Chapter 5: Problem 10
Compute and make a polar plot of the differential scattering cross section for a perfectly rigid sphere when \(k a=\frac{1}{2}\), using the first three partial waves \((l=0,1,2)\). What is the total cross section in this case, and what is the approximate accuracy of this result when the three terms are used?
Short Answer
Expert verified
The polar plot will use \(f(\theta) = -2 - \frac{15}{2}(\cos^2\theta - \frac{1}{3})\). The total cross section is computed by integrating the differential cross section. The calculation uses the first three partial waves providing a reasonable approximation given \(k a = \frac{1}{2}\).
Step by step solution
01
Understanding the Differential Scattering Cross Section
To compute the differential scattering cross section, we need to use the formula for the scattering amplitude for a perfectly rigid sphere. The scattering amplitude can be given by partial wave expansion: \[\frac{d\sigma}{d\Omega} = |f(\theta)|^2\]
02
Applying Partial Wave Expansion
For a rigid sphere, the partial wave expansion for the scattering amplitude is:\[f(\theta) = \sum_{l=0}^{\infty} (2l + 1) i^l [S_l - 1] P_l(\cos\theta)\]where \(S_l\) is the scattering matrix element for the \(l\)-th partial wave.
03
Using the Given Condition
Given that \(k a = \frac{1}{2}\), where \(k\) is the wave number and \(a\) is the radius of the sphere, we need to use the first three partial waves \((l=0,1,2)\).
04
Calculating the Scattering Matrix Elements
For a perfectly rigid sphere, the scattering matrix elements are approximate as \(S_l = (-1)^{l+1}\). Therefore:\[S_0 = -1, \quad S_1 = 1, \quad S_2 = -1\]
05
Constructing the Scattering Amplitude
Substitute the given \(S_l\) values back into the partial wave expansion:\[f(\theta) = \sum_{l=0}^{2} (2l + 1) i^l [S_l - 1] P_l(\cos\theta)= (2(0) + 1) i^0 [-1 - 1] P_0(\cos\theta) + (2(1) + 1) i^1 [1 - 1] P_1(\cos\theta) + (2(2) + 1) i^2 [-1 - 1] P_2(\cos\theta)\]This simplifies to:\[f(\theta) = -2 P_0(\cos\theta) - 10 P_2(\cos\theta)\]
06
Calculating the Differential Scattering Cross Section
Since \(P_0(\cos\theta) = 1\) and \(P_2(\cos\theta) = \frac{1}{2}(3\cos^2\theta - 1)\) we get:\[f(\theta) = -2 - \frac{15}{2}(\cos^2\theta - \frac{1}{3})= -2 - \frac{15}{2}(\frac{3}{4}\cos^2\theta - \frac{1}{4})\]Then the differential cross section is:\[\frac{d\sigma}{d\Omega} = |f(\theta)|^2 = (-2 - \frac{15}{2}(\cos^2\theta - \frac{1}{3}))^2\]
07
Computing the Total Cross Section
The total cross section is calculated by integrating the differential cross section over all solid angles:\[\sigma_{total} = \int_{0}^{2\pi} d\phi \int_{0}^{\pi} \frac{d\sigma}{d\Omega} \sin\theta d\theta\].
08
Estimating Accuracy
To estimate the accuracy of the result using the first three terms, compare the total cross section with known values for higher partial sums or exact solutions. Approximation accuracy typically depends on the small parameter \(k a\), ensuring \(k a \ll 1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
partial wave expansion
Partial wave expansion is a technique used in quantum mechanics to analyze scattering problems. It breaks down a wave function into a series of simpler radial components. Each of these components corresponds to a different angular momentum quantum number, denoted as \(l\).
The scattering amplitude for a rigid sphere can be expressed using partial wave expansion:
\[ \displaystyle f(\theta) = \sum_{l=0}^{\infty} (2l+1) i^l \left[S_l - 1\right] P_l(\cos \theta) \]
The sum runs over all possible \(l\) values, with \(S_l\) representing the scattering matrix element for the \(l\)-th partial wave, and \(P_l(\cos \theta)\) representing the Legendre polynomials. In practical scenarios, we often truncate the sum to a finite number of terms to make the computations manageable.
The scattering amplitude for a rigid sphere can be expressed using partial wave expansion:
\[ \displaystyle f(\theta) = \sum_{l=0}^{\infty} (2l+1) i^l \left[S_l - 1\right] P_l(\cos \theta) \]
The sum runs over all possible \(l\) values, with \(S_l\) representing the scattering matrix element for the \(l\)-th partial wave, and \(P_l(\cos \theta)\) representing the Legendre polynomials. In practical scenarios, we often truncate the sum to a finite number of terms to make the computations manageable.
scattering amplitude
The scattering amplitude \(f(\theta)\) is a measure of how an incoming wave is modified due to scattering by an obstacle. It plays a crucial role in determining the differential scattering cross section, which further helps in predicting the angular distribution of scattered particles.
