21. Neutron Transport

MIT OpenCourseWare

20 Sept 201954:22

EducationalLearning

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TLDRThe video script is a detailed lecture on the development of the neutron transport equation, a fundamental concept in nuclear engineering. The lecturer, Michael Short, begins by expressing enthusiasm for delving into the complex mathematics involved in tracking neutron populations within a nuclear reactor. The script outlines the process of formulating the equation by identifying variables such as position, energy, direction of travel, and time. It also discusses the simplification of the equation by eliminating angular dependence, resulting in fewer variables. The lecture covers various neutron interactions, including scattering, absorption, fission, and the concept of neutron multiplication. It further explores the neutron birth spectrum, the energy distribution of neutrons released from fission, and the role of different cross-sections in neutron interactions. The script concludes with an introduction to the diffusion approximation, a simplification technique for solving the neutron transport equation, and an invitation for students to engage with the computational reactor physics group for practical experience.

Takeaways

📚 The neutron transport equation is a complex mathematical model used to track the population of neutrons in a nuclear reactor, taking into account their position, energy, direction, and time.
🔍 The equation is developed by considering both gain and loss terms, which represent the production and removal of neutrons within a given volume.
🧮 Key variables in the equation include the neutron flux, neutron population, energy groups, and direction of travel, with many variables being angularly dependent or independent.
📈 The neutron transport equation is initially complex and is simplified over time to create a more manageable model for reactor calculations.
🤔 The physical processes considered in the equation include fission, scattering, absorption, and neutron multiplication, each with its own cross-section and spectrum.
🚀 External sources, such as californium, are used to initiate the fission process in nuclear reactors, acting as a 'kickstarter' to start the chain reaction.
🧬 Neutrons can also be produced through photo fission, where a gamma ray interacts with a nucleus, leading to fission and the subsequent release of neutrons.
💥 Inelastic scattering is considered a form of absorption because it results in a compound nucleus formation and the emission of a neutron from a different energy level.
🚫 Neutrons are not stable particles and can undergo decay, although this is a rare event with a mean-free path of approximately 10^8 centimeters.
🚧 The leakage term in the equation accounts for neutrons moving out of the control volume without undergoing any interaction.
⚙️ The divergence theorem is applied to convert surface integrals into volume integrals, simplifying the mathematical treatment of the neutron transport equation.

Q & A

What is the primary goal of the neutron transport equation?
-The primary goal of the neutron transport equation is to track the population of neutrons at a specific position, energy, direction, and time in a nuclear reactor.
What are the key variables involved in the neutron transport equation?
-The key variables involved in the neutron transport equation include the position vector r, energy E, direction omega, and time t, along with the neutron population or number of neutrons N(r, E, omega, t).
How does the neutron transport equation account for neutron gain and loss?
-The neutron transport equation accounts for neutron gain and loss by incorporating terms that represent the production and removal of neutrons due to various interactions such as fission, scattering, absorption, and leakage from the control volume.
What is the significance of the neutron multiplication factor (nu) in the neutron transport equation?
-The neutron multiplication factor (nu) represents the average number of neutrons produced during each fission event and is important for understanding the overall neutron population dynamics within a reactor.
What is the role of the scattering kernel in the neutron transport equation?
-The scattering kernel describes the probability of a neutron starting at a different energy and direction and then scattering into the energy and direction group being tracked by the equation.
How does the neutron transport equation handle neutron decay?
-Neutron decay is acknowledged as a process where a neutron spontaneously undergoes beta decay into a proton and an electron. However, due to its short half-life and the mean-free path involved, it is often not considered in the neutron transport equation for practical reactor calculations.
What is photo fission and how does it contribute to the neutron transport equation?
-Photo fission is a process where a gamma ray interacts with a nucleus and causes fission. It contributes to the neutron transport equation by adding neutrons to the population through the interaction of gamma rays with the reactor's materials.
What is the difference between the microscopic cross-section (sigma) and the macroscopic cross-section (Sigma)?
-The microscopic cross-section (sigma) represents the probability of interaction between a single neutron and a single particle, while the macroscopic cross-section (Sigma) represents the total probability of interaction with all particles present in the material.
How does the divergence theorem simplify the leakage term in the neutron transport equation?
-The divergence theorem allows the conversion of a surface integral of the neutron flux into a volume integral of the divergence of the flux. This simplification is crucial for transforming the equation into a more manageable form that can be solved with algebraic methods.
What is the diffusion approximation used in simplifying the neutron transport equation?
-The diffusion approximation is an assumption that treats neutrons as a diffusing gas, which simplifies the neutron transport equation from a second-order differential or integral equation into a simpler algebraic equation, making it easier to solve.

