Neutron diffraction by a crystal can be used to make a...

Neutron diffraction by a crystal can be used to make a velocity selector for neutrons. Suppose the spacing between the relevant planes in the crystal is $d=0.20 \mathrm{~nm}$. A beam of neutrons is incident at an angle $\theta=10.0^{\circ}$ with respect to the planes. The incident neutrons have speeds ranging from 0 to $2.0 \times 10^{4} \mathrm{~m} / \mathrm{s}$. (a) What wavelength(s) are strongly reflected from these planes? [Hint: Bragg's law, Eq. (25-15), applies to neutron diffraction as well as to x-ray diffraction.] (b) For each of the wavelength(s), at what angle with respect to the incident beam do those neutrons emerge from the crystal?

Key Concepts

Bragg's Law

Bragg's Law is a fundamental equation in diffraction phenomena that relates the wavelength of waves (such as neutrons or X-rays) to the spacing between crystal lattice planes and the angle of incidence. It states that strong diffraction (or constructive interference) occurs when n? = 2d sin?, where ? is the wavelength, d is the distance between adjacent planes, ? is the angle of incidence relative to the planes, and n is an integer representing the order of the diffraction. This principle is crucial for analyzing diffraction patterns and determining structural properties of crystals.

Neutron Diffraction

Neutron diffraction is a technique similar to X-ray diffraction that exploits the wave-like nature of neutrons to probe the internal structure of materials. Neutrons, due to their de Broglie wavelength and interaction with atomic nuclei, can provide detailed information about the arrangement of atoms within a crystal. This method is particularly useful for studying light elements or magnetic properties that are less discernible with X-ray diffraction.

de Broglie Wavelength

The de Broglie wavelength represents the wave-like behavior of particles, establishing that every particle with momentum p has an associated wavelength given by ? = h/p, where h is Planck's constant. This concept is fundamental in quantum mechanics and underpins diffraction techniques by linking a particle’s velocity to its wavelength, thereby allowing the selection or analysis of particles based on their speed when they interact with periodic structures such as crystal lattices.

Diffraction Geometry

Diffraction geometry refers to the spatial arrangement and angles involved when a beam of particles or waves interacts with a periodic structure like a crystal. Understanding how the incident beam, crystal planes, and reflected beams relate through angles is crucial for predicting the directions in which constructive interference occurs. This geometric analysis enables techniques such as velocity selection, where only particles with specific wavelengths (and thus velocities) are efficiently diffracted in predetermined directions.

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