For a rigid sphere and using partial wave expansion, the scattering amplitude can be approximated by using the first few partial waves. In this example, we use the first three partial waves: \(l=0, 1, 2\). The simplified scattering amplitude is:
\[ f(\theta) = -2 - 10 P_2(\cos \theta) \]
This is derived after considering the contributing elements of the partial waves.
For a rigid sphere and using partial wave expansion, the scattering amplitude can be approximated by using the first few partial waves. In this example, we use the first three partial waves: \(l=0, 1, 2\). The simplified scattering amplitude is:
\[ f(\theta) = -2 - 10 P_2(\cos \theta) \]
This is derived after considering the contributing elements of the partial waves.
scattering matrix elements
The scattering matrix elements \(S_l\) are quantities that describe how each partial wave interacts with the scattering potential for a given angular momentum \(l\). They form the core of the partial wave expansion. For a rigid sphere, these elements have specific values:
\[ S_l = (-1)^{l+1} \]
For example, for \(l = 0, 1, \text{and} 2\), the elements are:
\[ S_0 = -1, \quad S_1 = 1, \quad S_2 = -1\]
These values are crucial for constructing the scattering amplitude and therefore calculating the differential scattering cross section.
\[ S_l = (-1)^{l+1} \]
For example, for \(l = 0, 1, \text{and} 2\), the elements are:
\[ S_0 = -1, \quad S_1 = 1, \quad S_2 = -1\]
These values are crucial for constructing the scattering amplitude and therefore calculating the differential scattering cross section.
Legendre polynomials
Legendre polynomials \( P_l(\cos \theta) \) are a set of orthogonal polynomials that arise in the solutions to Legendre's differential equation. They are widely used in physics, especially in spherical coordinate systems.
In the context of scattering, they are used to express the angular part of the scattered wave. For the first three partial waves, the Legendre polynomials are:
\[ \begin{align*} P_0(\cos \theta) &= 1, \ P_1(\cos \theta) &= \cos \theta, \ P_2(\cos \theta) &= \frac{1}{2}(3\cos^2 \theta - 1)\end{align*} \]
These polynomials appear as coefficients in the partial wave expansion of the scattering amplitude.
In the context of scattering, they are used to express the angular part of the scattered wave. For the first three partial waves, the Legendre polynomials are:
\[ \begin{align*} P_0(\cos \theta) &= 1, \ P_1(\cos \theta) &= \cos \theta, \ P_2(\cos \theta) &= \frac{1}{2}(3\cos^2 \theta - 1)\end{align*} \]
These polynomials appear as coefficients in the partial wave expansion of the scattering amplitude.
wave number
The wave number \(k\) is related to the wavelength \(\lambda\) of a wave and is defined as \(k = \frac{2\pi}{\lambda}\). It represents the number of wavelengths per unit distance.
In scattering problems, the wave number is an important parameter that describes the momentum of the incident wave on the scatterer. For a rigid sphere with radius \(a\), the wave number is given as \(k a = \frac{1}{2}\). This parameter helps in determining the scattering matrix elements and evaluating the partial wave expansion terms.
In scattering problems, the wave number is an important parameter that describes the momentum of the incident wave on the scatterer. For a rigid sphere with radius \(a\), the wave number is given as \(k a = \frac{1}{2}\). This parameter helps in determining the scattering matrix elements and evaluating the partial wave expansion terms.
total cross section
The total cross section \(\sigma_{\text{total}}\) gives a measure of the likelihood of scattering events by integrating the differential cross section over all angles. It is computed as:
\[ \sigma_{\text{total}} = \int_{0}^{2\pi} d\phi \int_{0}^{\pi} \frac{d\sigma}{d\Omega} \sin \theta d\theta \]
In simpler terms, it sums the contributions of all scattered waves radiating in all directions.
The expression \(\frac{d\sigma}{d\Omega}\) represents the differential cross section, which in turn is derived from the scattering amplitude:
\[ \frac{d\sigma}{d\Omega} = |f(\theta)|^2 \]
After completing these integrals for the rigid sphere case at \(k a = \frac{1}{2}\), one can determine the total cross section and discuss its accuracy by considering the truncated partial wave sum used in the calculations.
\[ \sigma_{\text{total}} = \int_{0}^{2\pi} d\phi \int_{0}^{\pi} \frac{d\sigma}{d\Omega} \sin \theta d\theta \]
In simpler terms, it sums the contributions of all scattered waves radiating in all directions.
The expression \(\frac{d\sigma}{d\Omega}\) represents the differential cross section, which in turn is derived from the scattering amplitude:
\[ \frac{d\sigma}{d\Omega} = |f(\theta)|^2 \]
After completing these integrals for the rigid sphere case at \(k a = \frac{1}{2}\), one can determine the total cross section and discuss its accuracy by considering the truncated partial wave sum used in the calculations.