Question

In an absorbing medium, the neutron density obeys the differential equation $$\kappa {\mathrm{\nabla }}^{2}n-\frac{n}{T}=\frac{\partial n}{\partial t}$$ where $T$ is a constant. At $t=0$, a burst of neutrons is produced on the $yz$-plane of an infinite medium, so that $$n(\mathbf{x},0)=\delta (x)[\mathrm{not}\delta (\mathbf{x})]$$ Find the neutron density, and the total number of neutrons (per unit area), for later times.

Short answer

The neutron density is $$n(x,t)=\sqrt{\frac{T}{4\pi \kappa t}}{e}^{-\frac{x^{2}T}{4\kappa t}}{e}^{-\frac{t}{T}}$$ and the total number of neutrons per unit area is $$N(t)=e^{-\frac{t}{T}}$$

Step-by-step solution

Identify the type of differential equation

The given differential equation is: \ \ $$\kappa abl{a}^{2}n-\frac{n}{T}=\frac{\partial n}{\partial t}$$ Recognize that this is a partial differential equation (PDE) involving both space and time variables.

Simplify using Fourier Transform

To solve the equation, apply the Fourier Transform with respect to the spatial coordinates. Let $$\hat{n}(\mathbf{k},t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}n(\mathbf{x},t)e^{-i\mathbf{k}\cdot \mathbf{x}}d\mathbf{x}$$ Using properties of Fourier transforms: $$abl{a}^{2}n\to -k^2\hat{n}$$ and $$\frac{\partial n}{\partial t}\to \frac{\partial \hat{n}}{\partial t}$$ The PDE in the Fourier domain becomes: $$\kappa (-k^2)\hat{n}-\frac{\hat{n}}{T}=\frac{\partial \hat{n}}{\partial t}$$

Solve the transformed differential equation

Rearrange the transformed equation to: $$\frac{\partial \hat{n}}{\partial t}+(\frac{1}{T}+\kappa k^2)\hat{n}=0$$ This is a first-order linear differential equation, and its general solution is: $$\hat{n}(\mathbf{k},t)=\hat{n}(\mathbf{k},0)e^{-(\frac{1}{T}+\kappa k^2)t}$$

Apply the initial condition

Given: $$n(\mathbf{x},0)=\delta (x)$$ The initial condition in the Fourier domain is: $$\hat{n}(\mathbf{k},0)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)e^{-i\mathbf{k}\cdot \mathbf{x}}d\mathbf{x}=1$$ Thus, we have: $$\hat{n}(\mathbf{k},t)=e^{-(\frac{1}{T}+\kappa k^2)t}$$

Inverse Fourier Transform to find spatial solution

Obtain the neutron density in real space by taking the inverse Fourier Transform: $$n(\mathbf{x},t)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-(\frac{1}{T}+\kappa k^2)t}{e}^{i\mathbf{k}\cdot \mathbf{x}}dk$$ The integral solution gives: $$n(x,t)=\sqrt{\frac{T}{4\pi \kappa t}}{e}^{-\frac{x^{2}T}{4\kappa t}}{e}^{-\frac{t}{T}}$$

Compute the total number of neutrons per unit area

Integrate the neutron density over all space to find the total number of neutrons per unit area: $$N(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}n(x,t)dx$$ Here, $$N(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\sqrt{\frac{T}{4\pi \kappa t}}{e}^{-\frac{x^{2}T}{4\kappa t}}{e}^{-\frac{t}{T}}dx$$ Since the integral of the Gaussian function is 1, $$N(t)=e^{-\frac{t}{T}}$$

Key concepts

Fourier Transform

The Fourier Transform is a mathematical technique used to transform a function of time (or space) into a function of frequency. Essentially, it decomposes a signal into its constituent frequencies. For a given function, the Fourier Transform is defined as:
$$\hat{f}(k)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-ikx}dx$$ In this exercise, it helps us convert the partial differential equation into a simpler algebraic form by focusing on frequency components instead of direct spatial variations. This allows for easier manipulation and solving of the differential equations. The inverse Fourier Transform is then used to convert the solution back to the spatial domain.

Neutron Density

Neutron density describes the number of neutrons per unit volume in a given space. In the context of the exercise, we are looking at the neutron density resulting from a burst of neutrons on a specific plane (the yz-plane) at time t = 0. The differential equation given in the problem helps us model how this neutron density changes over time and space. The time evolution of neutron density is governed by factors such as diffusion and absorption, represented in the equation by the terms involving $abl{a}^{2}n$ and $\frac{n}{T}$, respectively. This model is crucial in understanding and predicting the behavior of neutrons in various physical environments.

Diffusion Equation

The diffusion equation is a partial differential equation that describes the spread of particles, energy, or other physical quantities over time. It’s used to model processes like heat conduction, pollutant spread, and in this case, neutron absorption and movement. The general form of the diffusion equation is a combination of Laplace's operator (${\text{abla}}^2$) and a time derivative:
$$\kappa {\text{abla}}^{2}n-\frac{n}{T}=\frac{\text{/}\partial n}{\text{/}\partial t}$$ Here, $n$ represents the neutron density, $\kappa$ is the diffusion coefficient, and $T$ is a characteristic time constant. The equation balances the diffusion of neutrons through the medium (represented by $\kappa {\text{abla}}^{2}n$) with their absorption (represented by $\frac{n}{T}$), explaining how neutron density changes over time.

Gaussian Integral

The Gaussian integral is fundamental in probability theory and various fields of physics and engineering. It refers to the integral of the Gaussian function e^{-x^2}, which is characteristic of a normal distribution. The Gaussian integral is:
$${\int }_{-fty}^{fty}{e}^{-\frac{x^2}{2}}dx=\sqrt{2\pi }$$ In this exercise, the neutron density solution involves a Gaussian function due to the initial condition (a delta function). The integral's properties simplify the calculation of neutron density over space and time. Consequently, the computed total number of neutrons per unit area integrates this Gaussian function, making use of its well-known result to simplify the integral computation.