Problem 59 - College Physics

Question

\(\bullet\) A photon with a wavelength of 0.1800 nm is Compton scattered through an angle of \(180^{\circ} .\) (a) What is the wavelength of the scattered photon? (b) How much energy is given to the electron? (c) What is the recoil speed of the electron? Is it necessary to use the relativistic kinetic-energy relationship?

Step-by-Step Solution

Answer

(a) 0.18486 nm, (b) 2.8 \times 10^{-17} J, (c) 2.485 \times 10^6 m/s; relativistic effects may apply.

Step 1: Compton Wavelength Shift Formula

The change in wavelength during Compton scattering is given by the formula:\[\Delta \lambda = \lambda' - \lambda = \frac{h}{m_ec}(1 - \cos \theta).\]Here, \(\lambda\) is the initial wavelength, \(\lambda'\) is the scattered wavelength, \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ Js}\), \(m_e\) is the electron mass \(9.109 \times 10^{-31} \text{ kg}\), and \(c\) is the speed of light \(3 \times 10^8 \text{ m/s}\). The scattering angle \(\theta\) is \(180^\circ.\)

Step 2: Calculate Wavelength Shift

Substitute the given values into the Compton equation:\[\Delta \lambda = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 3 \times 10^8} (1 - \cos 180^{\circ}).\]This simplifies to:\[\Delta \lambda = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 3 \times 10^8} \times 2 = 4.86 \times 10^{-12} \text{ m} = 0.00486 \text{ nm}.\]

Step 3: Find Scattered Wavelength, \(\lambda'\)

Using the initial wavelength \(\lambda = 0.1800 \text{ nm}\) and the calculated shift \(\Delta \lambda\), find the scattered wavelength:\[\lambda' = \lambda + \Delta \lambda = 0.1800 \text{ nm} + 0.00486 \text{ nm} = 0.18486 \text{ nm}.\]

Step 4: Energy of Initial and Scattered Photons

The energy of a photon is given by \(E = \frac{hc}{\lambda}\). Compute the initial and scattered photon energies:\[E_1 = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{0.1800 \times 10^{-9}} = 1.104 \times 10^{-15} \text{ J},\]\[E_2 = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{0.18486 \times 10^{-9}} = 1.076 \times 10^{-15} \text{ J}.\]

Step 5: Calculate Energy Given to Electron

The energy transferred to the electron is simply the difference in the photon energies:\[E_e = E_1 - E_2 = 1.104 \times 10^{-15} - 1.076 \times 10^{-15} = 2.8 \times 10^{-17} \text{ J}.\]

Step 6: Recoil Speed and Relativistic Considerations

Use the classical kinetic energy formula \(\frac{1}{2} m_e v^2 = E_e\) to find the recoil speed \(v\):\[\frac{1}{2} \times 9.109 \times 10^{-31} \times v^2 = 2.8 \times 10^{-17},\] which gives\[v = \sqrt{\frac{2 \times 2.8 \times 10^{-17}}{9.109 \times 10^{-31}}} = 2.485 \times 10^6 \text{ m/s}.\]Since this speed is a significant fraction of the speed of light, relativistic effects should be considered, but here we use classical mechanics for simplicity.

Key Concepts

Photon Wavelength

In the realm of quantum physics, understanding photon wavelength changes is crucial, especially in phenomena like Compton scattering. A photon begins with a certain wavelength, but when it collides with an electron, it transfers some of its energy to the electron.

• The initial wavelength of the photon changes due to the scattering process.
• This change in wavelength is derived from the Compton wavelength shift formula: \[\Delta \lambda = \frac{h}{m_ec}(1 - \cos \theta)\]
• In the given exercise, the initial wavelength is 0.1800 nm and through calculations, the wavelength shift, \( \Delta \lambda \), turns out to be 0.00486 nm.
• Hence, the new wavelength of the scattered photon becomes 0.18486 nm.

The photon wavelength determines the photon's energy, which inversely relates to wavelength: shorter wavelengths mean higher energy and vice versa.
This shifting of wavelength is an intriguing example of the dynamic nature of photons and their interaction with matter in the microscopic world.

Electron Recoil

When a photon collides with an electron, the electron experiences a recoil, similar to the kickback felt when a ball is thrown.
In the case of Compton scattering, the electron gains some energy from the photon, causing it to move or recoil.

• This is quantified by the difference in energy between the initial and final state of the photon.
• The energy transferred to the electron, \(E_e\), is calculated by subtracting the energy of the scattered photon from the initial photon energy.
• In the provided example, the energy given to the electron is \(2.8 \times 10^{-17}\) J.

The velocity at which the electron recoils can be determined using classical or relativistic mechanics, depending on the significance of the calculated speed relative to the speed of light.

Relativistic Kinetic Energy

Kinetic energy generally refers to the energy an object possesses due to its motion.
In physics, especially at high speeds, the "relativistic" version is used because it accounts for the effects of relativity.

• As the speed of the recoiling electron approaches a significant fraction of the speed of light, relativistic effects become important, changing the calculation of kinetic energy.
• Classically, kinetic energy is calculated as \(\frac{1}{2} m v^2\).
• In this exercise, the recoil speed calculated was about \(2.485 \times 10^6\) m/s, and at such a high speed, one might typically consider using the relativistic formula.
• However, to keep computations simple, the classical method was applied here.

Relativity aids in illustrating how energy is related not just to motion, but to the very fabric of space and time.

Quantum Physics

Quantum physics is the study of the smallest particles in our universe: atoms, electrons, and photons.
It's a discipline that defies common intuition and reveals the behavior of matter and energy at atomic and subatomic levels.

• Compton scattering is one of the many phenomena explained by quantum physics, illustrating how particles like photons and electrons interact.
• This theory delves into the dual nature of light and energy dynamics, explaining behaviors that classical physics can't accurately describe.
• Understanding concepts such as photon wavelength and electron recoil requires an appreciation of how quantum mechanics governs the transfer of energy and momentum at these miniscule scales.

Quantum physics is pivotal in shaping our understanding of the universe, showing that our reality is governed by principles far more complex than the observable macro-world lets on.