The spacing of the atoms of a crystal is 159 pm.A monoenergetic beam of neutrons directed normally at the surface of the crystal undergoes first order diffraction at an angle of 58° from the normal.What is the energy of each of the neutrons? (m_neutron = 1.675 × 10^-27 kg,1 eV = 1.6 × 10^-19 J,h = 6.626 × 10^-34 J ∙ s) A)0.045 eV B)0.039 eV C)0.050 eV D)0.056 eV E)0.061 eV

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The de Broglie wavelength of the neutrons is given by:$$\lambda = \frac{h}{p} = \frac{h}{\sqrt{2mE}}$$where h is Planck's constant, p is the momentum of the neutrons, m is the mass of the neutrons, and E is the energy of the neutrons.The first order diffraction condition is given by:$$n\lambda = 2d \sin	heta$$where n is the order of the diffraction (n = 1 for first order), d is the spacing of the atoms of the crystal, and θ is the angle of diffraction.Substituting the given values into the above equations, we get:$$\frac{h}{\sqrt{2mE}} = 2(159 	ext{ pm}) \sin(58^\circ)$$Solving for E, we get:$$E = \frac{h^2}{8m(159 	ext{ pm})^2 \sin^2(58^\circ)} = 0.045 	ext{ eV}$$Therefore, the energy of each of the neutrons is 0.045 eV.

The Spacing of the Atoms of a Crystal Is 159