In the context of nuclear fusion, the cross-section refers to the measure of the likelihood that two atomic nuclei will undergo a fusion reaction when they collide. It is typically denoted by the symbol “σ” (sigma) and has units of area.
The cross-section for nuclear fusion depends on various factors, including the types of nuclei involved, their energy, and the conditions of the environment. The reaction cross-section is an important parameter for understanding the rate at which fusion reactions occur in a plasma, such as in a fusion reactor.
Different fusion reactions have different cross-sections, and they vary based on factors such as the charges of the nuclei and the energies at which they collide. For example, the cross-section for the most well-known fusion reaction, which occurs in stars like our Sun, involves the fusion of two hydrogen nuclei (protons) to form helium-2 nuclei:
4 protons -> 2 protons + 2 neutrons + energy.
This reaction is known as the proton-proton chain and is responsible for the energy production in stars. The cross-section for this reaction is relatively small because it involves positively charged protons overcoming the electrostatic repulsion between them to get close enough for the strong nuclear force to bind them together.
In fusion research for practical energy production, scientists are primarily interested in reactions that occur between isotopes of hydrogen, such as deuterium (a heavy isotope of hydrogen) and tritium (another heavy isotope of hydrogen). One of the most promising fusion reactions for energy production is the deuterium-tritium fusion reaction:
2 deuterium nuclei + 3 tritium nuclei -> 1 helium nucleus + 1 neutron + energy.
The cross-section for this reaction is higher than that of other reactions involving hydrogen isotopes, making it more feasible for controlled fusion reactions in experimental devices like tokamaks and stellarators.
The measurement and understanding of these cross-sections are critical for designing and operating fusion reactors, as they directly impact the rate of energy production and the efficiency of the fusion process.
The main nuclear fusion
Keep in mind that these values can vary based on factors such as energy, temperature, and the specific isotopes involved. The cross-sections are usually given in units of barns (1 barn = 10^-28 m²). Here are some examples of important fusion reactions and their cross-sections:
- Deuterium-Deuterium Fusion:
D + D -> He-3 + n (helium-3 nucleus + neutron)
Cross-section: Around 0.0001 barns at typical fusion temperatures. - Deuterium-Tritium Fusion:
D + T -> He-4 + n (helium-4 nucleus + neutron)
Cross-section: Peaks around 20-30 barns at fusion-relevant temperatures. - Deuterium-Helium-3 Fusion:
D + He-3 -> He-4 + p (helium-4 nucleus + proton)
Cross-section: Varies but generally lower than D-T fusion, around 1 barn at fusion temperatures. - Proton-Boron Fusion (aneutronic fusion):
p + B-11 -> 3 He-4
Cross-section: Quite low, typically on the order of microbarns or smaller. - Carbon-Nitrogen-Oxygen (CNO) Cycle (important in massive stars):
Various reactions involving C, N, and O isotopes.
Cross-sections: Vary depending on specific reactions, generally smaller compared to D-T fusion.
It’s important to note that the cross-section values are highly dependent on the energy of the particles involved. Fusion reactions in a controlled environment, like a fusion reactor, require the particles to have sufficient energy to overcome the Coulomb barrier (repulsive electrostatic force) and come close enough for the strong nuclear force to bind them together.
Also, the cross-sections mentioned above are approximate and can vary based on experimental conditions and isotopic compositions. Scientists and researchers continually refine these values through experiments and theoretical calculations to better understand fusion reactions and improve the design of fusion reactors.
Here is a live plot of the cross-sections for DT, DD and DHe3 fusion.
H.-S. Bosch and his colleagues made significant contributions to improving the accuracy of formulas for fusion cross-sections, particularly for the Deuterium-Tritium (D-T) fusion reaction. Their work led to more accurate and reliable predictions of fusion cross-sections in various energy ranges, which is crucial for understanding and designing fusion experiments and reactors.
One of the notable contributions by H.-S. Bosch and others is the formulation of the “Bosch-Hale fusion cross-section formula.” This formula provides a more accurate representation of the D-T fusion cross-section as a function of the incident particle energy. It takes into account the energy dependence of the cross-section and provides a more comprehensive description of the fusion process compared to the earlier empirical formulas.
The Bosch-Hale formula takes into account both the astrophysical S-factor and the Coulomb barrier effects, providing a more accurate description of the fusion cross-section over a wide range of energies. This formula has been widely used in fusion research and provides better agreement with experimental data than previous empirical fits.
It’s important to note that the Bosch-Hale formula is just one example of the improvements made to fusion cross-section calculations, and other refined models and data compilations have been developed by various researchers to enhance our understanding of fusion reactions.
Here is a live plot of the cross-sections for DT, DHe3 and DD fusion by Bosch-Hale formulas.
Reactivity is a measure of how much a nuclear reactor or fusion system deviates from its critical state. In a critical state, the rate of nuclear reactions (fission or fusion) is constant, meaning that the system is self-sustaining. Reactivity is often used to describe the degree to which a reactor or fusion system is operating above or below criticality.
In a nuclear fission reactor, reactivity can be used to control the rate of fission reactions. Positive reactivity indicates a condition where the reactor is supercritical, meaning that the rate of reactions is increasing over time. Negative reactivity indicates a subcritical condition, where the rate of reactions is decreasing. Reactivity control is crucial for maintaining the stability and safety of nuclear reactors.
In the context of nuclear fusion, reactivity is also important. Fusion reactors aim to achieve a self-sustaining fusion reaction, where the energy produced by fusion reactions is greater than the energy required to sustain the necessary conditions. Positive reactivity in a fusion reactor context implies that the fusion reactions are producing more energy than needed to maintain the plasma state and the required temperature and pressure. Negative reactivity indicates that the reactor is not sustaining the fusion reactions.
Reactivity can be controlled and adjusted using various methods, such as changing the density of the fuel, adjusting the geometry of the reactor, or using control materials like control rods in fission reactors.
Overall, reactivity is a fundamental concept in nuclear physics and engineering, and it plays a crucial role in maintaining safe and controlled nuclear reactions, whether in fission or fusion systems.
The fusion rate depends on the relative velocity of the reacting particles and the ions in the plasma have a velocity distribution. The fusion rate per volume is proportional to fusion reactivity `<sigma v>`:
`(dR)/dV=(n_i n_j)/(1+delta_{ij})<sigma v>`
Here is a live plot of the DT reactivity(`xx10^16`).
The Coulomb potential barrier, also known as the Coulomb barrier, is a concept in physics that refers to the energy barrier that must be overcome for two charged particles to interact or come close enough to undergo a specific process, such as nuclear reactions or particle interactions. It arises due to the electrostatic repulsion or attraction between charged particles, as described by Coulomb’s law.