Equation 20.3 is the mean free path of a particle through a gas of identical particles of equal radius. An electron can be thought of as a point particle with zero radius. b. Electrons travel 3.0 km through the Stanford Linear Accelerator. In order for scattering losses to be negligible, the pressure inside the accelerator tube must be reduced to the point where the mean free path is at least 50 km. What is the maximum possible pressure inside the accelerator tube, assuming T = 20℃? Give your answer in both Pa and atm.

Verified step by step guidance

1. Identify the formula for the mean free path (λ) of a particle in a gas, which is given by λ = \frac{1}{n \sigma}, where n is the number density of the gas molecules and \sigma is the collision cross-section.
2. Since the electron can be considered a point particle with zero radius, the effective diameter of the gas molecules (d) is the main contributor to the collision cross-section. The cross-section \sigma can be approximated by \sigma = \pi d^2.
3. Calculate the number density n of the gas using the ideal gas law, PV = nRT, where P is the pressure, V is the volume, R is the gas constant, and T is the temperature in Kelvin. Convert the given temperature from Celsius to Kelvin.
4. Substitute the expression for n from the ideal gas law and the expression for \sigma into the mean free path formula to find an expression for λ in terms of P, T, and d.
5. Solve the expression for P, ensuring that λ is at least 50 km, to find the maximum allowable pressure inside the accelerator tube. Convert this pressure from Pascals (Pa) to atmospheres (atm).

Key Concepts

Mean Free Path

The mean free path is the average distance a particle travels between collisions with other particles. It is influenced by the density and size of the particles in a gas. A longer mean free path indicates fewer collisions, which is crucial for applications like particle accelerators where minimizing scattering is essential for maintaining beam quality.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is fundamental in determining how changes in pressure and temperature affect the behavior of gases. In the context of the accelerator, it helps calculate the maximum pressure needed to achieve the desired mean free path.

Scattering Losses

Scattering losses occur when particles, such as electrons, collide with gas molecules, leading to a loss of energy and direction. In particle accelerators, minimizing these losses is critical for maintaining the integrity of the particle beam. Achieving a sufficient mean free path reduces the likelihood of such collisions, allowing for more efficient acceleration.