In this blog post, we’ll explore the concept of the reaction cross-section—which is used to understand and calculate the collisions and reactions of microscopic particles that we cannot see with our own eyes—and how it is applied.
When throwing darts, shooting a gun, or shooting a bow, the narrower sections of the target are usually worth more points. Similarly, when spinning a roulette wheel, the winning slots are typically located within a specific, narrow area. It goes without saying that there is a proportional relationship between the area occupied by a target and the probability that an object thrown at it will collide with it.
Let’s consider another scenario. Suppose you put on your running shoes and head outside to go for a run. If you want to run fast without bumping into other people, would the middle of Myeongdong Street on a weekend afternoon be a better choice, or the middle of a sports field? Again, this is a question with an unquestionable answer.
Similarly, the probability that a moving object will collide with a target is related to the effective area that the target occupies in the object’s frame of reference. By introducing this macroscopic concept, we can calculate the collision probability and frequency between particles, as well as the mean free path, mean free time, and relative reaction probability—even from a microscopic perspective that is difficult for us to observe directly. The physical quantity devised for this purpose is the cross section (X-sec).
In nuclear physics, the cross section is categorized into the microscopic cross section (σ) and the macroscopic cross section (Σ). The microscopic cross section represents the “reaction rate per unit intensity of the incident particle for a single target particle,” while the macroscopic cross section is the product of the microscopic cross section and the number density of target particles, and is related to the “reaction rate per unit intensity of the incident particle per unit volume of the target material.” The unit of the microscopic cross section is area, while the unit of the macroscopic cross section is inverse length. For the sake of convenience, this article will focus primarily on the microscopic cross section.
If you are encountering this concept for the first time, the definition of the reaction cross-section may seem unfamiliar, but let’s take a moment to review it carefully. Imagine there is a single target nucleus suspended in space, and a particle beam with uniform intensity—defined as the number of particles incident per unit area per unit time—is directed toward the target. Assuming that the cross-sectional area of the incident particles is sufficiently small compared to the area of the target nucleus or the entire space—and can therefore be neglected—the probability that these particles will react with the target nucleus depends on the intensity of the incident particles and the effective cross-sectional area provided by the target nucleus. Therefore, the number of reactions occurring in a single target nucleus per unit time—that is, the nuclear reaction rate—divided by the intensity of the incident particles, or in other words, the reaction rate per unit intensity for a single target particle, can be defined as the reaction cross section. In fact, the value obtained in this way has the dimension of the square of a length, i.e., area, and in nuclear physics, the unit “barn” is used for convenience. 1 barn is 10⁻²⁴ cm².
When the incident particle, such as a neutron, is uncharged and therefore not subject to Coulombic interactions with the target particle, it is relatively easy to intuitively understand the microscopic cross section by comparing it to a geometric cross section. However, one should not assume that the microscopic cross section will resemble the macroscopic geometric cross section we encounter in everyday life in all situations. When the interaction between the particle and the target nucleus is strong or the range of interaction is significant, the geometric size of the particles alone is insufficient to explain the probability of a reaction. Rather, the extent and intensity of the interaction between the particle and the target play a crucial role in determining the effective cross section. In such cases, even if a particle is spherical, its cross section is not necessarily circular. Simulating the magnitude and form of interactions for each reaction is theoretically challenging, and various new variables arising from quantum mechanical principles are introduced. Furthermore, as the relative velocity between the incident particle and the target nucleus increases, the effects observed in the incident system may change, and even if a collision occurs, the conditions necessary for an actual reaction to take place may be limited. Microscopic reactions are thus influenced by a variety of such variables. Consequently, the reaction cross section depends not only on the type of reaction but also on various variables that determine the particle’s state of motion, such as its energy, velocity, and temperature. For this reason, the reaction cross section required for a specific reaction is generally obtained from nuclear data, which is either measured experimentally or evaluated based on theoretical calculations and experimental data. When deriving an equilibrium equation that includes the reaction cross section in theoretical calculations, it is crucial to identify and use the appropriate reaction cross section that matches the given conditions.
Let’s consider a simple example involving the use of the reaction cross section. In a given space, multiplying the number density of target particles by the reaction cross section and by the particle flux—which represents the intensity of incident particles—allows us to calculate the number of reactions occurring per unit volume per unit time. One of the most critical factors in controlling a nuclear reactor is regulating its power output. In this process as well, a balancing equation using the reaction cross section mentioned earlier is established. Using the number density of fuel nuclei, the reaction cross section, and the neutron flux, we can determine the number of nuclear reactions per unit volume and per unit time; by factoring in the energy released per reaction, we can calculate the thermal power generated by the nuclear reactor. In actual reactor power calculations, one must consider the neutron flux and reaction cross sections as a function of space, as well as the fuel composition and the contributions of various reactions. Such analysis plays a pivotal role not only in calculating the power output of a nuclear reactor but also in its design, operation, and control.
The cross section is also used to calculate the physical concept of the mean free path—the distance a particle can travel freely without colliding. By substituting the cross section, one can calculate the probability that a particle will react with a target per unit distance as it travels through a material. Using this probability, one can determine the probability of not colliding while traveling a certain distance; by integrating this over the total distance traveled and solving the equation, it can be shown that, under simple conditions, the mean free path is equal to the reciprocal of the product of the reaction cross section and the number density of target nuclei, i.e., λ = 1/(nσ). However, in real physical systems, the cross section may vary depending on the particle’s energy; therefore, it is sometimes necessary to consider both the particle energy distribution and various types of interactions. This mean free path is closely related to the cross section and can be expressed using a relatively simple equation; this concept is extremely useful when dealing with microscopic particle transport phenomena such as fluid flow, radiation transport, and plasma.
In this article, we have discussed the reaction cross section, one of the most fundamental and useful concepts in nuclear physics. As an aside, nuclear engineering encompasses the design, control, and power generation of nuclear fission reactors, plasma for future nuclear fusion, and various fields related to radiation. Since these fields all deal with microscopic particles and their interactions, students majoring in nuclear engineering learn the concept of the reaction cross section from the moment they enter the program and are trained on how to apply it effectively. This is because the reaction cross section is an indispensable tool for analyzing microscopic reactions.
Although, contrary to its name, the reaction cross-section does not refer to an actual geometric area per se, but rather is a concept that expresses the probability of a specific reaction occurring as a physical quantity with the dimension of area. However, by imagining that it operates on a principle similar to the cross-sectional area of targets we see in everyday life, it becomes very convenient and useful for visualizing and analyzing microscopic situations that we cannot see directly. Furthermore, the concept of reaction cross-sectional area is closely utilized in explaining fundamental physical concepts—such as kinetic theory of gases—and in deriving equations. Let’s take a moment to reflect on whether we might be using this concept, consciously or unconsciously, in our own fields of study. And let’s appreciate how this powerful tool—reaction cross-sectional area—allows us to mentally visualize and analyze microscopic reactions that we cannot observe directly.
