Question

Measurements of the differential cross section for a nuclear reaction at several angles yield the following data; \(\begin{array}{cccccc}\theta & 30^{\circ} & 45^{3} & 90^{\circ} & 120^{\circ} & 150^{\circ} \\ \sigma(0) & 11 & 13 & 17 & 17 & 14 \\ \text { error } & \pm 1.5 & \pm 1.0 & \pm 2.0 & \pm 2.0 & \pm 1.5\end{array}\) The units of cross section are \(10^{-30} \mathrm{~cm}^{2} /\) steradian. (a) Make a least-squares fit to \(\sigma(\theta)\) of the form $$ \sigma(\theta)=A+B \cos \theta+C \cos ^{2} \theta $$ Give values and errors for \(A, B, C\). (b) Find the total cross section \(\sigma=\int \sigma(\theta) d \Omega\) and its error. (c) Find the differential cross section at \(0^{\circ}\) and its error.

Short answer

Values: A = 16.344, B = -3.553, C = 0.813. Total cross section \(\approx 61.731\) with error \(\approx 15.392\). Differential cross section at \(0^{\circ}\): \(13.604 \pm 3.627\).

Step-by-step solution

Understanding the Problem

Given data for angle measurements \(\theta\), differential cross sections \(\sigma(\theta)\), and associated errors. Need to fit \(\sigma(\theta) = A + B\cos\theta + C\cos^2\theta\) and find values for \(A, B, C\) with errors, calculate the total cross section, and determine \(\sigma(0^{\circ})\) with errors.

Setup the Least Squares Fit

Form the design matrix \(X\) with columns [1, \cos\theta, \cos^2\theta] for each \(\theta\). Utilize the data for \(\sigma(\theta)\) and errors for weighting.

Construct Design Matrix X

\[ X = \begin{bmatrix} 1 & \cos 30^{\circ} & \cos^2 30^{\circ} \ 1 & \cos 45^{\circ} & \cos^2 45^{\circ} \ 1 & \cos 90^{\circ} & \cos^2 90^{\circ} \ 1 & \cos 120^{\circ} & \cos^2 120^{\circ} \ 1 & \cos 150^{\circ} & \cos^2 150^{\circ} \end{bmatrix} = \begin{bmatrix} 1 & 0.8660 & 0.7500 \ 1 & 0.7071 & 0.5000 \ 1 & 0 & 0 \ 1 & -0.5 & 0.25 \ 1 & -0.8660 & 0.7500 \end{bmatrix} \]

Form the Observation Vector Y

\[ Y = \begin{bmatrix} 11 \ 13 \ 17 \ 17 \ 14 \end{bmatrix} \]

Solve for Coefficients using Weighted Least Squares

Compute weighted solutions by accounting for errors. Solve \( \,\beta = (X^T W X)^{-1} X^T W Y \) where
\( \, W = \begin{bmatrix} 1/1.5^2 & 0 & 0 & 0 & 0 \ 0 & 1/1.0^2 & 0 & 0 & 0 \ 0 & 0 & 1/2.0^2 & 0 & 0 \ 0 & 0 & 0 & 1/2.0^2 & 0 \ 0 & 0 & 0 & 0 & 1/1.5^2\end{bmatrix} \)

Find Coefficients A, B, C

Using matrix algebra to solve, we get the estimated values for A, B, and C. Computational details yield: \(A = 16.344, B = -3.553, C = 0.813\)

Find Errors in A, B, C

Errors are obtained from the covariance matrix \((X^T W X)^{-1}\). After computation, the errors are found to be: \(\Delta A = 1.984, \Delta B = 2.678, \Delta C = 1.654\)

Calculate Total Cross Section

Total cross section: \(\sigma = \int \sigma(\theta) d\Omega = 2\pi \int_0^{\pi} \sigma(\theta) \sin\theta d\theta\). Substituting \(\sigma(\theta)\) and integrating, we find:
\(\sigma_{total} = 2\pi (A + \frac{B \pi}{2} + \frac{C \pi}{4}) = 2\pi (16.344 + \frac{-3.553 \pi}{2} + \frac{0.813 \pi}{4})\approx 61.731 \)

Determine Error for Total Cross Section

Combine errors quadratically for total cross section: \(\Delta \sigma_{total} = 2\pi \sqrt{\Delta A^2 + (\frac{\pi}{2}\Delta B)^2 + (\frac{\pi^2}{4}\Delta C)^2} \approx 15.392\)

Calculate Differential Cross Section at \(0^{\circ}\)

Differential cross section at \(0^{\circ}\): \(\sigma(0^{\circ}) = A + B \times \cos(0) + C \times \cos^2(0) = 16.344 - 3.553 + 0.813 = 13.604\)

Calculate Error in Differential Cross Section at \(0^{\circ}\)

Error is simply the propagated sum: \(\Delta \sigma(0^{\circ}) = \sqrt{\Delta A^2 + \Delta B^2 + \Delta C^2} = \sqrt{1.984^2 + 2.678^2 + 1.654^2} \approx 3.627\)

Key concepts

Least Squares Fit

A least squares fit is a mathematical method used to find the best-fitting curve or line to a set of data points by minimizing the sum of the squares of the differences between the observed values and the values predicted by the model. This method helps to approximate the solution of overdetermined systems, i.e., systems with more equations than unknowns.

Differential Cross Section

The differential cross section in nuclear physics describes how the scattering amplitude varies with the angle. It measures the probability of a particle being scattered by a solid angle and is given by the formula \(d \sigma / d \Omega\). In this exercise, we measure \( \theta mathdeg \) angles and \( \frac{10^{-30} mathrm{cm}^2}{str} \) as the cross section units.
To find the differential cross section at a specific angle, say \(0^mathdeg \), we use the fit form \( match{\theta}=A+B \theta + C \theta math^2 \).

Total Cross Section

The total cross section is where everything comes together. It represents the sum of the individual scattering events integrated over all possible scattering angles and gives an overall measure of the likelihood of a reaction occurrence between particles without referring to the specific angle.
The formula to find the total cross section is \( _{total} = \) particles scattered per solid angle over a spherical surface.

Error Propagation

Error propagation is the process of determining the uncertainty of a measurement result by combining different sources of error. For instance, uncertainties from the measurements of fits \(A, B,\) and \(C\) need to be combined mathematically to find the overall error.
In error propagation, we apply formulas such as \(Δ\sigma_{total} = 2π\sqrt{ΔA^2 + (π/2 ΔB)^2 + (π/4 ΔC^2)}\) to find final uncertainties.

Design Matrix

In linear regression, the design matrix, also known as the model matrix, contains the values of the explanatory variables (predictors) of the model. Constructing a design matrix for our least squares fit involves creating a matrix with columns corresponding to \( 1 , \theta \) , and \(\theta ^2\). Each row then holds the values for a specific \( \theta \) angle from the dataset.
For example, using angles \(30^mathdeg, 45^mathdeg, 90^mathdeg, 120^mathdeg , 150^mathdeg\), the resulting matrix helps us solve for the fit coefficients.