A beam of 188 -eV electrons is directed at normal incidence...

A beam of 188 -eV electrons is directed at normal incidence onto a crystal surface as shown in Fig. 39.3 $\mathrm{b} .$ The $m=2$ intensity maximum occurs at an angle $\theta=60.6^{\circ} .$ (a) What is the spacing between adjacent atoms on the surface? (b) At what other angle or angles is there an intensity maximum? (c) For what electron energy (in electron volts) would the $m=1$ intensity maximum occur at $\theta=60.6^{\circ} ?$ For this energy, is there an $m=2$ intensity maximum? Explain.

A beam of 188 -eV electrons is directed at normal incidence onto a crystal surface as shown in Fig. 39.3 $\mathrm{b} .$ The $m=2$ intensity maximum occurs at an angle $\theta=60.6^{\circ} .$ (a) What is the spacing between adjacent atoms on the surface? (b) At what other angle or angles is there an intensity maximum? (c) For what electron energy (in electron volts) would the $m=1$ intensity maximum
occur at $\theta=60.6^{\circ} ?$ For this energy, is there an $m=2$ intensity maximum? Explain.

Key Concepts

Constructive Interference and Diffraction Orders

Constructive interference in a diffraction pattern occurs when the path difference between waves scattered by adjacent atoms is an integer multiple of the electron’s wavelength. This condition is mathematically expressed as m? = d sin ?, where m is the diffraction order, ? is the de Broglie wavelength, d is the spacing between atoms, and ? is the diffraction angle. Understanding this principle is key to linking the observed diffraction angles with the physical structure of the crystal.

Energy-Wavelength Relationship

The relationship between the energy of an electron and its de Broglie wavelength is essential in electron diffraction experiments. Since the kinetic energy (E) is related to momentum (p) by E = p²/(2m), changes in electron energy affect the wavelength, thus shifting the conditions for constructive interference and the locations of the diffraction maxima. This concept is fundamental when adjusting experimental conditions to observe different diffraction orders.

De Broglie Wavelength

The de Broglie wavelength is the concept that all matter exhibits wave-like properties, with the wavelength (?) given by ? = h/p, where h is Planck’s constant and p is the momentum of the particle. This idea is critical in understanding phenomena such as electron diffraction, as the wavelength determines how electrons interfere after scattering from a periodic atomic structure.

Electron Diffraction

Electron diffraction refers to the scattering pattern resulting from the interaction of electron waves with a crystalline material. Because the atoms in a crystal are arranged in a regular, periodic manner, the electrons interfere constructively and destructively at specific angles, allowing for the determination of the spacing between atoms on the crystal's surface.

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