The Mathematics of Neutron Diffusion and Critical Mass in Uranium
Written on
Chapter 1: Introduction to Neutron Diffusion
This article serves as the third installment in a series focused on the mathematics and physics underpinning the evolution of nuclear weapons during World War II. For a foundational understanding and historical context, I recommend starting with the initial article in this series.
In that first article, we examined Werner Heisenberg's inaccurate estimates of uranium's critical mass—the necessary quantity for initiating and maintaining a nuclear fission chain reaction, integral for constructing an atomic bomb. His miscalculations significantly impacted the war's outcome. We also explored the mathematical framework behind his random walk model, which incorrectly projected a critical mass of 13 tonnes of uranium.
The second article introduced the mathematical principles behind the refined neutron diffusion method used by Allied physicists. This approach led to an accurate estimation of a critical mass of only 60 kg, enabling the development of the first atomic bomb in 1945. We also derived the neutron diffusion partial differential equation relevant to a sphere of uranium-235.
Now, we will conclude this series by employing a standard technique for solving partial differential equations known as separation of variables to resolve the neutron diffusion equation and determine the critical mass of a sphere of uranium-235.
Section 1.1: Understanding the Neutron Diffusion Equation
Before diving into the solution, let's revisit the core concept of the neutron diffusion method for addressing the critical mass problem and reintroduce the neutron diffusion equation we established in the previous article. The essence of the neutron diffusion approach hinges on one fundamental principle:
If the rate of neutron production inside a sphere of uranium-235 surpasses the rate at which neutrons escape, an exponential increase in fissions will occur, resulting in a self-sustaining nuclear fission chain reaction.
Consequently, the challenge of identifying the critical radius translates to finding the radius at which neutron production and loss are perfectly balanced.
To achieve this, we require a dynamic equation that outlines neutron production and diffusion throughout the sphere. This dynamic equation is the neutron diffusion equation, which we derived earlier.
Section 1.2: Transforming to Spherical Coordinates
Given our focus on a spherical mass of uranium-235, let's transform the neutron diffusion equation into a spherical coordinate system. This necessitates recognizing that neutron density shifts from relying on Cartesian coordinates N(x, y, z, t) to spherical coordinates N(r, θ, ϕ, t). In spherical coordinates, the Laplacian adapts accordingly.
However, since our sphere exhibits spherical symmetry, the neutron density does not depend on the azimuthal angle ϕ or the polar angle θ. Thus, neutron density is solely dependent on the radius r and time t, denoted as N(r, t). The Laplacian then simplifies:
We can now derive the spherical form of the neutron diffusion equation.
Chapter 2: Solving the Neutron Diffusion Equation
The first video titled "How to calculate an atomic bomb's critical mass" delves into the calculations needed to determine the critical mass for nuclear reactions, providing valuable insights into the underlying principles.
To linearize the equation while substituting variables, we recognize that the last term on the right side introduces a non-linear element, complicating the solution. We can simplify this by employing a variable substitution.
Substituting this relation into the neutron diffusion equation leads us to a linear form of the equation.
We can now solve for U(r, t) using the separation of variables method, ultimately reverting back to find an expression for neutron density N(r, t).
Section 2.1: Applying the Separation of Variables Technique
Separation of variables operates on the premise that our solution U(r, t) can be expressed as a product of a time function and a radius function:
Substituting this into the linearized spherical form of the neutron diffusion equation allows us to reorganize the equation so that one side is dependent on time and the other solely on radius.
The equality of time-dependent and radius-dependent terms implies that both sides must equal a constant, termed the separation constant. For our analysis, we express this constant in terms of the average time between fission events 𝜏.
This separation enables us to break the partial differential equation into two ordinary differential equations (ODEs)—one tied to time t and the other to radius r.
Section 2.2: Resolving the Time-Dependent ODE
Solving a first-order ordinary differential equation is relatively straightforward. By employing a trial solution for our time-dependent ODE, we derive an exponential function as the solution.
The next step is to resolve the radius-dependent ODE. Here, we can apply a similar method, using a suitable trial solution.
After substituting this Ansatz into the radius-dependent ODE, we compile the terms into a single γ term, yielding a general solution.
As we expand each complex exponential via Euler's formula, we arrive at a refined expression for U(r, t).
Combining our findings, we derive a neutron density equation that functions based on radius and time.
Section 2.3: Determining Critical Mass
By analyzing our neutron diffusion equation, we observe that neutron density either exponentially grows or decays based on the separation constant 𝛼. The critical point occurs when 𝛼 equals zero.
To ascertain the critical radius, we question: "What radius satisfies the criticality condition 𝛼R = 0?"
We apply boundary conditions to our sphere's surface, denoting this radius as R. A simple Dirichlet boundary condition suggests that neutron density at the surface is zero.
Alternatively, a Neumann boundary condition could provide a more accurate estimate, but for simplicity, we will proceed with the Dirichlet condition.
By substituting our boundary conditions into the neutron diffusion equation, we can derive an expression for the critical radius.
Finally, by incorporating the diffusion coefficient derived earlier, we can represent everything in terms of the mean free path length 𝜌 and neutron number σ.
The Critical Radius and Mass
Plugging in experimental values, we find the neutron number σ to be approximately 2.3, and the mean free path length is around 6 cm. This leads us to a critical radius estimation, and knowing the density of uranium-235 (approximately 18.95 g cm⁻³), we can ascertain an approximate critical mass of 60 kg.
The second video titled "23. Solving the Neutron Diffusion Equation, and Criticality Relations" elaborates on the mathematical intricacies of neutron diffusion and criticality, helping to visualize these concepts in practice.
To visualize the radial distribution of neutron density at criticality, we can examine the time-independent distribution.