X rays with an initial wavelength of $0.900\times {10}^{-10}$ 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?

Verified step by step guidance

1. Determine the relationship between the change in wavelength (Δλ) and the scattering angle (θ) using the Compton wavelength shift formula: Δλ = (h / (m_e * c)) * (1 - cos(θ)), where h is Planck's constant, m_e is the electron mass, and c is the speed of light.
2. Calculate the change in wavelength (Δλ) as 1.0% of the initial wavelength: Δλ = 0.01 * 0.900 × 10^-10 m.
3. Substitute the known values of Δλ, h, m_e, and c into the Compton formula to isolate the term (1 - cos(θ)): (1 - cos(θ)) = Δλ / (h / (m_e * c)).
4. Solve for cos(θ) using the equation: cos(θ) = 1 - (Δλ / (h / (m_e * c))).
5. Finally, calculate the scattering angle θ by taking the inverse cosine (arccos) of the result: θ = arccos(cos(θ)).

Key Concepts

Compton Scattering

Compton scattering is a phenomenon where X-rays or gamma rays collide with matter, resulting in a change in the wavelength of the radiation. This effect occurs due to the interaction between photons and electrons, leading to the transfer of energy and momentum. The change in wavelength is dependent on the scattering angle, which can be calculated using the Compton wavelength shift equation.

Wavelength Shift

The wavelength shift in Compton scattering refers to the difference in wavelength between the incident and scattered X-rays. It can be quantified using the formula Δλ = λ' - λ = (h/m_ec)(1 - cos(θ)), where h is Planck's constant, m_e is the electron mass, c is the speed of light, and θ is the scattering angle. A 1% increase in wavelength indicates a specific relationship between the initial and final wavelengths that must be calculated.

Scattering Angle

The scattering angle is the angle at which a photon is deflected after colliding with an electron. It plays a crucial role in determining the change in wavelength during Compton scattering. By analyzing the relationship between the scattering angle and the resulting wavelength shift, one can derive the angle that corresponds to a specific increase in wavelength, such as the 1% increase mentioned in the question.