Outlines

00:00

📚 Introduction to Neutron Transport Equation

The paragraph introduces the concept of the neutron transport equation, emphasizing its complexity and importance in tracking neutron populations within a nuclear reactor. The discussion outlines the need to understand the equation's terms and how they relate to the physical behavior of neutrons, including the simplification process that will occur in subsequent classes.

05:03

🔬 Defining Cross-Sections and Neutron Interactions

This section delves into the various cross-sections associated with neutron interactions, such as the micro-cross-section and macroscopic cross-section. It also explores different types of neutron interactions with matter, including scattering, absorption, fission, and neutron decay, highlighting the significance of these processes in the context of the neutron transport equation.

10:08

🚀 External Sources and Neutron Multiplication

The paragraph discusses external sources of neutrons, such as californium, which can be used to initiate a nuclear reaction in a reactor. It also touches on neutron multiplication, the average number of neutrons produced per fission event, and the energy dependence of this process.

15:10

🤔 Scattering Kernel and Neutron Entry Mechanisms

This part of the script introduces the scattering kernel, a probability function that describes the change in energy and direction of neutrons as they scatter. It also addresses the various ways neutrons can enter a particular energy group, including from fission, external sources, and scattering from adjacent volumes.

20:10

🧠 Neutron Loss Mechanisms and Leakage

The paragraph outlines the loss mechanisms of neutrons, such as scattering out of the energy group, capture, and fission. It also explains the concept of leakage, where neutrons leave the control volume due to their continuous motion, and how it is treated differently in the transport equation.

25:13

🧮 Gain and Loss Terms in Neutron Transport

This section focuses on the mathematical representation of gain and loss terms in the neutron transport equation. It discusses how to describe the production of neutrons from fission, external sources, and photo fission, as well as the loss of neutrons due to reactions and leakage. The paragraph emphasizes the importance of integrating over all relevant variables to account for all possible interactions.

30:21

🔍 Simplifying the Neutron Transport Equation

The final paragraph discusses the process of simplifying the neutron transport equation using the divergence theorem and making the diffusion approximation. This simplification transforms the complex equation into a more manageable form that can be solved with algebra, making it easier to understand and apply in reactor physics.

Mindmap

Keywords

💡Neutron transport equation

The neutron transport equation is a fundamental equation in nuclear engineering that describes the behavior and interactions of neutrons within a nuclear reactor or other system. It is used to track the population of neutrons as a function of position, energy, direction, and time. In the video, the development of this equation is the central focus, with the goal of understanding the complex dynamics of neutron behavior in a reactor environment.

💡Fission

Fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts, releasing a large amount of energy and often producing free neutrons. In the context of the neutron transport equation, fission is a key source term that represents the production of neutrons from the fission process. The script explains how the fission term is developed by considering the reaction rate, neutron multiplication factor, and the energy distribution of neutrons produced.

💡Cross-section

In nuclear physics, the cross-section is a measure of the probability that a particular nuclear reaction will occur when a particle, such as a neutron, interacts with a target nucleus. It is typically expressed in units of area (barns) and is used to calculate the rate of reactions like fission, absorption, or scattering. The script discusses various micro-cross-sections and macroscopic cross-sections, which are essential parameters in the neutron transport equation.

