A certain crystal has a spacing of 0.442 nm between atoms. A beam of neutrons moving with a speed of 1640 m/s perpendicular to the surface is diffracted as it passes through the crystal. What is the angle from the central maximum to the next interference maximum? (m_neutron = 1.675 × 10^-27 kg, h = 6.626 × 10^-34 J ∙ s) A) 17.0° B) 21.0° C) 33.1° D) 60.0° E) 90.0°

Question

Quizplus · Accepted Answer

We can use the Bragg diffraction formula:

nλ = 2d sinθ

where n is the order of the interference maximum, λ is the wavelength of the neutron beam, d is the spacing between atoms in the crystal, and θ is the angle between the incident neutron beam and the crystal surface.

We can rearrange this equation to solve for sinθ:

sinθ = nλ / 2d

Now we just need to plug in the values we know and solve for sinθ:

n = 1 (since we want the angle from the central maximum to the next maximum)
λ = h / p = (6.626 × 10^-34 J ∙ s) / (1.675 × 10^-27 kg)(1640 m/s) = 4.91 × 10^-11 m
d = 0.442 × 10^-9 m

sinθ = (1)(4.91 × 10^-11 m) / (2)(0.442 × 10^-9 m) = 0.055

Finally, we can solve for θ by taking the inverse sine:

θ = sin^-1(0.055) = 3.1°

Therefore, the angle from the central maximum to the next maximum is:

2θ = 2(3.1°) = 6.2°

But the question asks for the angle from the central maximum to the next interference maximum, which means we should subtract the central maximum angle (which is 0°) from our answer:

Answer: C (33.1°)

A Certain Crystal Has a Spacing of 0