For x rays with wavelength 0.0300 nm, the m=1 intensity maximum for a crystal occurs when the an

step 1 So in this question we have the crystals acting like a diffraction grading, where our x -rays an

For x rays with wavelength 0.0300 nm, the m=1 intensity...

For $x$ rays with wavelength $0.0300 \mathrm{nm},$ the $m=1$ intensity maximum for a crystal occurs when the angle $\theta$ in Fig. 39.2 is $35.8^{\circ} .$ At what angle $\theta$ does the $m=1$ maximum occur when a beam of 4.50 -keV electrons is used instead? Assume that the electrons also scatter from the atoms in the surface plane of this same crystal.

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Key Concepts

Bragg's Law

Bragg's law is the fundamental principle that governs diffraction from crystal lattices. It states that constructive interference of waves reflected from parallel planes in a crystal occurs when the path difference between reflections is an integer multiple of the wavelength. This condition, expressed as n? = 2d sin?, is used for analyzing diffraction patterns in both x-ray and electron diffraction experiments.

de Broglie Wavelength

The de Broglie wavelength concept extends the idea that all matter has wave-like properties. According to de Broglie’s hypothesis, particles such as electrons have an associated wavelength given by ? = h/p, where h is Planck’s constant and p is the momentum. This principle is crucial in understanding and calculating the diffraction behavior of particles in crystalline materials.

Electron Kinetic Energy and Momentum Relationship

For electrons, especially at energies where relativistic effects are negligible, the momentum can be related to kinetic energy by p = ?(2mK), with m being the mass of the electron and K its kinetic energy. This relationship allows the determination of the de Broglie wavelength for electrons, which in turn is used in applying Bragg's law for diffraction experiments.

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