💡Flux

Flux, in the context of nuclear engineering, refers to the rate at which particles, such as neutrons, pass through a unit area perpendicular to their direction of motion. It is a measure of the particle flow and is a critical variable in tracking the movement and interactions of neutrons within a reactor. The script discusses how the flux of neutrons is used in the neutron transport equation to calculate the reaction rates and the overall neutron balance.

💡Energy group

In the neutron transport equation, an energy group refers to a specific range or 'bin' of neutron energies being considered in the analysis. The script discusses tracking the population of neutrons within different energy groups, which is crucial for understanding how neutrons interact with the reactor material and how their energy distribution changes over time.

💡Angular dependence

Angular dependence refers to the variation of a physical quantity with respect to the direction of a particle's motion. In the neutron transport equation, it is important to consider the direction of neutron travel, as their interactions with materials can depend on the angle of incidence. The script discusses simplifying the equation by removing angular dependence to make it more manageable, which is a common approximation in certain types of reactor analysis.

💡Leakage

Leakage, in the context of the neutron transport equation, refers to the loss of neutrons from a control volume due to their movement out of the volume without undergoing a reaction. It represents the neutrons that 'escape' the region of interest and is an essential term in the equation to ensure the conservation of neutrons. The script explains how leakage is treated differently from other terms in the equation, as it involves the movement of neutrons rather than their interaction with materials.

💡Diffusion approximation

The diffusion approximation is a simplification used in the neutron transport equation that assumes neutrons behave like a diffusing gas, moving randomly and independently of each other. This approximation allows for the transformation of the transport equation into a more manageable form that can be solved with algebraic methods rather than complex differential equations. It is a key concept in reactor physics as it simplifies the analysis of neutron behavior.

💡Neutron birth spectrum

The neutron birth spectrum describes the distribution of energies at which neutrons are produced or 'born' during fission events. It is a probability distribution that indicates the likelihood of neutrons being released at various energies. Understanding the neutron birth spectrum is important for accurately modeling the behavior of neutrons in a reactor, as it affects the energy balance and the overall nuclear reaction dynamics.

💡In-scattering

In-scattering is a process in nuclear physics where a neutron that is moving towards a particular energy group is scattered into that group from a different energy group. This term is significant in the neutron transport equation as it represents the gain of neutrons in a particular energy group due to scattering from other energy groups. It is a key mechanism by which the energy distribution of neutrons within a reactor is maintained and changed.

Highlights

The neutron transport equation is a fundamental tool for tracking the population of neutrons in a nuclear reactor.

The development of the neutron transport equation involves complex mathematics and is a significant part of nuclear engineering education.

The neutron transport equation is used to understand neutron behavior in terms of position, energy, direction, and time within a reactor.

The flux of neutrons is defined as the product of the neutron population and their velocity.

The neutron transport equation simplifies by removing angular dependence, reducing the number of variables in the equation.

Cross-sections, such as micro-cross-section and macroscopic cross-section, are essential parameters in the neutron transport equation.

The neutron multiplication factor (nu) represents the average number of neutrons produced per fission event.

The neutron birth spectrum (chi of E) describes the energy distribution of neutrons released from fission.

Different types of interactions, such as scattering, absorption, fission, and capture, are considered in the neutron transport equation.

Neutrons can undergo elastic and inelastic scattering, which are treated differently in the equation.

The total cross-section represents the sum of all possible interactions a neutron can have with matter.

Neutrons are not stable particles and can undergo beta decay, although this is a rare event and often not considered in reactor physics calculations.

The scattering kernel is a probability function that describes the likelihood of a neutron scattering into a specific energy group.

The neutron transport equation accounts for neutron gains from fission, external sources, and scattering, as well as losses due to reactions and leakage.

The divergence theorem is used to convert surface integrals into volume integrals, simplifying the neutron transport equation.

The diffusion approximation is an upcoming topic that simplifies the neutron transport equation by treating neutrons as a diffusing gas.

The neutron transport equation is crucial for reactor calculations and is applicable to both fission and fusion reactors.

The course aims to prepare students for advanced studies in reactor physics, including computational reactor physics.

Transcripts

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