Nuclei in a radium-226 radioactive source emit photons whose...

Nuclei in a radium-226 radioactive source emit photons whose energy is $186 \mathrm{keV}$. These photons are scattered by the electrons in a metal target; a detector measures the energy of the scattered photons as a function of the angle $\theta$ through which they are scattered. Find the energy of the photons scattered through $\theta=90.0^{\circ}$ and $180.0^{\circ}$.

Key Concepts

Photon Energy and Wavelength Relationship

The relationship between photon energy and wavelength is governed by the equation E = hc/?, where E is energy, h is Planck’s constant, c is the speed of light, and ? is the wavelength. This inverse relationship means that an increase in wavelength corresponds to a decrease in energy, which is essential when analyzing the energy loss of a photon after scattering events such as Compton scattering.

Compton Scattering

Compton scattering is the inelastic collision between a photon and a free or loosely bound electron, resulting in a decrease in photon energy and an increase in its wavelength. This phenomenon is a fundamental demonstration of the particle nature of light, where the scattered photon loses energy to the electron in a manner that depends on the angle of scattering, and is described quantitatively by the Compton formula.

Role of Scattering Angle

The scattering angle in Compton scattering plays a crucial role in determining the energy transferred from the photon to the electron. According to the Compton formula, the change in wavelength — and hence the change in photon energy — depends on the cosine of the scattering angle. Larger angles, approaching 180 degrees, lead to a greater shift (i.e., lower scattered photon energy), while smaller angles result in a lesser shift, making the scattering angle a key parameter in predicting the outcome of the interaction.

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