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Somak Raychaudhurywww.sr.bham.ac.uk/~somak/Y2SiU/
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
Lecture 11Guest Lecturer: Dr W J Chaplin
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
dP(r)
drGM(r)
r2 (r)
Atomic nucleiAtomic 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.
12C has 6 protons, 6 neutrons and 6 electrons)
1u=1.66054x10-27 kg = 931.494 MeV/c2
um
um
um
e
n
p
0005486.0
00866.1
00728.1
• The mass of 1 proton + 1 electron is 1.0078285u• Note 6 p + 6 n + 6 e- = 12.099.
Binding energyBinding energy
There is also a binding energy associated with the nucleons themselves.
• The mass difference is 0.099u, equivalent to 92.22 MeV!• This is the binding energy of the C-12 atom
• The mass of the Carbon-12 atom: 6 p + 6 n + 6 e- = 12.099.
The mass of an atom (protons+neutrons+electrons) is not equal to the mass of the individual particles.
Energy is released in fusion reaction if the sum of masses of initial nuclei is larger that the mass of the final nucleus
mp + mp
MD + me < 2 mpDeuterium
Positron (antielectron)
neutrino
Deuterium has larger binding energy than protons (more tightly bound)
M = 2 mp- MD - meEnergy released E = M c2
Einstein’s relation: E = mc2
hydrogen
hydrogen
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
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 couldstill last for 1010 yr if powered by nuclear fusion energy. The Sun is currently 4.5 x 109 yr old.
Nuclear energy: fusionNuclear energy: fusion
Nuclear energy is sufficient to sustain the Sun’s luminosity. But can it actually occur naturally in the Sun?
Coulomb repulsionCoulomb repulsion
r
eZZUC
221
04
1
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, 10-15 m
• What temperature is
required to overcome the Coulomb barrier?
Statistical mechanicsStatistical 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:
dvvekT
mndvn kT
mv
v22
2/3
42
2
The most probable velocity:m
kTv 22
The average kinetic energy: kTvm2
3
2
1 2
Overcoming the Coulomb barrierOvercoming 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
Maxwell-Boltzmann doesn’t helpMaxwell-Boltzmann doesn’t help
Could the protons at the tail end of the Maxwell-Boltzmann distribution of energies have sufficient kinetic energy to overcome the Coulomb barrier?
The relative fraction of protons with thermal energy of 1 MeV is only
The central temperature of the Sun is
The KE of a proton at this temperature is ≈ 2 keV
The electrostatic PE of two protons 10-15 m apart is 1 MeV
Energy
Quantum mechanics to the rescueQuantum mechanics to the rescue
The answer lies in quantum physics. The uncertainty principle states that momentum and position are not precisely defined:
2
xpx
•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
Quantum mechanics to the rescueQuantum mechanics to the rescue
What temperature is required for two protons to come within one de Broglie wavelength of each other?
deBroglie h
3kTmH
Quantum tunnelingQuantum tunneling
•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.
Approximately: tunneling is possible if the protons come within 1 de Broglie wavelength of each other:
h
ph
v
h
3kTmH
For two protons, at T~107 K m
T7
3
1057.31013.1
Nuclear reactionsNuclear 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.
Proton-proton chain (PPI)Proton-proton chain (PPI)
HHeHeHe
HeHH
eHHH e
11
42
32
32
32
11
21
21
11
11
2
The net reaction is: 2224 42
11
eeHeH
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:
eenp Energy
This occurs via the weak force. The rate of this reaction determines the rate of Helium production
Proton-proton chain (PPII and PPIII)Proton-proton chain (PPII and PPIII)
HeHLi
LieBe
BeHeHe
e
42
11
73
73
74
74
42
32
2
Alternatively, helium-3 can react with helium-4 directly:
HeBe
eBeB
BHBe
e
42
84
84
85
85
11
74
2
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.
In the Sun, this reaction occurs 31% of the time; PPI occurs 69% of the time.
The triple-alpha processThe triple-alpha process
The burning of helium occurs via the triple alpha process:
CHeBe
BeHeHe126
42
84
84
42
42
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
kgWeTY T /1009.51
8027.4438
325
113
W/kg1085.3 0.418
325
83 TY
Note the very strong temperature dependence. A 10% increase in T increases the energy generation by a factor 50.
NucleosynthesisNucleosynthesis
At the temperatures conducive to helium burning, other reactions can take place by the capturing of -particles (He atoms).
NeHeO
OHeC2010
42
168
168
42
126
NucleosynthesisNucleosynthesis
The binding energy per nucleon describes the stability of a nucleus. It is easier to break up a nucleus with a low binding energy.
The Davis experiment
400,000 liters of perchlorethyleneburied 1 mile deep in a gold mine
About 1 Chlorine atom per day is converted into Argon as a result ofinteraction with solar neutrino
Much more difficult than finding a needle in a haystack!!
There are 1032 Cl atoms in a tank!