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Consider Compton scattering of a photon by a moving electron. Before the collision the photon has wavelength \(\lambda\) and is moving in the \(+x\) -direction, and the electron is moving in the \(-x\) -direction with total energy \(E\) (including its rest energy \(m c^{2}\) ). The photon and electron collide head-on. After the collision, both are moving in the \(-x\) -direction (that is, the photon has been scattered by \(180^{\circ}\) ). (a) Derive an expression for the wavelength \(\lambda^{\prime}\) of the scattered photon. Show that if \(E \gg m c^{2}\), where \(m\) is the rest mass of the electron, your result reduces to $$ \lambda^{\prime}=\frac{h c}{E}\left(1+\frac{m^{2} c^{4} \lambda}{4 h c E}\right) $$ (b) A beam of infrared radiation from a \(\mathrm{CO}_{2}\) laser \((\lambda=10.6 \mu \mathrm{m})\) collides head-on with a beam of electrons, each of total energy \(E=10.0 \mathrm{GeV}\left(1 \mathrm{GeV}=10^{9} \mathrm{eV}\right) .\) Calculate the wavelength \(\lambda^{\prime}\) of the scattered photons, assuming a \(180^{\circ}\) scattering angle. (c) What kind of scattered photons are these (infrared, microwave, ultraviolet, etc.)? Can you think of an application of this effect?

Step by step solution

01

Derive the expression for the scattered photon's wavelength

For Compton scattering, we can apply the principle of conservation of energy and momentum. The total initial energy is the sum of the photon's and electron's energies, given by \(E + h c / \lambda\), and the total final energy is given by \(m c^{2} + h c / \lambda^{\prime}\), wherein \(\lambda^{\prime}\) is the wavelength of the scattered photon. Solving the energy conservation equation \(E + h c / \lambda = m c^{2} + h c / \lambda^{\prime}\) gives us the formula for \(\lambda^{\prime}\).

02

Simplify the formula for high energy electron

Assuming \(E \gg m c^{2}\), we can simplify the formula obtained in step 1 to \(\lambda^{\prime} = h c / E (1 + m^{2} c^{4} \lambda / 4 h c E)\).

03

Apply the formula for a given scenario

Given the initial wavelength of a photon (\(\lambda = 10.6 µm\)) and the total energy of an electron (\(E = 10 GeV\)), we can substitute these values into the formula obtained in step 2.

04

Determine the scattered photon's type

By comparing the scattered photon's wavelength with the range of wavelengths for different types of photon, we can ascertain the type of scattered photon. For instance, if the wavelength is in the range for infrared light, then the scattered photon is an infrared photon.

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Key Concepts

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

Conservation of Energy and Momentum

In Compton scattering, an essential principle to understanding the behavior of the photon and electron during the collision is the conservation of energy and momentum. This principle states that within a closed system, the total energy and momentum before an event must equal the total energy and momentum afterwards.

When a photon with an initial wavelength \( \lambda \) collides head-on with a moving electron, two critical things occur. Firstly, the energy of the photon partially transfers to the electron, increasing its kinetic energy. Secondly, as momentum is a vector quantity, meaning it has both magnitude and direction, the momentum of the photon and electron are affected by their directions of motion.

The formula derived from these principles allows us to predict the wavelength \( \lambda' \) of the photon after the collision. Hence, the concept is fundamental not just to solving theoretical problems but to applications like particle accelerators and understanding quantum physics phenomena. By drawing from these solid physical principles, we simplify seemingly complex interactions into solvable mathematical problems.

Scattered Photon Wavelength

The wavelength of the scattered photon in Compton scattering is a reflection of the energy exchange between the photon and the electron. The photon's initial energy, which is inversely proportional to its wavelength, is partially transferred to the electron. As a result, the photon loses energy, which manifests as an increase in its wavelength, known as 'Compton shift'.

The mathematical expression derived from conservation laws allows us to quantify this 'Compton shift'. In Compton scattering problems, we often express the scattered photon's wavelength \( \lambda' \) in terms of the initial wavelength of the photon and the energy of the electron. Specifically, for high-energy electrons where \( E \gg mc^{2} \), the simplified expression becomes particularly useful to calculate the new wavelength quickly. It illuminates the direct relationship between electron energy and the resultant photon wavelength, which has applications in spectroscopy and medical imaging technologies.

High Energy Electron Physics

High energy electron physics deals with electrons that have kinetic energies much greater than their rest mass energy \( mc^{2} \). In situations where \( E \gg mc^{2} \)—the case for many particle physics experiments—the dynamics of electrons can be quite different compared to low energy scenarios.

For instance, in the Compton scattering problem at hand, assuming that the electron is highly energetic simplifies the calculations significantly. The assumption allows us to neglect terms in the equations that are small compared to the electron's kinetic energy. This simplification is crucial in practical applications, such as in high-resolution electron microscopes, accelerators, and detecting cosmic rays.

This area of physics is not just theoretical—knowledge of high-energy electron behavior is used to probe the fundamental aspects of matter and has paved the way for advances in various scientific fields, including the development of radiation therapies in medicine and exploration of quantum mechanics.