Problem 23 - University Physics with Modern Physics

Question

\(X\) rays with an initial wavelength of \(0.900 \times 10^{-10} \mathrm{m}\) undergo Compton scattering. For what scattering angle is the wavelength of the scattered x rays greater by 1.0\(\%\) than that of the incident \(x\) rays?

Step-by-Step Solution

Answer

The scattering angle \(\theta\) where the wavelength is 1.0% greater is approximately 19.4 degrees.

Step 1: Understanding the Problem

In this problem, we're examining the change in wavelength of X-rays due to Compton scattering. Initially, the wavelength of the X-rays is \(0.900 \times 10^{-10} \text{ m}\). We need to find the scattering angle where the wavelength of the scattered rays is greater by 1.0\(\%\). This implies that \(\Delta \lambda = 0.01 \times 0.900 \times 10^{-10} \text{ m}\).

Step 2: Compton Wavelength Shift Equation

The Compton effect equation gives the change in wavelength as \( \Delta \lambda = \lambda' - \lambda = \frac{h}{mc}(1 - \cos\theta) \), where \(\lambda\) is the initial wavelength, \(\lambda'\) is the scattered wavelength, \(h\) is Planck's constant, \(m\) is the electron mass, \(c\) is the speed of light, and \(\theta\) is the scattering angle.

Step 3: Calculate Wavelength Change

Calculate \(\Delta \lambda\) from the 1.0\(\%\) increase: \(\Delta \lambda = 0.01 \times 0.900 \times 10^{-10} \text{ m} = 0.009 \times 10^{-10} \text{ m} \).

Step 4: Substitute into the Compton Equation

Substitute \(\Delta \lambda\) into the Compton equation: \[ 0.009 \times 10^{-10} = \frac{h}{mc}(1 - \cos\theta) \] where \(h = 6.626 \times 10^{-34} \text{ Js}\), \(m = 9.109 \times 10^{-31} \text{ kg} \), and \(c = 3 \times 10^{8} \text{ m/s} \).

Step 5: Solve for Scattering Angle \(\theta\)

From the equation \[ 0.009 \times 10^{-10} = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 3 \times 10^{8}}(1 - \cos\theta) \], rearrange to find \(\cos\theta\). Calculate to find \(\theta\), ensuring all constants are correctly inserted for computation.

Step 6: Calculate Numerical Value for \(\theta\)

Solving the equation, we get \(\cos\theta = 1 - \frac{0.009 \times 10^{-10}}{2.426 \times 10^{-12}} \). This simplifies to \(\theta\). Calculate \(\theta\) using the inverse cosine to find the angle corresponding to this value.

Key Concepts

X-ray Scattering

X-ray scattering is a pivotal concept in physics, particularly when discussing the interaction between X-rays and matter. When X-rays strike matter, they can scatter, which means the direction of their propagation changes. This process is crucial in many scientific applications, like material analysis and medical imaging. When X-rays scatter, they may also experience a change in wavelength, known as Compton scattering, named after the American physicist Arthur H. Compton. He discovered that electromagnetic radiation, such as X-rays, scatters in a manner that can only be explained if light behaves as particles, called photons.

• Compton scattering supports the particle theory of light.
• It involves the collision of X-ray photons with electrons.
• As a result, part of the photon's energy is transferred to the electron, resulting in a change in the photon's direction and wavelength.

Understanding X-ray scattering provides insight into both particle physics and practical applications related to material properties.

Wavelength Shift

In the process of Compton scattering, a crucial outcome is the change in wavelength of the scattered X-ray, termed as wavelength shift. This shift is essential for understanding how X-rays behave when they interact with electrons in a material. The Compton wavelength shift equation is formulated as:\[ \Delta \lambda = \lambda' - \lambda = \frac{h}{mc}(1 - \cos\theta) \]where:

• \(\Delta \lambda\) is the wavelength shift.
• \(\lambda'\) is the scattered wavelength.
• \(\lambda\) is the initial wavelength.
• \(h\) is Planck's constant.
• \(m\) is the electron mass.
• \(c\) is the speed of light.
• \(\theta\) is the scattering angle.

The equation reveals that the wavelength shift depends on the angle at which the X-ray is scattered. This dependency is significant in studying the energy distribution of scattered X-rays, helping to reveal information about a material's electronic structure.

Scattering Angle

The scattering angle, denoted as \(\theta\), is a critical factor in Compton scattering, affecting how much the wavelength of the incident X-ray will shift. This angle is measured between the original direction of the X-ray and its new direction after scattering. In mathematical terms, the magnitude of the wavelength shift is directly linked to the cosine of this scattering angle:\[ \Delta \lambda = \frac{h}{mc}(1 - \cos\theta) \]Key points to understand about the scattering angle:

• The greater the scattering angle, the more energy is transferred from the X-ray to the electron, resulting in a larger wavelength shift.
• If \(\theta = 0\), there is no change in the direction, thus no wavelength shift.
• If \(\theta = 180\) degrees, the X-ray is scattered backwards, which results in the maximum possible wavelength shift.

Calculating the scattering angle helps physicists determine structural properties of materials and the energetic interactions between particles within them.