Question

There is a fairly good one-to-one correspondence between the energy of a fast proton and its range in a given material; that is, the distance traveled in that material before the proton is slowed down and stops. There is, of course, some straggling, or variation in path lengths, because the slowing-down process is a statistical one. Suppose a beam of moncenergetic protons is stopped in a bubble chamber and \(N\) individual tracks are found 10 have lengths \(x_{1}, x_{2}\), \(\ldots x_{N}\). Assuming the probability distribution for track length is Gaussian, use the dita to lind maximum likelihood values of the range \(R\) (mean length) and the straggling parameter (standard deviation) \(\delta\). Find the errors in both \(R\) and \(\delta\).

Short answer

R is the sample mean: \( R = \frac{1}{N} \sum_{i=1}^N x_i \). \( \delta \) is the sample standard deviation: \( \delta = \sqrt{\frac{1}{N} |\continuous_summationi=1^N (x_i - R)^2}\). Errors are: \( \sigma_R^{2} = \frac{\delta^2}{N} \) and \( \sigma_\delta^{2} = \frac{2\delta^2}{N} \).

Step-by-step solution

- Write Down the Gaussian Probability Distribution

The Gaussian or normal distribution for track length is given by: \[ P(x; R, \, \delta) = \frac{1}{\sqrt{2\pi \delta^2}} \exp\left(-\frac{(x - R)^2}{2 \delta^2}\right) \]

- Write the Likelihood Function

The likelihood function is the product of the probabilities of each observation: \[ L(R, \delta) = \prod_{i=1}^N P(x_i; R, \, \delta) \] Substituting the Gaussian distribution, this becomes: \[ L(R, \delta) = \left( \frac{1}{\sqrt{2\pi \delta^2}} \right)^N \exp\left(-\sum_{i=1}^N \frac{(x_i - R)^2}{2 \delta^2} \right) \]

- Take the Logarithm of the Likelihood Function

To simplify differentiation, take the natural logarithm of the likelihood function: \[ \ln L = -N \ln \delta - \frac{N}{2} \ln(2\pi) - \frac{1}{2 \delta^2} \sum_{i=1}^N (x_i - R)^2 \]

- Differentiate with Respect to (w.r.t) R

Find the partial derivative of \( \ln L \) w.r.t. \( R \): \[ \frac{\partial \ln L}{\partial R} = \frac{1}{\delta^2} \sum_{i=1}^N (x_i - R) = 0 \] Solving, we get the mean range (\( R \)): \[ R = \frac{1}{N} \sum_{i=1}^N x_i \]

- Differentiate with Respect to (w.r.t) \delta

Find the partial derivative of \( \ln L \) w.r.t. \( \delta \): \[ \frac{\partial \ln L}{\partial \delta} = -\frac{N}{\delta} + \frac{1}{\delta^3} \sum_{i=1}^N (x_i - R)^2 = 0 \] Solving for \( \delta \), the standard deviation (\( \delta \)) is: \[ \delta = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - R)^2} \]

- Calculate Uncertainties in R and \delta

The uncertainty (variance) in the mean range \( R \) can be found from the Cramer-Rao lower bound: \[ \sigma_R^2 = \frac{\delta^2}{N} \] For the standard deviation \( \delta \), the uncertainty (variance) is: \[ \sigma_\delta^2 = \frac{2\delta^2}{N} \]

Key concepts

Gaussian Distribution

The Gaussian distribution, also known as the normal distribution, is a foundational concept in statistics. It describes how data tends to be distributed around a mean value. The probability density function of a Gaussian distribution is defined as: \[ P(x; R, \, \delta) = \frac{1}{\text{square root}(2\text{pi} \delta^2)} \exp\bigg(-\frac{(x - R)^2}{2 \delta^2}\bigg) \]Here, \(R\) represents the mean (or the expected value) and \(\delta\) is the standard deviation, which measures the spread of the distribution. Values closer to the mean are more likely, with decreasing probabilities for values farther from the mean. In the context of the exercise, we analyze the lengths of proton tracks using the Gaussian distribution because it naturally models variations around an average value.

Proton Range

The proton range refers to the distance a proton travels before it stops in a given material. The exercise states that there is a one-to-one correspondence between the proton's energy and its range. Different protons can have slightly different ranges due to statistical variations in their energy loss processes. In the bubble chamber experiment, the task is to determine the most likely mean range of these protons (\(R\)) and the standard deviation (\(\delta\)) using observed track lengths. This is done by analyzing a sample of tracks and fitting a Gaussian distribution to the observed data.

Statistical Analysis

Statistical analysis helps us make sense of varying data by deriving meaningful conclusions. In this exercise, we perform statistical analysis using maximum likelihood estimation (MLE). MLE is a method used to infer parameters of a statistical model that makes the observed data most probable. Here, we derive the likelihood function for track lengths and optimize it to find the most probable values of \(R\) and \(\delta\). The approach involves:

• Defining a likelihood function based on Gaussian probability
• Taking the natural logarithm of the likelihood for easier computation
• Finding derivatives with respect to parameters

This way, we efficiently estimate the proton range and the straggling parameter from the track length data.

Standard Deviation

The standard deviation (\(\delta\)) measures the amount of variation or dispersion in a set of values. For the proton track lengths, it signifies how much individual lengths deviate from the mean range (\(R\)). The formula to estimate \(\delta\) using the maximum likelihood method is: \[ \delta = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - R)^2} \]This calculation involves finding the average of squared differences between each observed length and the mean. A larger standard deviation indicates more spread in the lengths, while a smaller standard deviation implies that the lengths are more tightly clustered around the mean.

Error Calculation

Error calculation is crucial for understanding the precision of our estimates. For the mean range (\(R\)) and the standard deviation (\(\delta\)), we calculate uncertainties based on the Cramer-Rao lower bound. The variances are given by:

• For \(R\), \( \sigma_R^2 = \frac{\delta^2}{N} \)

• For \(\delta\), \( \sigma_\delta^2 = \frac{2 \delta^2}{N} \)

These formulas help us quantify the expected error in our estimates. A smaller error implies a more reliable estimate, while a larger error indicates more uncertainty. Proper error calculation allows us to gauge the confidence in our statistical analysis results.