55 keV x-ray photons are incident on a target. At what scattering angle do the scattered photons have an energy of 50 keV?

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Understand the problem: This is a Compton scattering problem, where an x-ray photon scatters off a target, and we need to find the scattering angle (θ) when the energy of the scattered photon is given. The relationship between the initial and scattered photon energies is governed by the Compton wavelength shift formula.

Write the Compton wavelength shift formula: Δλ = λ' - λ = (h / (m_e * c)) * (1 - cos(θ)), where λ is the wavelength of the incident photon, λ' is the wavelength of the scattered photon, h is Planck's constant, m_e is the electron mass, c is the speed of light, and θ is the scattering angle.

Relate energy to wavelength: Use the formula E = h * c / λ to express the wavelengths of the incident and scattered photons in terms of their energies. For the incident photon, λ = h * c / E₁, and for the scattered photon, λ' = h * c / E₂, where E₁ = 55 keV and E₂ = 50 keV.

Substitute the expressions for λ and λ' into the Compton formula: Δλ = (h / (m_e * c)) * (1 - cos(θ)). Replace Δλ with (h * c / E₂) - (h * c / E₁). Simplify the equation to isolate cos(θ).

Solve for θ: Rearrange the equation to find θ = cos⁻¹(1 - Δλ * (m_e * c / h)), where Δλ = (h * c / E₂) - (h * c / E₁). Substitute the known constants (h, m_e, c) and the given energies (E₁ = 55 keV, E₂ = 50 keV) to calculate θ.

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주요 개념

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Compton Scattering

Compton scattering is a phenomenon where X-ray or gamma-ray photons collide with electrons, resulting in a change in the photon's energy and direction. This effect is significant in understanding how photons interact with matter, particularly in the context of high-energy photons like X-rays. The energy loss of the photon depends on the scattering angle, which is crucial for solving problems involving energy changes in scattered photons.

Energy Conservation

Energy conservation is a fundamental principle in physics stating that the total energy in a closed system remains constant. In the context of scattering, the initial energy of the photon must equal the sum of the energy of the scattered photon and the kinetic energy gained by the electron. This principle allows us to relate the initial and final energies of the photons involved in the scattering process.

Scattering Angle

The scattering angle is the angle at which a photon is deflected after colliding with an electron. It plays a critical role in determining the energy of the scattered photon, as described by the Compton formula. By analyzing the relationship between the scattering angle and the energy of the photons, one can calculate the angle at which a specific energy is observed post-collision.