Problem 19 - University Physics with Modern Physics

Question

\(\mathrm{X}\) rays with initial wavelength 0.0665 \(\mathrm{nm}\) undergo Compton scattering. What is the longest wavelength found in the scattered \(x\) rays? At which scattering angle is this wavelength observed?

Step-by-Step Solution

Answer

The longest scattered wavelength is 0.07136 nm at a 180° angle.

Step 1: Understand Compton Scattering Formula

Compton scattering describes the increase in wavelength (shift) of X-rays or gamma rays when they are scattered. The formula for the Compton wavelength shift is given by \[ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \]where \(\lambda\) is the initial wavelength, \(\lambda'\) is the scattered wavelength, \(h\) is Planck’s constant (\(6.626 \times 10^{-34} \mathrm{m^2kg/s}\)), \(m_e\) is the electron rest mass (\(9.11 \times 10^{-31} \mathrm{kg}\)), \(c\) is the speed of light in vacuum, and \(\theta\) is the scattering angle.

Step 2: Calculate the Compton Wavelength Shift Constant

The Compton wavelength shift constant, \(\frac{h}{m_e c}\), is fundamental in finding the change in wavelength due to scattering. Calculate this constant:\[ \frac{h}{m_e c} = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 3.00 \times 10^8} = 2.43 \times 10^{-12} \mathrm{m} = 0.00243 \mathrm{nm} \]

Step 3: Determine Maximum Wavelength Longest

The longest wavelength in scattered X-rays corresponds to the maximum wavelength shift, which occurs when \(1 - \cos \theta = 2\) (i.e., \(\theta = 180^\circ\)). Therefore, the maximum wavelength \(\lambda'\) is given by:\[ \lambda' = \lambda + \Delta \lambda = \lambda + 0.00243 \times 2 = 0.0665 + 0.00486 = 0.07136 \mathrm{nm} \]

Step 4: Conclusion on Scattering Angle

The longest wavelength observed in this scattered X-ray is at a scattering angle \(\theta = 180^\circ\), where the wavelength shift \(\Delta \lambda\) is at its maximum value. This means the wavelength increases due to the backscattering effect at this angle.

Key Concepts

X-ray wavelength

X-rays are a form of electromagnetic radiation with very short wavelengths, typically measured in nanometers (nm). The X-ray wavelength indicates the distance between successive peaks in the waveform of the X-ray. In the context of Compton scattering, the initial X-ray wavelength is critical because it provides the baseline for determining how much the wavelength has shifted after scattering.
The shorter the X-ray wavelength, the higher the energy of the X-ray photon. This energy is sufficient to cause scattering when it interacts with electrons. In the given exercise, the initial X-ray wavelength is 0.0665 nm. This value is used to measure the change in wavelength that occurs due to scattering, known as the Compton wavelength shift.

• The initial wavelength serves as a crucial starting point for calculations.
• Any change in wavelength after scattering indicates energy being partially transferred or altered by interaction with electrons.

Understanding the properties of X-ray wavelengths allows us to analyze the scattering process and predict the behavior of these rays after encountering material like electrons.

scattering angle

In Compton scattering, the scattering angle, denoted as \(\theta\), is the angle at which the X-ray is deflected as it scatters off an electron. The scattering angle is significant because it directly impacts the change in wavelength of the X-ray.
The relationship between the scattering angle and wavelength shift is given by \(\Delta \lambda = \frac{h}{m_e c}(1 - \cos \theta)\). Here, \(\theta\) modifies the equation to represent how much the wavelength changes depending on the direction in which the X-rays scatter. For instance, in the given exercise, the maximum shift in wavelength occurs at a scattering angle of \(180^\circ\). This angle, known as backscattering, represents the greatest possible change.

• Scattering angle determines the extent of wavelength change.
• Maximum wavelength shift occurs at \(180^\circ\), indicating a full backward deflection.

Knowing the scattering angle helps to understand and calculate the effect of Compton scattering on X-ray characteristics.

Compton wavelength shift

The Compton wavelength shift refers to the change in wavelength of X-rays or gamma rays when they scatter off electrons. This shift is calculated using the Compton scattering formula: \(\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta)\).
The exercise solution demonstrated that the constant \(\frac{h}{m_e c}\), known as the Compton wavelength shift constant, is crucial for such calculations and has a value of 0.00243 nm.
It's important to remember that the longer the wavelength shifts, the more energy has been transferred to the electron from the X-ray photon.
In this context, "shift" essentially means that the scattered X-ray has a different wavelength compared to the original. Using this fundamental understanding, we calculate the new wavelength after scattering and understand phenomena like energy transfer and backscattering.

• Compton shift is a measure of the photon energy transferred during scattering.
• It is dependent on the scattering angle; it's maximized at \(180^\circ\) as shown.

Understanding the nature and implications of the Compton wavelength shift is essential for analyzing the scattering outcomes and understanding compensations in wavelength due to physical interactions.