An x-ray diffraction experiment using 16 -keV x -rays is...

An x-ray diffraction experiment using 16 -keV $x$ -rays is repeated using electrons instead of x-rays. What should the kinetic energy of the electrons be in order to produce the same diffraction pattern as the x-rays (using the same crystal)?

Key Concepts

Photon Energy-Wavelength Relationship

This concept involves the relation between the energy of a photon and its wavelength, given by E = hc/?, where E is the photon energy, h is Planck’s constant, c is the speed of light, and ? is the wavelength. It is essential for determining the wavelength of x-rays from their energy, which in turn governs their diffraction behavior in experiments.

de Broglie Hypothesis

The de Broglie hypothesis posits that every particle exhibits wave-like properties, with a wavelength given by ? = h/p, where h is Planck’s constant and p is the momentum of the particle. This concept is fundamental in relating the kinetic properties of electrons to a wavelength that can be compared with that of x-rays in diffraction experiments.

Kinetic Energy and Electron Wavelength

For electrons, especially under non-relativistic conditions, the momentum is related to kinetic energy by p = ?(2m_eE), with m_e being the electron mass and E its kinetic energy. Consequently, the de Broglie wavelength of an electron can be expressed as ? = h/?(2m_eE). Matching this wavelength to that of x-rays determines the electron’s required kinetic energy for a similar diffraction pattern.

Diffraction Condition Equivalence

In diffraction experiments, the observed pattern is dictated by the wavelength of the incident radiation or particles. To replicate an x-ray diffraction pattern using electrons on the same crystal, the electrons must have a de Broglie wavelength equal to the wavelength of the 16-keV x-rays. This requirement connects the photon energy-wavelength relationship with the particle wavelength-energy relationship.

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