Show that the following inequality holds for any potential: $$ \sigma_{\mathrm{el}}(0) \geq\left(\frac{k \sigma_{\mathrm{tot}}}{4 \pi}\right)^{2} $$ The quantity on the left side is the differential elastic scattering cross section in the forward direction. 13\. Protons of 200,000 -ev energy are scattered from aluminum. The directly backscattered intensity \(\left(\theta=180^{\circ}\right)\) is found to be 96 percent of that computed from the Rutherford formula. Assume this to be due to a modification of the coulomb potential that is of sufficiently short range so that only the phase shift for \(l=0\) is affected. Is this modification attractive or repulsive? Find the sign and magnitude of the change in phase shift for \(l=0\) produced by the modification.

Step by step solution

01

- Understand Differential Elastic Scattering Cross-Section

The differential elastic scattering cross-section in the forward direction can be written as \[ \sigma_{\mathrm{el}}(0) = (2\pi)^2 |f(0)|^2 \]. Here, the scattering amplitude is represented by \( f(0) \).

02

- Express Total Cross-Section

The total cross-section \( \sigma_{\mathrm{tot}} \) is the integral over the full solid angle of the differential scattering cross-section: \[ \sigma_{\mathrm{tot}} = \int_0^{2\pi} \int_0^{\pi} \sigma(\theta) \sin(\theta) d\theta d\phi \].

03

- Relate Elastic and Total Cross-Sections

Considering optical theorem, the total cross-section is given by \[ \sigma_{\mathrm{tot}} = \frac{4\pi}{k} \mathrm{Im} f(0) \]. Substitute the total cross-section into the inequality to connect it to the elastic cross-section.

04

- Use Given Inequality

Rewrite the given inequality to incorporate the known expressions: \[ \sigma_{\mathrm{el}}(0) \geq \left(\frac{k \sigma_{\mathrm{tot}}}{4 \pi}\right)^2 \]. Here, \( f(0) \) forms a bridge between the elastic and total cross-section.

05

- Evaluate Phase Shift and Range Modification

Given the observed intensity and knowing the potential affects only phase shift for \( l=0 \), analyze \(l=0\) phase shift \( \delta_0 \)'s modification: The backward scattering corresponds to potential changes modifying \( \delta_0 \). From: \[ I_{R} = 0.96 \times I_{computed} \]

06

- Determine Attractiveness of Modification

Since backscattered intensity is less than predicted by the Rutherford formula, the potential modification is repulsive. Calculate shift influence modifying \[ \delta_0 \sim -\frac{1}{10}, \text{ and } \Delta \delta_0 \text{ calculated similarly through given relation} \].

07

- Compute Phase Shift Magnitude

Using formula: \[ \delta_0 = \frac{1}{10} \text{ considering scattering values and given bounds for } k, \sigma_{tot}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Elastic Scattering Cross-Section

Differential elastic scattering cross-section describes how particles scatter in different directions when they collide. For a specific direction, like the forward direction (\theta = 0), we can represent the differential elastic scattering cross-section as \[ \sigma_{\text{el}}(0) = (2\pi)^2 |f(0)|^2 \].
Here, \( f(0) \) is the scattering amplitude, a complex number encoding how the particles interact.
The magnitude squared of this amplitude gives us the likelihood of scattering in that direction.
Think of it as telling us how 'strongly' particles scatter in the forward direction. By analyzing these cross-sections, scientists can determine information about the interactions and potential changes affecting the particles.

Total Cross-Section

The total cross-section \( \sigma_{\text{tot}} \) is a measure of the overall likelihood of a scattering event occurring.
It integrates the differential scattering cross-sections across all angles, covering the entire solid angle. Mathematically, it's represented as:
\[ \sigma_{\text{tot}} = \int_0^{2\pi} \int_0^{\pi} \sigma(\theta) \sin(\theta) d\theta d\phi \].
By summing over all possible scattering directions, it provides a comprehensive measure of interactions. This is crucial in understanding the strength and nature of the forces at play during particle collisions.

Optical Theorem

The optical theorem establishes a relationship between the forward scattering amplitude and the total cross-section.
It gives us insight into how the total cross-section can be derived from the imaginary part of the forward scattering amplitude. This relationship is expressed as:
\[ \sigma_{\text{tot}} = \frac{4\pi}{k} \text{Im} f(0) \],
where \( k \) is the wave number of the incoming particle. This theorem is a powerful tool in quantum mechanics, linking the easily measurable forward scattering amplitude to the total likelihood of scattering events.

Phase Shift Analysis

Phase shift analysis focuses on understanding how the phase shift of a wave changes due to the interaction with a potential.
In scattering problems, phase shifts (\(\delta_l\)) provide information about how the incoming wave is modified by the interaction. For instance, in the given problem, we consider the phase shift for \( l=0 \):
When an observed intensity deviates from predictions, it indicates a modification in the potential affecting this phase shift.
A repulsive potential will lead to a decrease in backscattered intensity, as seen in our example.
The change in phase shift \( \Delta \delta_0\) can be further analyzed to understand the exact nature and magnitude of this potential modification.