12. Numerical Examples of Activity, Half Life, and Series Decay

The video begins with an introduction to MIT OpenCourseWare, emphasizing the importance of donations for maintaining free educational resources. The lecturer, Michael Short, introduces a numerical example related to radioactive decay, specifically using cobalt-60 as a case study. He discusses the initial activity of the source, its calibration in March 2011, and a subsequent measurement in October 2016. The concept of significant figures is highlighted, and the uncertainty in the source's activity level is calculated, revealing a range between 0.5 to 1.5 microcuries.

The lecturer delves into the specifics of cobalt-60's half-life, which is 1,925 days, and uses this information to calculate the decay of the radioactive source from March 2011 to October 2016. The audience is walked through the process of finding the decay constant (lambda) and using it to determine the original activity of the source. The calculation results in an initial activity of approximately 1.07 microcuries, which is deemed intuitive given the elapsed time and the half-life of cobalt-60.

Short shifts the focus to calculating the mass of cobalt-60 originally present in the source. He uses the Curie to becquerel conversion and the decay constant to find the initial number of atoms. By applying Avogadro's number, the number of moles, and the atomic mass of cobalt-60, he calculates the mass of cobalt-60 to be 0.95 nanograms. This segment emphasizes the potency of small amounts of radioactive material.

The discussion moves to the calculation of disintegrations per second from the current activity of the cobalt-60 source, which is found to be about 19,000 becquerels. The lecturer clarifies that cobalt-60 primarily undergoes beta decay but is used for its characteristic gamma rays. By examining the decay diagram, it is determined that each disintegration event produces, on average, two gamma rays. This information is crucial for calculating the efficiency of a Geiger counter used to measure the source.

Short engages the audience in setting up the differential equation model for radioactive decay and production. Using cobalt-59 as an example, he explains how to formulate the production and destruction rates for cobalt-60. The audience is encouraged to identify the correct cross-sections for neutron interactions from the Janis Database, emphasizing the need for the right cross-section to accurately model the nuclear reactions.

The video segment focuses on the concept of cross-sections in nuclear physics. Short explains that cross-sections, measured in barns (a unit of area), are a theoretical construct representing the probability of interaction between particles. He differentiates between total and elastic cross-sections and how they relate to neutron scattering and fission. The segment also touches on the calculation of cross-sections from quantum mechanics and their determination through experimentation.

The lecturer presents a differential equation for the production of cobalt-60 in a reactor, considering both natural decay and artificial decay induced by neutron flux. By applying an integrating factor, the equation is solved to find the amount of cobalt-60 present as a function of time. The solution is then adjusted based on the initial condition of no cobalt-60 at the start of the reactor operation.

Short introduces a homework problem where students are tasked with determining the optimal运行时间 for a reactor to maximize profit from cobalt-60 production. He provides parameters such as the cost of running the reactor, the price of cobalt-60, and the initial amount of cobalt-59. The segment concludes with a theoretical solution that will be used to make a value judgment on the most profitable operational time.

The video concludes with a numerical analysis of the derived equation for cobalt-60 production, using specific values for the constants involved. Short uses Desmos to illustrate the production curve over time and discusses the concept of integrating over the full energy range for a more realistic model. He emphasizes that while simulations are useful, mathematical modeling is essential for making value judgments in reactor operation.

In the final segment, Short asks for feedback on the usefulness of the example covered in the lecture. He acknowledges the voluntary overtime stay of the audience and reiterates the significance of the covered material, which allows students to start making value judgments on complex, real-world scenarios like reactor operation.