Understanding Radioactive Decay Through Data Analysis

Radioactive decay is a fundamental concept in nuclear physics, describing the process by which an unstable atomic nucleus loses energy by emitting radiation. This blog post delves into the analysis of Barium-137m decay, examining the relationship between time and radioactivity count using both linear and logarithmic models.

Introduction to Radioactive Decay

Radioactive decay follows a predictable pattern where the amount of radioactive substance decreases over time. This decay can be quantified and visualized using scatterplots and regression analysis. In this post, we will explore the decay of Barium-137m, a common isotope used in medical imaging and scientific research.

Data Collection and Initial Observations

We collected data on the decay of Barium-137m, recording the radioactivity count at various time intervals. Here is the initial dataset:

Scatterplot Analysis

We begin by plotting the raw counts against time to observe the decay trend. The scatterplot reveals a strong negative correlation between time and radioactivity count.

Figure 1: Barium-137m Decay

Linear and Logarithmic Models

To better understand the decay pattern, we applied both linear and logarithmic regression models to the data.

Linear Regression Model

The linear regression model for the raw data is given by: y = −74.7x + 602.8

This model shows a clear downward slope, indicating a decrease in radioactivity over time.

Figure 2: Linear Regression of Barium-137m Decay

Logarithmic Regression Model

When we log-transform the radioactivity counts, the relationship becomes more linear: ln⁡(y) =−0.2606x + 6.5931. This transformation helps in making the distribution linear, providing a clearer picture of the decay process.

Residual Analysis

To validate our models, we examine the residuals, which are the differences between observed and predicted values.

Figure 4: Residual Plot for Linear Model

Figure 5: Residual Plot for Logarithmic Model

The residuals for the log-transformed data are much smaller and more evenly distributed around zero, indicating a better fit compared to the linear model.

Least Squares Regression

We performed least squares regression for both models to quantify the fit:

Linear Model:

· Correlation: r = −0.999

· Sum of Squared Differences: 8440.2

Logarithmic Model:

· Correlation: r = −0.9642

· Sum of Squared Differences: 0.004054

The logarithmic model clearly provides a better fit for the data, as evidenced by the lower sum of squared differences and a higher correlation coefficient.

Conclusion

Our analysis of Barium-137m decay demonstrates the effectiveness of using logarithmic transformations to linearize data and improve model accuracy. The log-transformed regression model provides a more accurate representation of the decay process compared to the linear model.

By understanding these models, we can better predict the behaviour of radioactive substances, which is crucial in fields like nuclear medicine and environmental science.