A student wants to measure the shift in wavelength predicted...

A student wants to measure the shift in wavelength predicted by Compton's formula (4.25). She finds that she cannot measure a shift of less than $5 \%$ and that she cannot conveniently measure at angles greater than $\theta=150^{\circ}$. What is the longest X-ray wavelength that she can use and still observe the shift?

Key Concepts

Scattering Angle

The scattering angle in Compton scattering is the angle between the incident and scattered photon trajectories. The wavelength shift depends on this angle through the term (1 - cos?) in the Compton formula; larger angles yield a larger shift. Constraints on the maximum practical scattering angle in an experiment, like limiting ? to 150°, directly affect the maximum observable shift in wavelength.

Measurement Sensitivity

Measurement sensitivity refers to the experimental capability to detect small changes in wavelength. In the context of Compton scattering experiments, the minimum measurable wavelength shift (for example, 5% of the original wavelength) sets a limit on the incident photon wavelengths that can be used, ensuring that the predicted shift is large enough to be observed despite experimental constraints.

Compton Wavelength

The Compton wavelength, defined as h/(m_e*c) for an electron, is a fundamental constant that sets the scale of the wavelength shift during Compton scattering. It represents the maximum possible wavelength change when a photon scatters at an angle of 180° (backscattering), and it forms the proportionality factor in the Compton shift formula.

Compton Scattering

Compton scattering is the phenomenon where a photon collides with a free or loosely bound electron, resulting in a scattered photon with a longer wavelength and a recoiling electron. This effect is critical in understanding the particle-like behavior of light and provides experimental evidence for the conservation of energy and momentum in quantum processes.

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