Problem 35 - Fundamentals of Physics

Question

A certain radionuclide is being manufactured in a cyclotron at a constant rate \(R\). It is also decaying with disintegration constant \(\lambda\). Assume that the production process has been going on for a time that is much longer than the half-life of the radionuclide. (a) Show that the number of radioactive nuclei present after such time remains constant and is given by \(N=R / \lambda .\) (b) Now show that this result holds no matter how many radioactive nuclei were present initially. The nuclide is said to be in secular equilibrium with its source; in this state its decay rate is just equal to its production rate.

Step-by-Step Solution

Answer

The number of radioactive nuclei at equilibrium is \(N = R / \lambda\), independent of initial amount.

Step 1: Understand the Problem

We need to find the number of radioactive nuclei, \(N\), produced at a constant rate, \(R\), and decaying at a rate proportional to the number of nuclei present. Also, the nuclei has reached a state called secular equilibrium, where its production rate equals its decay rate.

Step 2: Define the Decay Process

The decay of the nuclei is governed by the decay law, which is \(dN/dt = - \lambda N\). Here, \(\lambda\) is the disintegration constant and \(N\) is the number of nuclei at any time \(t\).

Step 3: Define the Production Process

The production adds nuclei at a constant rate \(R\). So, the total change in \(N\) with time \(t\) can be given by the equation \(dN/dt = R - \lambda N\).

Step 4: Find the Steady State Condition

In secular equilibrium, the number of decaying nuclei equals the number being produced, so \(dN/dt = 0\). Thus, \(R - \lambda N = 0\).

Step 5: Solve the Equation for N

Solving \(R - \lambda N = 0\) gives \(N = \frac{R}{\lambda}\). This represents the constant number of nuclei in secular equilibrium.

Step 6: Consider Initial Condition Independence

Initially, any number of nuclei might have been present, but over a time long compared to the half-life, the initial number becomes insignificant due to decay, showing \(N = R/\lambda\) solely determined by current production and decay rates.

Key Concepts

Cyclotron Production

Cyclotron production is a fascinating process used to create radionuclides, which are atoms with an unstable nucleus that loses energy by emitting radiation. In a cyclotron, charged particles such as protons or deuterons are accelerated using a magnetic field. These high-speed particles collide with a target, causing nuclear reactions that produce radionuclides specific to medical or scientific applications.

• These radionuclides are crucial for medical imaging and cancer therapy.
• The target material inside the cyclotron determines which radionuclide will be produced.
• Cyclotrons can create short-lived isotopes that decay quickly, making them suitable for direct application nearby, as transportation would result in significant loss of activity.

In the exercise, radionuclides are produced at a constant rate, denoted by the symbol \(R\). Understanding this process is essential for controlling the quantity of radionuclide available for various applications and ensuring its production is sustainable over long periods.

Radioactive Decay

Radioactive decay is a naturally occurring process by which an unstable atomic nucleus loses energy by emitting radiation. It is governed by the exponential decay law, which states that the quantity of undecayed nuclei decreases at a rate proportional to the number of remaining undecayed nuclei. This relationship is fundamental in various fields such as physics, chemistry, and medicine.

• Different types of decay include alpha, beta, and gamma decay, each involving the emission of specific particles or radiation.
• The rate of decay can be expressed by the differential equation \(\frac{dN}{dt} = - \lambda N\), where \(N\) is the number of radioactive nuclei and \(\lambda\) is the disintegration constant.
• The time it takes for half of a sample's nuclei to decay is known as its half-life.

Understanding the decay process allows scientists to predict how long a material will remain active and how it will behave over time, which is crucial for applications ranging from nuclear power to medical therapies.

Disintegration Constant

The disintegration constant, \(\lambda\), is a vital parameter in describing radioactive decay. It provides a measure of the probability per unit time that a single nucleus will decay. A higher \(\lambda\) indicates a faster decay process, leading to a shorter half-life.

• The decay equation \(\frac{dN}{dt} = - \lambda N\) shows that the number of remaining undecayed nuclei decreases exponentially over time.
• The half-life, \(t_{1/2}\), is related to the disintegration constant by the equation \(t_{1/2} = \frac{\ln(2)}{\lambda}\).
• In the scenario described in the exercise, the disintegration constant, \(\lambda\), is key to balancing the production and decay rates to achieve secular equilibrium, where \(N = \frac{R}{\lambda}\).

Understanding \(\lambda\) is crucial for accurately modeling nuclear processes and for various practical applications, including safety assessments and the design of medical and industrial techniques.