An x -ray photon of initial frequency 3.0 ×10^19 Hz collides with a free electron at rest; the

step 1 For this exercise, we have a photon, an incident photon, with a frequency of 3 times 10 to the 1

An x -ray photon of initial frequency 3.0 ×10^19 Hz...

An $\mathrm{x}$ -ray photon of initial frequency $3.0 \times 10^{19} \mathrm{~Hz}$ collides with a free electron at rest; the scattered photon moves off at $90^{\circ} .$ What is the frequency of the scattered photon?

Key Concepts

Relativistic Considerations in High-Energy Photon Interactions

When dealing with high-energy photons (such as x-rays), it becomes necessary to apply relativistic mechanics to accurately describe the collision process. The use of relativistic equations ensures that the energy and momentum principles are correctly applied, particularly when the photon energies are comparable to the rest mass energy of the electron.

Conservation of Energy and Momentum

During the scattering process, both energy and momentum are conserved. This means the energy lost by the photon is gained by the recoiling electron while the overall energy and momentum of the system remain constant. These principles are essential for analyzing and predicting the outcomes of photon-electron collisions.

Photon Energy-Wavelength Relationship

The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength, as described by the equation E = hf, where h is Planck's constant. This relationship is critical in understanding how changes in a photon's energy, such as those arising from scattering events, are reflected as changes in its frequency and wavelength.

Compton Scattering

Compton scattering is a phenomenon where a photon collides with a free electron, causing the photon to lose energy (and thus increase in wavelength) while the electron recoils. The change in the photon's wavelength depends on the scattering angle and is described by the Compton formula, which relates the change in wavelength to the scattering geometry and fundamental constants.

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