Question

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

Short answer

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

Step-by-step solution

Understand Compton Scattering Formula

The Compton scattering formula is given by \( \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \), where \( \lambda \) is the initial wavelength, \( \lambda' \) is the wavelength after scattering, \( h \) is Planck's constant, \( m_e \) is the electron mass, \( c \) is the speed of light, and \( \theta \) is the scattering angle. Our goal is to find the maximum scattered wavelength \( \lambda' \).

Calculate the Maximum Wavelength Shift

The maximum change in wavelength \( \Delta \lambda \) occurs when \( (1 - \cos \theta) \) reaches its maximum value. As \( \theta \) ranges from 0 to 180 degrees, \( 1 - \cos \theta \) is maximized to 2 at \( \theta = 180^{\circ} \). Therefore, the maximum wavelength shift is \( \Delta \lambda_{max} = \frac{2h}{m_e c} \).

Substituting Constants

Substitute the values of constants to calculate \( \Delta \lambda_{max} \):- Planck's constant \( h = 6.626 \times 10^{-34} \) Js- Electron mass \( m_e = 9.11 \times 10^{-31} \) kg- Speed of light \( c = 3 \times 10^8 \) m/s.Thus, \( \Delta \lambda_{max} = \frac{2 \times 6.626 \times 10^{-34} }{9.11 \times 10^{-31} \times 3 \times 10^{8} } \approx 4.86 \times 10^{-12} \) meters or 0.00486 nm.

Calculate the Longest Wavelength

Using the initial wavelength \( \lambda \), the longest wavelength \( \lambda' \) is given by \( \lambda' = \lambda + \Delta \lambda_{max} \). Thus, \( \lambda' = 0.0665 \text{ nm} + 0.00486 \text{ nm} = 0.07136 \text{ nm} \).

Conclusion for Scattering Angle

The longest wavelength \( 0.07136 \) nm occurs at a scattering angle of \( 180^{\circ} \). This angle provides the maximum wavelength shift, confirming that \( \theta = 180^{\circ} \) corresponds to backscatter, where the shift in wavelength is greatest.

Key concepts

Wavelength Shift

Compton scattering involves the interaction between photons, typically X-rays or gamma rays, and electrons. One of the fundamental concepts here is the change in wavelength, known as the wavelength shift (\( \Delta \lambda \)). This shift occurs due to the transfer of energy from the photon to the electron. Since wavelength and energy are inversely related, the scattering process results in an increase in wavelength (and thus a decrease in energy) of the photon.

The Compton scattering formula helps us to calculate this shift:\[\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta)\]Using this formula, we can determine how much longer the photon's wavelength becomes after the collision. The term \( \lambda - \lambda' \) represents the change from the initial wavelength \( \lambda \) to the scattered wavelength \( \lambda' \).

Knowing that the maximum wavelength shift occurs at a scattering angle of \( 180^{\circ} \), we can deduce the longest wavelength using the initial wavelength and the constant values of Planck's constant, electron mass, and speed of light.

Scattering Angle

The scattering angle (\( \theta \)) is critical when analyzing Compton scattering. It represents the angle at which a photon scatters after colliding with an electron. The scattering angle affects the degree of energy transferred from the photon to the electron, thus altering the wavelength of the emergent photon.

In Compton scattering, as \( \theta \) increases, the photon loses more energy, resulting in a larger wavelength shift. The maximized angle of \( 180^{\circ} \), also known as backscatter, corresponds to the maximum transfer of energy and, hence, the maximum increase in wavelength. This is why the formula:\[\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)\]reaches its maximum value, \( \Delta \lambda_{max} = \frac{2h}{m_e c} \), at this angle, making it a crucial point in calculations related to Compton scattering.

Understanding the scattering angle allows physicists and students alike to predict and evaluate the behavior of photons during scattering experiments. Such knowledge can be applied to practical scenarios in fields like astrophysics and medical imaging.

Planck's Constant

Planck's constant (\( h \)) is a pivotal constant in quantum mechanics and plays a major role in the Compton scattering formula. It relates the energy of a photon to its frequency, and subsequently, to its wavelength. The expression for energy in terms of Planck's constant is given by \( E = h u \), where \( u \) is the frequency of the photon.

Within the context of Compton scattering, Planck's constant is part of the equation that describes the wavelength shift:\[\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)\]Here it appears in the numerator, indicating its influence over the magnitude of change in wavelength during scattering.

Planck's constant is crucial not only for quantifying energy, but also for connecting microscopic phenomena with observable outcomes. This dimensionful constant effectively bridges the gap between the quantum world and macroscopic observations, enabling scientists to precisely calculate and predict various physical interactions, such as those occurring in Compton scattering.