Nuclear Fusion: The Power Behind Stars
Deuterium-Tritium Fusion
- Consider the fusion of deuterium ($ ^2\text{H} $) and tritium ($ ^3\text{H} $): $$^2\text{H} + ^3\text{H} \rightarrow ^4\text{He} + ^1\text{n}$$
- In this reaction, a helium nucleus ($ ^4\text{He} $) and a neutron ($ ^1\text{n} $) are produced.
- The mass of the reactants is slightly greater than the mass of the products, and this mass difference ($ \Delta m $) is converted into energy: $$Q = \Delta m c^2$$
- For this reaction, the energy released is approximately 17.6 MeV (million electron volts) per fusion event, an immense amount of energy at the atomic scale.
What is the role of the mass defect in determining the energy released during nuclear fusion?
Calculate the energy released in the reaction $^2\text{H} + ^3\text{H} \rightarrow ^4\text{He} + ^1\text{n}$, given the following atomic masses:
- $^2\text{H}$: 2.014102 u
- $^3\text{H}$: 3.016049 u
- $^4\text{He}$: 4.002602 u
- $^1\text{n}$: 1.008665 u
Solution
- The mass defect is: $$\Delta m = (2.014102 + 3.016049) - (4.002602 + 1.008665)$$ $$ = 0.018884 , \text{u}$$
- Convert this mass into energy using $1 \, \text{u} = 931.5 \, \text{MeV c}^{-2}$: $$Q = \Delta m c^2 $$ $$= 0.018884 \times 931.5 = 17.6 \, \text{MeV}$$
Conditions for Fusion to Occur
Fusion requires extreme conditions to overcome the Coulomb repulsion between positively charged nuclei. Let’s break down the three key conditions:
High Temperature
- At high temperatures (millions of kelvin), nuclei move at very high speeds.
- This increases the likelihood that they will collide with enough energy to overcome the repulsive electrostatic force between them.
In stars, core temperatures often exceed 10 million K, making fusion possible.
High Density
A high density of nuclei ensures that there are enough collisions happening per second to sustain a chain of fusion reactions.
In stars, the core is incredibly dense, with billions of particles packed into a small volume.
Confinement Time
The conditions of high temperature and density must be maintained for a sufficiently long time to allow significant fusion to occur.
In stars, gravitational pressure ensures that the core remains stable and confined.
- Many students think high temperature alone is enough for fusion.
- However, without high density and confinement, the nuclei would simply fly apart after collisions, preventing sustained fusion.
High Pressure
- High pressure plays a crucial role in fusion by forcing nuclei closer together, increasing the probability of collisions.
- In stars, gravitational pressure compresses the plasma, ensuring a high density of nuclei and sustaining fusion reactions.
High pressure increases the likelihood of fusion by bringing nuclei closer together, enhancing the collision rate.
Quantum Tunneling
- Even at extreme temperatures, classical physics suggests that most nuclei do not have enough energy to overcome the Coulomb barrier.
- However, due to quantum mechanics, there is a small probability that nuclei can "tunnel" through the energy barrier instead of going over it.
- This effect allows fusion to occur at lower energies than expected and is essential for sustaining fusion reactions in stars.
Quantum tunneling allows nuclei to fuse even when they don’t have enough classical energy to overcome electrostatic repulsion.
Why Does Fusion Release Energy?
The energy released in fusion comes from the binding energy of atomic nuclei.
Binding energy
Binding energy is the energy required to hold the nucleus together.
- Heavier nuclei like helium have a higher binding energy per nucleon than lighter nuclei like hydrogen.
- When light nuclei fuse to form a heavier nucleus, the total binding energy increases, and the excess energy is released.
- The binding energy per nucleon peaks at iron ($ ^{56}\text{Fe} $), which is why fusion processes in stars stop at iron.
- Beyond this, energy must be added to fuse heavier elements.
Why does the binding energy per nucleon determine whether a fusion reaction releases or absorbs energy?
Fusion in Stars: The Energy Source of the Universe
- Stars are nature’s perfect fusion reactors.
- In their cores, immense gravitational pressure creates the high temperatures and densities needed for fusion.
- Let’s examine how this works:
The Proton-Proton Chain
- In stars like our Sun, the dominant fusion process is the proton-proton chain, which converts hydrogen nuclei ($ ^1\text{H} $) into helium ($ ^4\text{He} $): $$^1\text{H} + ^1\text{H} \rightarrow ^2\text{H} + e^+ + \nu_e$$ $$^2\text{H} + ^1\text{H} \rightarrow ^3\text{He} + \gamma$$ $$^3\text{He} + ^3\text{He} \rightarrow ^4\text{He} + 2 ^1\text{H}$$
- The net result is the fusion of four hydrogen nuclei into one helium nucleus, releasing energy in the form of photons, positrons, and neutrinos.
- The mass of a star determines its core temperature and, therefore, the dominant fusion process.
- More massive stars rely on the CNO cycle, which requires higher temperatures than the proton-proton chain.
