Problem 15 - Classical Mechanics - An Undergraduate Text

Question

A uniform flux of particles, each of mass \(m\) and speed \(V\), is incident upon a fixed scatterer that exerts the repulsive radial force \(\boldsymbol{F}=\left(m \gamma^{2} / r^{3}\right) \widehat{\boldsymbol{r}}\). Find the impact parameter \(p\) as a function of the scattering angle \(\theta\), and deduce the differential scattering cross section. Find the total back- scattering cross-section.

Step-by-Step Solution

Answer

The impact parameter \(p\) in terms of the scattering angle \(\theta\) is given by \(p=\gamma / \sqrt{\theta}\). The differential scattering cross section is \(d \sigma / d \Omega=m^{2} / 16 \pi V^{2} \gamma^{4} \theta^{4}\), and the total back scattering cross-section is \(\sigma_{\text{total}}=m^{2} / 16 V^{2} \gamma^{4}\).

Step 1: Conservation of Angular Momentum

For a problem of particle scattering, like this one, we can use the conservation of angular momentum to relate the impact parameter to the classical turning point. The angular momentum about the scattering center is \(L=m V p\). The angular momentum must also equal \(m r^{2} d \phi / d t=m r V_{\perp}\) where \(V_{\perp}\) is the component of velocity perpendicular to the radius and \(d \phi / d t\) is the angular velocity. We equate these to find an expression for the impact parameter, \(p=r V_{\perp} / V\).

Step 2: Centripetal Force

Next, we use the centripetal force definition to associate the radial force with the dynamics of radial motion. At the classical turning point the radial velocity component is zero and the centripetal force equals the force exerted by the scatterer, i.e. \(m V_{\perp}^{2} / r=F\). We substitute for the force given in the problem: \(m \gamma^{2} / r^{3}\) and also substitute the variable \(V_{\perp}=r d \phi / d t\). After some re-arrangements, it yields \(r^{5}=\gamma^{2} / \left(p^{2} d \phi / d t^{2}\right)\).

Step 3: Scattering Angle

We can find an expression for the differential of the scattering angle, \(d \theta\), as \(-d \phi\), given the geometry of the problem. Integrating this expression over time from \(t=-\infty\) (when \(\phi=0\)) to \(t=+\infty\) (when \(\phi=\theta\)) yields a relation between the impact parameter \(p\) and the scattering angle \(\theta\). We substitute for \(r\) from the previous step, and integrate to find \(p=\gamma / \sqrt{\theta}\).

Step 4: Differential Scattering Cross Section

The differential scattering cross section is defined as the ratio of the number of particles scattered into an angle \(\theta\) and the incident flux. Mathematically \(\frac{d \sigma}{d \Omega}=\frac{p}{\sin \theta}abs\left(\frac{d p}{d \theta}\right)\). Substituting the expression for \(p\) from the previous step and simplifying, we obtain the scattered flux \(d \sigma / d \Omega=m^{2} / 16 \pi V^{2} \gamma^{4} \theta^{4}\).

Step 5: Total Back-Scattering Cross Section

The total back-scattering cross-section is obtained by integrating this result over all solid angles such that \(\theta>\pi / 2\). It gives the total back-scattering cross-section as \(\sigma_{\text{total}}=m^{2} / 16 V^{2} \gamma^{4}\).

Key Concepts

Angular Momentum Conservation

In physics, conservation laws play a crucial role in analyzing motion. One key conservation law is that of angular momentum, especially in scattering problems. Angular momentum is a measure of the momentum of a particle moving around a point. For a particle of mass \(m\), moving with velocity \(V\) at a distance \(r\) from a scatterer, its angular momentum \(L\) is given by \(L = mVp\), where \(p\) is the impact parameter. The impact parameter represents the shortest distance to the scatterer's path.
To solve problems like particle scattering, we equate the initial angular momentum \(mVp\) with the dynamic angular momentum \(mr^2 (d\phi/dt)\). This relationship helps us rearrange and solve for the impact parameter as it relates to other dynamic conditions.

• Impact Parameter: The perpendicular distance from the scatterer's path.
• Angular Momentum Formula: \(L = mr^2 \frac{d\phi}{dt} = mVp\).

Radial Forces in Physics

Radial forces act along the line joining the center of motion and the object. In this scenario, the radial force \(\mathbf{F} = \frac{m\gamma^2}{r^3} \hat{\mathbf{r}}\) implies a repulsive force that weakens with distance.
Radial forces are pivotal because they determine the trajectory and ultimate motion of the particle around a scatterer. These forces set the boundary conditions for the centripetal forces that originate due to circular-like motions.

• Nature of Radial Force: Inversely proportional to \(r^3\).
• Repulsive Behavior: The force decreases sharply with distance, pushing away the particle.

Centripetal Force Dynamics

Centripetal force is what keeps an object moving along a curved path. It is directed radially inward toward the center of the curved path. In scattering problems involving radial forces, the centripetal force equates to the given radial force when radial velocity is zero.
In this scenario, we equate \(mV_{\perp}^2/r\) with the radial force provided to find conditions at the classical turning point. This point occurs where radial velocity (speed towards or away from the scatterer) is zero, causing pure circular motion.

• Turning Point: Where radial motion ceases and only angular motion persists.
• Centripetal Balance: Set equal to the given radial force to find dynamics.

Back-Scattering Cross Section

Back-scattering is a phenomenon where particles bounce back almost towards their origin. The cross-section measures how effectively a target scatters incoming particles backward. Mathematically, it involves integrating the differential scattering cross section across angles greater than \(\pi/2\).
This integration provides insight into the reflective qualities of a scatterer and captures probabilistic scattering events. The back-scattering cross-section reflects the scattering efficiency for observed phenomena.

• Angle of Reflection: Typically bounces back beyond \(\pi/2\) radians.
• Cross-Section Calculation: Derives from angular integration over suitable limits.

Particle Scattering Theory

Particle scattering theory explores how structures interact with particles. It investigates how particles like photons, atoms, or molecules deviate paths after interacting with targets, typically encrypting information about both the particle and scatterer involved.
The theory encompasses calculating cross sections, employing conservation laws, and understanding the forces at play. It unravels interactions that dictate experimental and theoretical analyses of scattered particles.

• Scope of Understanding: Covers interactions of particles with various forces.
• Applications: Quantum mechanics, nuclear physics, and material science.