Question

X-rays of wavelength \(\lambda=0.120 \mathrm{nm}\) are scattered from carbon. What is the Compton wavelength shift for photons detected at \(90.0^{\circ}\) angle relative to the incident beam?

Short answer

Answer: The Compton wavelength shift for X-rays scattered at a 90-degree angle is 2.43 nm.

Step-by-step solution

Given information and Compton scattering formula

We are given the following information: - Initial wavelength of the X-rays, \(\lambda = 0.120 \, \mathrm{nm}\) - Scattering angle, \(\theta = 90.0^{\circ}\) The Compton scattering formula is given by: $$ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos{\theta}) $$ Here, \(h\) is the Planck's constant \((6.626 \times 10^{-34} \, \mathrm{J \cdot s})\), \(m_e\) is the electron's mass \((9.109 \times 10^{-31} \, \mathrm{kg})\), \(c\) is the speed of light \((2.998 \times 10^8 \, \mathrm{m/s})\), and \(\Delta \lambda\) is the Compton wavelength shift. We need to find the value of \(\Delta \lambda\).

Convert given wavelength to meters

Before we proceed with the calculation, let's convert the given wavelength to meters: $$ \lambda = 0.120 \, \mathrm{nm} \times \frac{1 \, \mathrm{m}}{10^9 \, \mathrm{nm}} = 1.20 \times 10^{-10} \, \mathrm{m} $$

Calculate the Compton wavelength using the formula

Now, let's substitute the given values into the Compton scattering formula and compute \(\Delta \lambda\): $$ \Delta \lambda = \frac{h}{m_e c} (1 - \cos{90.0^{\circ}}) $$ $$ \Delta \lambda = \frac{(6.626 \times 10^{-34}\, \mathrm{J\cdot s})}{(9.109 \times 10^{-31} \, \mathrm{kg})(2.998 \times 10^8 \, \mathrm{m/s})}(1 - \cos{90.0^{\circ}}) $$ Since \(\cos{90.0^{\circ}} = 0\), the formula simplifies to: $$ \Delta \lambda = \frac{h}{m_e c} = \frac{(6.626 \times 10^{-34}\, \mathrm{J\cdot s})}{(9.109 \times 10^{-31}\, \mathrm{kg})(2.998 \times 10^8\, \mathrm{m/s})} $$ Calculating the value, we get: $$ \Delta \lambda = 2.43 \times 10^{-12} \, \mathrm{m} $$

Convert the Compton wavelength shift to nanometers

Finally, let's convert the Compton wavelength shift to nanometers: $$ \Delta \lambda = 2.43 \times 10^{-12} \, \mathrm{m} \times \frac{10^9 \, \mathrm{nm}}{1 \, \mathrm{m}} = 2.43 \, \mathrm{nm} $$ The Compton wavelength shift for photons detected at a \(90.0^{\circ}\) angle relative to the incident beam is \(\boldsymbol{2.43\, \mathrm{nm}}\).

Key concepts

Photon Scattering

Photon scattering refers to the interaction of photons with matter, which results in a change of direction and sometimes energy of the photons. This occurs frequently when photons pass through materials and collide with particles such as electrons. One significant type of photon scattering is known as Compton scattering, particularly relevant in the realm of X-rays and gamma rays interacting with electrons. Here are some key points of photon scattering:

• When a photon hits an electron, it can transfer some of its energy to the electron, causing the photon to scatter at a different angle with a lower energy.
• This scattering process also leads to a shift in the wavelength of the photon, a unique phenomenon of particular importance in medical imaging and astrophysics.
• The degree of scattering—how much the direction of the photon changes—depends on the energy of the incoming photon and the angle of interaction.

Understanding photon scattering is crucial for various technologies and scientific research areas, from designing better medical diagnostic tools to studying the structure of the universe.

Wavelength Shift

The wavelength shift in Compton scattering occurs due to the change in energy of the photons after they collide with electrons. According to the Compton effect, when X-rays or other high-energy photons hit a target, their wavelength increases after scattering. This shift in wavelength (\(\Delta \lambda\)) can be determined using the Compton scattering formula.Here’s more on how this shift comes about:

• Initially, the photon has a certain wavelength \(\lambda\) and after interacting with an electron, its wavelength becomes \(\lambda'\).
• The shift, \(\Delta \lambda = \lambda' - \lambda\), is calculated using the formula: \(\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)\), where \(h\) is Planck’s constant, \(m_e\) is the mass of the electron, \(c\) is the speed of light, and \(\theta\) is the scattering angle.
• This means the shift is directly dependent on the angle of scattering, and hence, different angles cause different shifts in the wavelength of the scattered photon.

The importance of understanding the wavelength shift lies mainly in various scientific and technological advancements, such as improving laser and X-ray devices, and facilitating research in quantum physics.

Physics Problem Solving

Approaching physics problems like Compton scattering involves a clear understanding of formulas and proper conversion of given units. This branch of physics teaches us how to systematically approach a problem and apply known physics principles to find solutions. Here’s a brief guide on problem-solving steps used in physics: 1. **Understand the Problem:** Start by identifying what is given and what needs to be found. For instance, knowing the initial wavelength and scattering angle is crucial for solving Compton wavelength shift problems.
2. **Use Appropriate Formula:** Utilize the correct equations that relate to the problem at hand, like the Compton scattering formula in this specific exercise.
3. **Check Units:** Physics often involves working with standard units. Make sure all quantities are in compatible units before performing calculations, like converting wavelengths from nanometers to meters.
4. **Solve Step by Step:** Break down calculations into manageable steps, substituting values into equations and simplifying systematically.
5. **Verify and Wrap Up:** After arriving at a solution, verify by checking if the value makes physical sense and falls within expected ranges.
Mastering these problem-solving strategies is essential for effectively tackling physics exercises, leading to a deeper understanding of physical phenomena and their mathematical representation.