Problem 40 - University Physics with Modern Physics

Question

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 \(mc^2\)). 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 \(\lambda'\) of the scattered photon. Show that if \(E \gg mc^2\), where m is the rest mass of the electron, your result reduces to $$\lambda' = {hc \over E} (1 + {m^2c^4\lambda \over 4hcE}) $$ (b) A beam of infrared radiation from a CO\(_2\) laser (\(\lambda = 10.6 \mu{m}\)) collides head-on with a beam of electrons, each of total energy \(E\) = 10.0 GeV (1 GeV = 10\(^9\) eV). Calculate the wavelength \(\lambda'\) 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

Answer

(a) Use the derived formula. (b) Calculate using given values and units. (c) The scattered photons are determined by their wavelength type.

Step 1: Understand Compton Scattering Formula

Compton scattering is described by the equation:\[\lambda' - \lambda = \frac{h}{mc}(1 - \cos(\theta))\]where \(\lambda'\) is the wavelength after scattering, \(\lambda\) is the initial wavelength, \(h\) is Planck's constant, \(m\) is the electron's rest mass, \(c\) is the speed of light, and \(\theta\) is the scattering angle. In this problem, \(\theta = 180^\circ\).

Step 2: Simplify for 180-degree Scattering

For 180-degree scattering, \(\cos(180^\circ) = -1\). Therefore, the formula simplifies to:\[\lambda' - \lambda = \frac{2h}{mc}\]Thus, the basic expression for the scattered wavelength is the initial wavelength plus an additional term related to the electron properties.

Step 3: Incorporate Relativistic Effects

When the electron moves at high speed, it's necessary to factor in relativistic energy changes. The relativistic energy is \(E = \gamma mc^2\) where \(\gamma\) is the Lorentz factor. The change in wavelength is affected by this total energy. For cases where \(E \gg mc^2\), simplifications can be applied.

Step 4: Derive the Approximation for High Energy Electrons

Assuming \(E \gg mc^2\), the scattered wavelength \(\lambda'\) from the energy conservation and considering the electron's motion can be expressed as:\[\lambda' = \frac{hc}{E} \left(1 + \frac{m^2c^4\lambda}{4hcE}\right)\]This expression considers the energy ratio \(E/mc^2\) to approximate \(\lambda'\).

Step 5: Calculate Scattered Wavelength for Given Values

Given \(\lambda = 10.6 \, \mu{m}\) and \(E = 10.0 \, \text{GeV}\), let's convert to consistent units: \(1 \, \mu{m} = 10^{-6} \, ext{m}\) and \(1 \, ext{GeV} = 10^9 \, ext{eV}\). Use the derived formula:\[\lambda' = \frac{hc}{E} \left(1 + \frac{m^2c^4\lambda}{4hcE}\right)\]Substitute constants and convert units to find the numerical value of \(\lambda'\).

Step 6: Determine Photon Type and Potential Application

Evaluate the wavelength \(\lambda'\) to classify the photon type (infrared, microwave, ultraviolet, etc.). Use known wavelength ranges for each type. Potential applications of this specific Compton scattering effect include materials analysis and medical imaging, given the high energy involved in the interaction.

Key Concepts

Photon Scattering

Photon scattering is an important process in the field of physics, where a photon collides with a particle and changes its direction. This change is described by Compton scattering in this context. Compton scattering examines how photons interact specifically with electrons. As they collide, the photon's direction and energy change based on its interaction angle, noted by \(\theta\). It helps in understanding fundamental interactions between light (photons) and matter (particles).

In Compton scattering, only the photon's wavelength is altered from \(\lambda\) (initial) to \(\lambda'\) (scattered), while some energy is transferred to the electron. The classical equation to describe this is \(\lambda' - \lambda = \frac{h}{mc}(1 - \cos(\theta))\). Changes in scattering angles like 180° result in maximum energy transfer.

Relativistic Energy

Relativistic energy takes into account the effects of high velocities, which are significant when those velocities approach the speed of light. Unlike classical energy, it includes the rest energy of a particle (\(mc^2\)) and energy due to its motion. This is represented as \(E = \gamma mc^2\), where \(\gamma\) is the Lorentz factor, reflecting the increase in energy as speeds increase.

When dealing with high-speed electrons in a photon scattering scenario, the relativistic energy of the electron becomes crucial. It plays a key role in predicting how much energy will transfer to a photon during a Compton scattering process. In scenarios with very high electron energies (\(E \gg mc^2\)), simplifications like \(\lambda' = \frac{hc}{E} (1 + \frac{m^2c^4\lambda}{4hcE})\) can help estimate the resultant photon wavelength.

Wavelength Calculation

Calculating the new wavelength of a scattered photon is a fundamental aspect of studying Compton scattering. In our scenario, the calculation involves the change in wavelength, referred to as the Compton shift. For exact scenarios, where the scattering angle is 180°, the difference \(\lambda' - \lambda = \frac{2h}{mc}\) can be directly calculated, providing us the change in wavelength.

When dealing with high energies, the wavelength is expressed with terms balancing electron energies and photon interactions: \(\lambda' = \frac{hc}{E} (1 + \frac{m^2c^4\lambda}{4hcE})\). This considers both the initial energy level of the electron and the interaction conditions, i.e., the angle and existing wavelengths.

Infrared Radiation

Infrared radiation falls within the electromagnetic spectrum and has longer wavelengths than visible light but shorter than microwave radiation. Typically, it ranges from 700 nm to 1 mm. In this exercise, the initial infrared wavelength \(\lambda = 10.6 \, \mu{m}\) fits perfectly within this classification.

Infrared light is often associated with heat, as these wavelengths can be emitted by hot objects. When photons of this wavelength undergo Compton scattering, they may change classification based on the new wavelength it achieves post-collision. Understanding such transformations is vital, especially when dealing with laser beams like those from CO\(_2\) lasers, used extensively in fields like material processing.

Photon Classification

Photons can be classified based on their wavelengths into categories like infrared, visible, ultraviolet, etc. After a photon undergoes Compton scattering, its new wavelength determines its new classification.

The initial photon in this exercise was infrared. Post-scattering, you calculate the new wavelength \(\lambda'\). If \(\lambda'\) results in a wavelength fitting into a lower band (such as visible or ultraviolet), the classification changes accordingly. This knowledge helps in figuring out the energy and possible applications.

• Infrared: 700 nm to 1 mm
• Visible Light: 380 to 740 nm
• Ultraviolet: 10 to 400 nm

By analyzing the scattered wavelength, we can explore different applications like medical imaging or materials testing, which exploit specific ranges of photon energies.