A beam of neutrons that all have the same energy scatters...

A beam of neutrons that all have the same energy scatters from atoms that have a spacing of 0.0910 $\mathrm{nm}$ in the surface plane of a crystal. The $m=1$ intensity maximum occurs when the angle $\theta$ in Fig. 39.2 is $28.6^{\circ} .$ What is the kinetic energy (in electron volts) of each neutron in the beam?

Key Concepts

Energy Unit Conversion

In many physical problems, energy derived from calculations in SI units (joules) needs to be converted to electron volts (eV) for easier interpretation, particularly in atomic and nuclear physics. This conversion is performed using the relation 1 eV = 1.602×10?¹? joules.

Kinetic Energy and Momentum Relationship

The kinetic energy of a particle is related to its momentum through the equation KE = p²/2m, where m is the mass of the particle. When using the de Broglie relation, this formula allows one to determine the kinetic energy from the wavelength associated with the particle.

de Broglie Wavelength

This concept relates a particle’s momentum to its associated wave-like behavior, given by the relation ? = h/p, where h is Planck’s constant and p is the momentum. It is central to understanding how particles such as neutrons can exhibit diffraction and interference phenomena similar to light waves.

Bragg Diffraction

Bragg diffraction describes the constructive interference of waves scattered by a periodic array of scatterers, such as atoms in a crystal lattice. The condition for the mth order maximum is given by n? = 2d sin?, where d is the spacing between lattice planes, ? is the scattering angle, and m is the order of the diffraction maximum. This principle is used to determine structural properties and energy of particles via their wavelengths.

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