Exploring the Neutron Diffusion Equation in Nuclear Physics
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Chapter 1: Introduction to Neutron Diffusion
This article serves as the second installment in a series exploring the mathematics and physics that contributed to the development of nuclear weaponry during World War II. For a foundational overview and historical context, I recommend reviewing the preceding article in this series.
In the earlier discussion, we examined Werner Heisenberg's significantly flawed estimation of the critical mass of uranium, which is the quantity needed to initiate and sustain a nuclear fission chain reaction (essentially, the mass required for creating an atomic bomb). We also analyzed how this miscalculation impacted the war's outcome. Subsequently, we explored the mathematics behind Heisenberg's random walk model, which inaccurately estimated a critical mass of 13 tonnes of uranium.
The following articles will provide an overview of the more precise neutron diffusion methodology developed by Allied physicists, which enabled the Western Allies to create the first atomic bomb in 1945. Initially, we will derive the neutron diffusion equation for a sphere of uranium-235 and then utilize it to determine the critical mass of uranium-235.
The Neutron Diffusion Methodology
The neutron diffusion approach, rooted in a 1940 memorandum by physicists Otto Frisch and Rudolf Peierls, allowed Allied nuclear scientists to estimate the critical mass of approximately 60 kg of uranium-235. This method is based on a fundamental concept:
If the neutron production rate within a sphere of uranium-235 surpasses the rate at which neutrons escape, an exponential increase in fissions occurs, leading to a self-sustaining nuclear fission chain reaction.
Consequently, the challenge of determining the critical radius transforms into identifying the spherical radius where internal neutron production and neutron loss are perfectly balanced.
To ascertain this balance, we must first establish a mathematical framework to describe how neutrons are produced and, more critically, how they diffuse from the sphere. This requires a dynamic equation.
Chapter 2: Deriving the Neutron Diffusion Equation
To understand the production and diffusion of neutrons throughout a large sphere of uranium-235, let us first consider an infinitesimal region at an arbitrary position (x, y, z) within the sphere. We will refer to this as a control volume (CV) with dimensions Δx, Δy, and Δz.
We aim to formulate an expression for the rate of change in the total neutron count within this control volume. Neutrons will flow into and out of the CV in each dimension, and some will also be generated internally. Thus, the dynamic equation will take the following form:
Before constructing the equation, we need to define three key variables and clarify their units:
- N(x,y,z,t): neutron density (number of neutrons per unit volume) [# m⁻³]
- J(x,y,z,t): neutron 'current' density or neutron flux (number of neutrons passing through a unit area per second) [# s⁻¹ m⁻²]
- v: average neutron velocity [m s⁻¹]
- σ: neutron number (the average number of neutrons released during fission)
The Rate of Neutron Accumulation
The expression for the total neutron count change is straightforward. The total number of neutrons is the product of neutron density N and the volume of the CV, V = ΔxΔyΔz. Thus, we have:
The Rate of Internal Neutron Production
Neutron production rates depend on various factors. Primarily, neutrons are generated through fission events, and the occurrence of fissions is directly proportional to the total number of neutrons in the CV that can initiate them.
However, not every neutron in the CV leads to a fission event. There exists a probability factor that quantifies the likelihood of a neutron triggering a fission. This can be expressed as 1/𝜏, where 𝜏 is the average time between fission events. Each fission results in one neutron being absorbed and σ neutrons being released.
By integrating these factors, we derive the net flow rate of neutrons in each dimension by analyzing the number of neutrons moving into and out of the CV.
The first video titled "Solving the Neutron Diffusion Equation and Criticality Relations" offers insights into the mathematical foundation of neutron diffusion in nuclear engineering.
Relating Flux Density J to Neutron Density N
Revisiting a concept from the previous article, the mean free path 𝛌 represents the average distance a neutron travels before colliding with a uranium-235 nucleus.
To find the expression for flux density J through a surface in one dimension, consider the neutron sources located at ±𝛌. Neutrons emit from each source at an average velocity v, with half traveling left and half to the right. The resulting flux density is the product of neutron density N and velocity v.
Thus, our relationship between flux density and neutron density can be expressed as:
This relationship can be generalized for a three-dimensional surface, leading to a comprehensive equation.
Putting It All Together
At this point, we can merge the expressions for total neutron accumulation, internal production rates, and net flow rates to derive the neutron diffusion partial differential equation.
Next Up…
In the following article, we will utilize the method of separation of variables to solve the neutron diffusion partial differential equation, leading to an analytical expression for the neutron density N(x, y, z, t) in a sphere of uranium-235. We will then apply boundary conditions to determine the critical radius and ultimately the critical mass of uranium-235.
The second video, titled "NE410/510 - Lecture 8: The P1 Approximation and the Neutron Diffusion Equation," delves into the P1 approximation within the context of neutron diffusion.
References
[1] Bernstein, Jeremy. "Heisenberg and the critical mass." American Journal of Physics 70.9 (2002): 911–916.
[2] Logan, Jonothan. "The critical mass." American Scientist 84.3 (1996): 263–277.