The inter atomic spacing in a crystal of table salt is 0.282 nm . This crystal is being studied

step 1 We're told that we have a particle of mass M and a Debrilj wavelength of Lambda, and we're asked

The inter atomic spacing in a crystal of table salt is 0.282...

The inter atomic spacing in a crystal of table salt is 0.282 $\mathrm{nm}$ . This crystal is being studied in a neutron diffraction experiment, similar to the one that produced the photograph in Figure 29.12$a .$ How fast must a neutron (mass $=1.67 \times 10^{-27} \mathrm{kg} )$ be moving to have a de Broglie wavelength of 0.282 $\mathrm{nm}$ ?

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Key Concepts

Momentum

Momentum is a measure of a particle’s motion, typically defined as the product of its mass and velocity. In quantum mechanics, the momentum of a particle directly determines its de Broglie wavelength, linking its classical motion to its wave properties.

De Broglie Wavelength

The de Broglie wavelength is the wavelength associated with any particle due to its momentum, defined by the relation ? = h/p, where h is Planck’s constant. This concept illustrates that matter has wave-like properties and is crucial for understanding phenomena such as diffraction and interference at the quantum scale.

Wave-Particle Duality

Wave-particle duality is the principle that every particle or quantum entity exhibits both wave-like and particle-like characteristics. This dual nature is fundamental in quantum mechanics and helps explain experimental outcomes in phenomena like diffraction experiments.

Diffraction Experiments

Diffraction experiments exploit the wave nature of particles to study the structure of materials. In these experiments, the interference of matter waves, such as those from neutrons, provides information about interatomic spacings and the crystalline structure, revealing details of material properties at the atomic or molecular scale.

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