A monochromatic x-ray beam whose wavelength is 55.8 pm is scattered through 46 deg Find the wavelength of the scattered beam

To solve this problem, we can use the Compton scattering formula, which is given by:

Δλ = λ' - λ = h/(m_e * c) * (1 - cos θ)

where:

• Δλ is the change in wavelength,
• λ' is the wavelength of the scattered beam,
• λ is the initial wavelength,
• h is Planck's constant (6.62607015 × 10^-34 m^2 kg / s),
• m_e is the electron rest mass (9.10938356 × 10^-31 kilograms),
• c is the speed of light (299792458 m/s),
• θ is the scattering angle.

We are given:

• λ = 55.8 pm = 55.8 * 10^-12 m,
• θ = 46 degrees = 46 * π/180 radians = 0.80285 radians.

We need to find λ', the wavelength of the scattered beam.

First, we calculate Δλ using the Compton scattering formula:

Δλ = h/(m_e * c) * (1 - cos θ) = (6.62607015 × 10^-34 m^2 kg / s) / (9.10938356 × 10^-31 kilograms * 299792458 m/s) * (1 - cos(0.80285 radians)) = 2.4263102389 * 10^-12 m * (1 - 0.6845471059286886) = 2.4263102389 * 10^-12 m * 0.3154528940713114 = 7.6539828008 * 10^-13 m = 0.76539828008 pm.

Then, we find λ' by adding Δλ to the initial wavelength λ:

λ' = λ + Δλ = 55.8 pm + 0.76539828008 pm = 56.56539828008 pm.

So, the wavelength of the scattered beam is approximately 56.57 pm.