell-Boltzmann doesn’t help
The central temperature of the Sun is
The KE of a proton at this temperature is ≈ 2 keV
The electrostatic PE of two protons m apart is 1 MeV
Could the protons at the tail end of the Maxwell-Boltzmann distribution of energies have sufficient kinetic energy to overcome the Coulomb barrier?
Energy
The relative fraction of protons with thermal energy of 1 MeV is only

16. Quantum mechanics to the rescue
The answer lies in quantum physics. The uncertainty principle states that momentum and position are not precisely defined:
The uncertainty in the position means that if two protons can get close enough to each other, there is some probability that they will be found within the Coulomb barrier.
This is known as tunneling.
The effectiveness of this process depends on the momentum of the particle

17. Quantum mechanics to the rescue
What temperature is required for two protons to come within one de Broglie wavelength of each other?

18. Quantum tunneling
Approximately: tunneling is possible if the protons come within 1 de Broglie wavelength of each other:
For two protons, at T~107 K
So the protons don’t need to get anywhere near 10-15m before they can begin to tunnel past the barrier
Without this quantum effect, fusion would not be possible in the Sun and such high luminosities could never be achieved.

19. Nuclear reactions
So – what are the specific reactions are we talking about??
The probability that four H atoms will collide at once to form a single He atom is exceedingly small. Even this simple fusion reaction must occur via a number of steps.

20. Proton-proton cycle
Figure 7.17: The proton–proton chain combines four protons (at left) to produce one helium nucleus (at right). Energy appears as gamma rays and as positrons, which combine with electrons to convert their mass into energy. Neutrinos escape, carrying away about 2 percent of the energy.

21. Proton-proton chain (PPI)
The net reaction is:
But each of the above reactions occurs at its own rate. The first step is the slowest because it requires a proton to change into a neutron:
This occurs via the weak force. The rate of this reaction determines the rate of Helium production

22. Proton-proton chain (PPII and PPIII)
Alternatively, helium-3 can react with helium-4 directly:
In the Sun, this reaction occurs 31% of the time; PPI occurs 69% of the time.
Yet another route is via the collision between a proton and the beryllium-7 nucleus
This reaction only occurs 0.3% of the time in the Sun.

23. The triple-alpha process
The burning of helium occurs via the triple alpha process:
The intermediate product 8-beryllium is very unstable, and will decay if not immediately struck by another Helium. Thus, this is almost a 3-body interaction
Note the very strong temperature dependence. A 10% increase in T increases the energy generation by a factor 50.

24. Nucleosynthesis
At the temperatures conducive to helium burning, other reactions can take place by the capturing of a-particles (He atoms).

25. Nucleosynthesis
The binding energy per nucleon describes the stability of a nucleus. It is easier to break up a nucleus with a low binding energy.

28. The solar neutrino problem

29. Neutrino have zero or very small mass and almost do not interact with matter
10,000 years

30. Neutrino image of the Sun

31. 400,000 liters of perchlorethylene buried 1 mile deep in a gold mine
The Davis experiment
400,000 liters of perchlorethylene
buried 1 mile deep in a gold mine
About 1 Chlorine atom per day is
converted into Argon as a result of
interaction with solar neutrino
Figure 7.18: (a) The Davis solar neutrino experiment used cleaning fluid and could detect only one of the three flavors of neutrinos. (Brookhaven National Laboratory)
There are 1032 Cl atoms in a tank!
Much more difficult than finding
a needle in a haystack!!

32. Sudbury neutrino observatory: 1000 tons of heavy water D2O

33. 32,000 ton of ultra-pure water
13,000 detectors

34. Observed neutrino flux is 2 times lower than the theoretical prediction!

35. Neutrinos should have mass Particle physics models should be modified
The problem has been finally solved just recently:
Neutrinos “oscillate”!
They are converted into other flavors: mu and tau neutrinos
Neutrinos should have mass
Particle physics models should be modified