A beam of neutrons is used to study molecular structure through a series of diffraction experiments. A beam of neutrons with a wide range of de Broglie wavelengths comes from the core of a nuclear reactor. In a time-offlight technique, used to select neutrons with a small range of de Broglie wavelengths, a pulse of neutrons is allowed to escape from the reactor by opening a shutter very briefly. At a distance of \(16.4 \mathrm{m}\) downstream, a second shutter is opened very briefly 13.0 ms after the first shutter. (a) What is the speed of the neutrons selected? (b) What is the de Broglie wavelength of the neutrons? (c) If each shutter is open for 0.45 ms, estimate the range of de Broglie wavelengths selected.

Given that the distance between the two shutters is \(16.4\) m, and the time taken for the neutrons to travel this distance is \(13\) ms, we can find the speed of the neutrons by dividing the distance by time. Speed \(v = \frac{Distance}{Time}\) \(v = \frac{16.4 \mathrm{m}}{13.0 \times 10^{-3}\mathrm{s}}\) \(v \approx 1261.5 \mathrm{m/s}\)

Now that we know the speed of the neutrons, we can find their de Broglie wavelength using the formula: \(\lambda = \frac{h}{m v}\) Here, \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant (\(6.63 \times 10^{-34}\,\text{Js}\)), \(m\) is the mass of a neutron (\(1.674927 \times 10^{-27}\,\text{kg}\)), and \(v\) is the speed we calculated in Step 1. \(\lambda = \frac{6.63 \times 10^{-34}\,\text{Js}}{(1.674927 \times 10^{-27}\,\text{kg})(1261.5\,\text{m/s})}\) \(\lambda \approx 3.16 \times 10^{-10}\,\text{m}\)

We are given that each shutter is open for \(0.45\) ms. The range of speeds can be found using the formula: \(\Delta v = \frac{\Delta x}{\Delta t}\) Where \(\Delta x\) is the distance between the shutters (\(16.4\,\mathrm{m}\)), and \(\Delta t\) is the time each shutter is open (\(0.45\,\text{ms}\)). \(\Delta v = \frac{16.4\,\mathrm{m}}{0.45 \times 10^{-3}\,\text{s}}\) \(\Delta v \approx 36444.4\,\text{m/s}\) Now, we can estimate the range of de Broglie wavelengths using the formula: \(\Delta \lambda = \frac{h}{m (\Delta v)}\) \(\Delta \lambda = \frac{6.63 \times 10^{-34}\,\text{Js}}{(1.674927 \times 10^{-27}\,\text{kg})(36444.4\,\text{m/s})}\) \(\Delta \lambda \approx 9.27 \times 10^{-12}\,\text{m}\) Therefore, the range of de Broglie wavelengths selected is approximately \(9.27 \times 10^{-12}\,\text{m}\).